Volume Particle Theory (VP) Whitepaper v0.1.2

Young jae Lee
Independent researcher, Daegu, Korea
ORCID: 0009-0002-7535-8245
rego093@naver.com

Draft v0.1.2 (English edition) (2025-12-15)

Abstract

This document presents the core skeleton of Volume Particle (VP) theory as a long-form white paper and, within the same file, provides a full reproducibility specification (LOCK schema, derivation chain, Gate verdicts, cross-validation, and a DOI-anchored code/data bundle). VP theory models vacuum/space as a fully packed, infinitely rigid jammed lattice; observed “soft” phenomena are treated as event structures and failure rates rather than intrinsic softness of the VP.

The operational rule is the one-way flow LOCK \(\rightarrow\) derive \(\rightarrow\) Gate: (i) lock inputs and meanings (no post-hoc tuning), (ii) derive quantities using only transformations permitted by the LOCK, and (iii) restrict the admissible status of results through Gate verdicts PASS/FAIL/INCONCLUSIVE. This version locks the canonical length anchor \(D_{\mathrm{anch}}\), proton radius \(r_p\), realization scales \(a,\Delta t\), and rectification constants \((\alpha,\delta)\), and derives event-rate relations and length–time–mass/force scale mappings, including ideal-limit forms used in blackbody/Casimir-style calibration routes.

DOI (text + code + data bundle): 10.5281/zenodo.17932567.

Affiliation: Independent researcher, Daegu, Korea ORCID: 0009-0002-7535-8245 Email: rego093@naver.com
DOI: 10.5281/zenodo.17932567
Conflict of interest: No conflict of interest Funding: No funding

Keywords: Volume Particle Theory (VP); jamming lattice; discrete space lattice; vacuum rigidity; event counting; fine-structure constant; blackbody radiation; Planck law; Casimir effect; reproducibility; Zenodo DOI.
Recommended citation: Young jae Lee, “Volume Particle Theory (VP): A Jamming-Lattice White Paper with a LOCK \(\rightarrow\) Derive \(\rightarrow\) Gate Reproducibility Specification,” White paper, Zenodo, 2025-12-15, DOI: 10.5281/zenodo.17932567.

1 W. Executive Summary

1.1 W.1 One-page overview (what is locked, what is derived, and how verdicts are made)

1.2 W.2 Ledger: LOCK inputs vs derived outputs (summary)

LOCK inputs (frozen; change only by versioning) Derived outputs (derived; qualified by Gate)
Canonical length anchor \(D_{\mathrm{anch}}\) (canon_lock) Electron radius \(r_e=(D_{\mathrm{anch}}/2)\delta\) (e.g., Ch. 9.3, 13.5)
Proton radius \(r_p\) (canon_lock) Connection conventions (e.g., \(\lambda_C\)) and length-ratio invariants (e.g., Ch. 6.3, 13)
Realization length \(a\) (VP diameter; realization_lock) Lattice unit energy \(U_{\mathrm{lat}}:=h\,c_{\mathrm{ref}}/a\) (13.1)
Realization time \(\Delta t\) (realization_lock) Cross-consistency of event counting / time windows (RCROSS, Ch. 11–12)
Rectification constants \(\alpha=2/\pi\), \(\delta=1/\pi^2\) (universal regime) Mass scales \(m(\mathcal{O})=U_{\mathrm{lat}}/S(\mathcal{O})\) family (Ch. 13)
Rotation-drive length \(\ell_{\mathrm{rot}}\) (protocol/realization family) Anisotropy / SOC / jets (optional route; Ch. 17)

1.3 W.3 Gate verdicts and reproducibility artifacts (summary)

1.3.1 W.3.0 Claim Tiers

This document defines “rigor” not as rhetorical strength, but as the scope of a claim and the verification duty. That is, the stronger the conclusion, the larger the required Gate stack; without the corresponding Gate reports, the conclusion remains automatically UNLOGGED/INCONCLUSIVE.

1.3.1.1 Default scope of this version

This PDF (Draft v0.1.2 (English edition)) primarily presents conventions and derivation formulas within the Level-C0–C1 range. To use Level-C2+ (quantitative agreement with experiments; cosmological replacement) as conclusions, the corresponding Gate reports must be attached in the bundle and verified as PASS. Otherwise, related sentences are treated only as hypotheses / branch examples.

1.3.2 W.3.1 Gate Status Summary Table (key artifacts; Draft v0.1.2 (English edition))

The table below compresses onto a single page the key artifacts already appearing in the main text: (i) which LOCK inputs are used, (ii) which Gate stack must PASS for the result to be admissible as evidence, and (iii) where in the bundle the verdict log (evidence) must reside, so a reader can find it immediately.

Status notation (summary): PASS/FAIL/INCONCLUSIVE are confirmed only when a report exists under bundle_root/gate/reports/. If this PDF is distributed alone without attached Gate reports, the status is shown as UNLOGGED (= no evidence files attached).

Artifact (summary) LOCK used (summary) Required Gate (summary) Status Evidence path (standard)
\(a\) (VP diameter; SI realization) realization_lock: \(\lambda_{\mathrm{ref}}\), \(a\); analysis_lock: \(N\) G-SYM, G-LOCK, G-REG, G-NT, (G-RCROSS), (G-NUM) UNLOGGED gate/reports/gate_report_a.json
snapshot/manifest.(json|yaml)
\(\Delta t\) (time tick; realization) realization_lock: \(a\), \(c_{\mathrm{ref}}\); analysis_lock: \(A\) G-SYM, G-LOCK, G-REG, G-NT, (G-RCROSS), (G-NUM) UNLOGGED gate/reports/gate_report_dt.json
gate/logs/*dt*.(log|csv)
\(U_{\mathrm{lat}}:=h\,c_{\mathrm{ref}}/a\) (lattice unit energy) canon_lock: \(h\); realization_lock: \(c_{\mathrm{ref}},a\) G-SYM, G-LOCK, G-REG, G-NT, (G-NUM) UNLOGGED gate/reports/gate_report_Ulat.json
derived/claims.(tex|json)
\(r_e=(D_{\mathrm{anch}}/2)\,\delta\) (electron radius) canon_lock: \(D_{\mathrm{anch}}\); analysis_lock: \(\delta\) G-SYM, G-LOCK, G-REG, G-NT, (G-RECT), (G-NUM) UNLOGGED gate/reports/gate_report_re.json
derived/claims.(tex|json)
\(m_e\) (electron mass; \(m_e=U_{\mathrm{lat}}/S\)) realization_lock: \(U_{\mathrm{lat}}\); analysis_lock: \(S\); canon_lock: \(r_e\) G-SYM, G-LOCK, G-REG, G-NT, (G-RECT), (G-NUM) UNLOGGED gate/reports/gate_report_me.json
derived/tables/*mass*.csv
\(m_p\) (proton mass) canon_lock: \(r_p\); realization_lock: \(U_{\mathrm{lat}}\); analysis_lock: \(S_p\) or an event-rate mapping G-SYM, G-LOCK, G-REG, G-NT, (G-NUM) UNLOGGED gate/reports/gate_report_mp.json
derived/claims.(tex|json)
\(m_H\) (Higgs mass; e.g., \(U_{\mathrm{lat}}/(5\pi)\)) realization_lock: \(U_{\mathrm{lat}}\); analysis_lock: coefficient (e.g., \(5\pi\)) G-SYM, G-LOCK, G-REG, G-NT, (G-NUM) UNLOGGED gate/reports/gate_report_mH.json
derived/claims.(tex|json)
\(m_p/m_e\) (mass ratio; cross-consistency target) analysis_lock: \(\nu_{p,\mathrm{can}}\) etc.; derived: \(m_p,m_e\) G-SYM, G-LOCK, G-REG, G-NT, (G-RCROSS), (G-NUM) UNLOGGED gate/reports/gate_report_mp_me.json
gate/reports/gate_report_RCROSS.json
manifest+checksums+registry_snapshot (release integrity) protocol_lock & snapshot/* (sealing specification) G-SYM, G-LOCK, G-NT, (G-REP) UNLOGGED snapshot/manifest.(json|yaml|csv)
snapshot/checksums.(txt|json)

1.4 W.4 Known constraints / open items (explicit from a white-paper viewpoint)

2 0. Prologue (how to use this document)

Purpose of the chapter

This chapter declares the writing conventions for the entire treatise. The declaration covers: (i) the purpose of the chapter, (ii) the artifacts that the chapter and the document must output, (iii) the definition of internal traversal routes, and (iv) the one-way flow LOCK \(\rightarrow\) derive \(\rightarrow\) Gate that every claim must follow.

The exposition binds a single backbone (volume particle, infinite rigidity, full packing, jamming, throats, event rate, unit realization, cross-validation) into one skeleton. This skeleton keeps the same form throughout the document: it does not redefine the same items in other sections, nor does it repeat the same derivation to inflate “support.”

This chapter does not repeat the rules in narrative form. Items declared here will only be referenced later; the same content will not be re-expanded in prose.

Outputs

The outputs of this treatise are fixed as the following four bundles.

  1. Canonical (LOCK) bundle: axioms, definitions, symbol/unit conventions, operational anchors (reference points for unit realization), pre-registered thresholds and verdict rules, and the dependency (ordering) structure among them.

  2. Derived bundle: propositions, theorems, numerical results, scaling relations, ratio invariants, and dependency trace tables, obtained from the LOCK by allowed transformations only.

  3. Verification (Gate) bundle: the set of verdict conditions (PASS/FAIL/INCONCLUSIVE) used to evaluate derived results, cross-validation channels, sensitivity/robustness checks, failure-mode definitions, and verdict logs.

  4. Reproducibility (artifact) bundle: code, input files, environment information, seeds, logs, data, outputs (figures/tables/numerical summaries), hashes/checksums, manifests, release tags, and change logs.

Artifacts are cross-referenced inside the document; no single artifact qualifies a conclusion by itself. Numerical results must always carry both a LOCK provenance and a Gate verdict.

Definition of traversal routes

This treatise provides internal traversal routes that reorder the same body text for different purposes. A route is not advice about who should read what; it is a formal classification of possible paths along the document’s dependency graph.

The three routes share the same body; no route adds extra assumptions. Routes differ only in order and emphasis.

2.1 0.4 Declaration of the LOCK \(\rightarrow\) derive \(\rightarrow\) Gate flow

All conclusions in this treatise are generated only through the one-way flow LOCK \(\rightarrow\) derive \(\rightarrow\) Gate. Each stage is fixed to the following role.

2.1.0.1 (1) LOCK

LOCK is the set of items that, once fixed, are not modified or “corrected” later within the same document. LOCK includes axioms/definitions/conventions/operational anchors/thresholds/verdict rules, and records the dependency ordering among LOCK items. Changing a LOCK is not an “edit”; it exists only as a new version. Existing conclusions remain bound to their original LOCK version.

2.1.0.2 (2) Derivation

A derivation starts only from LOCK items and applies the permitted transformation rules and explicit closure rules to produce a predictable form. A derivation terminates in one or more of: (i) numerical values, (ii) scaling relations, (iii) ratio invariants, or (iv) verifiable decision expressions. Replacing definitions, reinterpreting units, or shifting reference points during a derivation is not allowed; if needed, such changes must be handled only by LOCK versioning.

2.1.0.3 (3) Gate

A Gate is a verdict device that assigns admissibility to a derived result. Gates include pre-registered thresholds, cross-validation channels, sensitivity checks, and failure-mode conditions. Gate outputs are restricted to PASS/FAIL/INCONCLUSIVE, where INCONCLUSIVE means deferral of a conclusion. Results with FAIL or INCONCLUSIVE verdicts are not used as evidence in later sections; to enable their use, a LOCK change or an additional pre-registered Gate is required.

2.1.0.4 (4) Form of conclusion sentences

In this treatise, every sentence that states a number or law must carry three elements: (i) identifiers of the LOCK items providing provenance, (ii) an identifier summarizing the derivation chain, and (iii) identifiers of the passed Gates (or verdict status). Statements without these three elements are treated as unclassified text, not conclusions, and are not promoted as evidence in later sections.

2.2 0.1 Claim Levels

2.2.1 0.1.1 Definition of claim levels

In this document, a “claim” means the qualification under which a sentence or an equation block is stated. The qualification is fixed to the following three classes.

  1. [F] Fact: Includes definitions, axioms, symbol/unit conventions, LOCK-frozen inputs, and results that have been reproduced after passing the required Gates. In this white paper, [F] is used only as an internal baseline; retroactively promoting [H] to [F] is not allowed.

  2. [H] Hypothesis: Includes closures, idealizations, approximations, model choices, and choices of computational procedures. An [H] must include all three of (i) the list of assumptions, (ii) the resulting prediction (in an observable form), and (iii) the scope of validity (regime condition). An [H] lacking these three elements is not recognized as a claim.

  3. [V] Verification: A record of how an [H] prediction was tested. A [V] must include (i) method, (ii) results (numbers/figures/tables), (iii) reproducibility information (inputs/seed/environment/error), and (iv) a verdict (PASS/FAIL/INCONCLUSIVE). An [H] without [V] is not promoted to evidence.

These three are a classification of sentence qualification and are applied uniformly regardless of the magnitude of the statement. A section may contain a mixture of [F]/[H]/[V], but only as separated labels; mixing (reinterpretation of definitions, shifting baselines, post-hoc correction) is forbidden.

2.2.2 0.1.2 What this document claims (claim list)

This document takes responsibility for the following claims. Each item is decomposed in the main text into [F]/[H]/[V] components; conclusions are not stated without their decomposition.

2.2.2.1 (C1) Fundamental constituents of space and a rigidity axiom

The basic unit of space is the volume-particle (VP), and the VP has the property of an internally incompressible “Stone.” This property is declared as infinite rigidity (incompressibility) and is used as the starting point of all rigidity/propagation/threshold logic.

2.2.2.2 (C2) Full packing and the existence of a jamming regime

VPs form a lattice/packing structure that fully fills space. The structure is separated into a jamming regime (a rigidity network spans the domain) and a non-jamming regime (no rigidity spanning). The boundary of “propagable/non-propagable” is defined as a jamming-regime verdict and is implemented as a Gate.

2.2.2.3 (C3) Fixation of geometric rectification constants

This document claims that rectification constants are determined under a specific geometric averaging/projection convention. The key rectification constants are \(\alpha=2/\pi\) and \(\delta=1/\pi^2\). They are derived at a single location, then locked (LOCK), and later sections only reference them (no re-derivation). These constants recur in event-rate rules and in chained derivations of length–time–mass/force scales.

2.2.2.4 (C4) A canonical length scale (anchor) and determinism of derived geometry

This document places an “anchor length” \(D_{\mathrm{anch}}\) as a canonical (CANON) LOCK input. Once the anchor is fixed, derived geometric quantities such as \(r_0=D_{\mathrm{anch}}/2\) are claimed to be determined deterministically. In addition, the “proton radius” \(r_p=\rproton\) is treated as a separate CANON LOCK input, and in combination with the anchor it becomes a key branching point for event-rate and mass derivations.

2.2.2.5 (C5) A separately locked rotation-drive length scale

For rotation-driven jamming experiments, this document defines a rotation-drive length unit \(\ell_{\mathrm{rot}}\) and records its value as a LOCK item. \(\ell_{\mathrm{rot}}\) must be locked “by definition”; changing it after observing results is judged as tuning and is forbidden. It is used as a standard experimental input for anisotropy, spin indices, and throat-distribution changes.

2.2.2.6 (C6) Operational definition of quantum events and canonical event rates

This document operationally defines an “event” in a form that is observable/loggable, and claims that event rates can be described by canonical rules combined with rectification constants. For the electron and proton, canonical event rates are presented as separate definition–derivation chains. A chain acquires conclusion status only through the coupling of (i) LOCK inputs, (ii) derivation rules, and (iii) Gate verdicts.

2.2.2.7 (C7) Fixing the VP length \(a\) and the time tick \(\Delta t\) via realization

This document claims a procedure to realize length/time defined in a dimensionless (simulation) world into SI units. The outputs of this procedure are \[a = \aVP, \qquad \Delta t = 1.86\times 10^{-21}\ \mathrm{s}\] and both values are recorded in the REALIZATION LOCK. The procedure also includes the condition that cross-validation collapses immediately unless the diameter/radius meaning and the cell geometry (e.g., cube) are locked together.

2.2.2.8 (C8) Definition of the lattice unit energy \(U_{\mathrm{lat}}\) and derivation of mass scales

After \(a\) is fixed, this document defines the unit energy directly exposed by one lattice cell (length \(a\)), \(U_{\mathrm{lat}}\), and derives mass scales from it. Mass is described in the form of a “geometric resistance” (effective cross-section / effective-length integrals). The document includes the numerical results that the proton mass \(m_p\) and electron mass \(m_e\) are obtained as \[m_p \approx 0.938\ \mathrm{GeV}, \qquad m_e \approx 0.511\ \mathrm{MeV}\] with the requirement that LOCK provenance, derivation chain, and Gate verdicts accompany the results.

2.2.2.9 (C9) Internal derivation (absolute scale) of the charge-interaction force scale

This document claims that the absolute scale of a “charge-interaction force” can be derived internally from lattice units (length, time, energy) and a geometric attenuation convention. Agreement with external numbers is not treated as justification but only as a Gate verdict item; if the verdict fails, no conclusion status is granted.

2.2.2.10 (C10) Conditional extensions (anisotropy, SOC, jets/tubes, throughput ceiling)

This document includes conditional extensions such as anisotropy (rotation drive), throat networks, SOC amplification, stream-tube/jet-like concentration, and throughput-based upper bounds. Here “jet” is not a borrowing of the fluid-mechanics term; it means a pattern in which event flux concentrates along specific directions of a backbone/throat network (17.2). Atmospheric/oceanic jets or astrophysical jets are mentioned only as examples of possible correspondences; no external theory, numbers, or equations are used as evidence. All extensions are conditional claims ([H]) with declared regimes and closures, and they acquire independent conclusion status only within the range where the designated Gates pass.

2.2.3 0.1.3 What this document does not claim (non-claim list)

The following items are fixed as explicit non-claims for which this document does not take direct responsibility.

  1. Unverifiable proclamations: Ultimate-reality proclamations, value judgments, or metaphysical conclusions that cannot be decided by observation/logging/reproduction are not stated as [F].

  2. Complete reconstruction of external systems: This document does not re-prove external theory frameworks end-to-end, nor does it import them as justification. External texts are mentioned only in a limited manner, e.g., as a correspondence table or a “non-use” declaration.

  3. Universalization across all regimes: Results that depend on initial/boundary conditions, dimension, protocol, or seed are not stated as universal laws.

  4. Justification of post-hoc correction: Changing definitions, baselines, locked values, meanings (diameter/radius), or verdict thresholds after seeing results is not justified. If a change is necessary, it exists only as a LOCK version-up.

  5. “Agreement implies truth” leaps: Agreement with external numbers is not used to declare axioms/definitions “true.” Agreement is recorded only as a Gate PASS condition or as an error-budget report.

2.2.4 0.1.4 Scope and non-scope (regime declaration)

The scope of this document is specified as a set of defined regimes. A regime is designated by a combination of the following elements, and each PART states its own regime on its first page.

  1. Dimension: the space dimension in which calculations/simulations/graph definitions are carried out (2D/3D, etc.), and explicit separation of how dimensionality affects conclusions.

  2. Boundary conditions: closed/open system, driven/undriven, fixed/free boundaries, and a declaration of which quantities are conserved or not under the chosen boundary.

  3. Initial conditions: declaration and log fixation of initial settings (near/above/below jamming, defect distribution, occupancy, spin distribution, etc.).

  4. Scale window: declaration of the applicable scale window (long/short wavelength, quasi-static/dynamic, linear/nonlinear, etc.).

  5. Protocol: procedural choices such as rotation drive (e.g., \(\ell_{\mathrm{rot}}\) input), percolation/throat definitions, sampling/averaging methods.

The non-scope range is fixed as “regime elements are undefined, or they are defined but judged as FAIL or INCONCLUSIVE by Gate.” In addition, if the diameter/radius meaning is not locked so that the same number can be interpreted as different geometries, the result is automatically classified as out-of-scope (no conclusion status).

2.2.5 0.1.5 Examples of prohibited interpretive sentences (short)

The following types of sentences do not have conclusion status and are not allowed even as [H] (reasons: non-verifiability, over-generalization, post-hoc justification, missing regime declaration).

  1. “Because this number matches, the entire axiom system is true.”

  2. “Because it appeared once under one setting, it holds under all conditions.”

  3. “Even if we change the definition (diameter/radius), the conclusion will be the same.”

  4. “The Gate says FAIL, but the interpretation is correct.”

  5. “It must hold intuitively even without verification.”

2.3 0.2 Reader guide: required route vs optional route

2.3.1 0.2.1 Definition of a route

A route is an ordering convention that preserves the direction of internal dependencies (LOCK items \(\rightarrow\) derivation chain \(\rightarrow\) Gate verdict). Routes are fixed to two types.

2.3.2 0.2.2 Composition of the required route (core derivations)

The required route is composed of the following step bundle. Each step requires internal completion of “LOCK fixation \(\rightarrow\) derivation \(\rightarrow\) Gate verdict.”

  1. Canonical convention step: lock symbol/unit conventions, diameter–radius meaning, canonical inputs (anchor length, reference radius, etc.), and the non-overloading rule.

  2. Axiom step: lock infinite rigidity (Stone), full packing, and the existence of a jamming regime.

  3. Rectification-constant step: unify the derivation location of geometric rectification constants (e.g., \(\alpha=2/\pi\), \(\delta=1/\pi^2\)) and lock them.

  4. Core-geometry step: combine canonical lengths and canonical radii to derive core geometry (radius ratios, area ratios, invariant ratios) and judge by Gate.

  5. Discrete-structure step: define core/shell structures (e.g., core integer structure, shell cancellation structure) and judge required structural invariants (sum/symmetry/cancellation rules) by Gate.

  6. Event step: give an operational definition of an event (quantum event), derive canonical event rates for electron/proton, and judge by Gate.

  7. Unit-realization step: realize and lock the length scale \(a\) and time tick \(\Delta t\) (operational anchors) with cross-consistency, and judge by Gate.

  8. Mass/force step: from unit energy and geometric resistance (effective cross-section / effective-length integrals), compute mass scales (electron/proton/other mass scales) and force scales, and judge by Gate.

2.3.3 0.2.3 Gate types that the required route must pass (declaration)

The required route has PASS of the following Gate types as an admissibility condition. Gate thresholds/decision expressions/log formats are locked in the corresponding sections.

  1. G-SYM (symbol/unit consistency Gate): decides symbol meanings (including diameter/radius), unit dimensions, and absence of definition collisions.

  2. G-LOCK (lock integrity Gate): decides that referenced LOCK items are fixed under the same version and show no post-hoc changes.

  3. G-REG (regime compatibility Gate): decides valid ranges of jamming/non-jamming, linear/nonlinear, long/short wavelength, boundary/initial conditions.

  4. G-RECT (rectification constant Gate): decides uniqueness of the definition/derivation location of rectification constants and internal consistency (including prohibition of re-derivation/re-definition).

  5. G-STR (structural invariant Gate): decides preservation of required invariants of discrete structure (sum/symmetry/cancellation rules).

  6. G-RCROSS (cross-consistency Gate): decides mutual consistency across two or more independent channels in unit realization.

  7. G-REP (reproducibility Gate): decides reproduction under the same conventions (code/inputs/environment/seed/logs/outputs).

2.3.4 0.2.4 Composition of the optional route (extensions)

The optional route adds the following extension bundles on top of required-route outputs. Each extension bundle has its own LOCK and its own Gate.

  1. Anisotropy extension: locks rotation-drive input (e.g., \(\ell_{\mathrm{rot}}\)) and derives direction-dependence of orientation distribution/fabric/throats.

  2. SOC amplification extension: locks event-cluster (avalanche) definitions and amplification coefficients, and derives distributions/scale invariance/pinning conditions.

  3. Stream-tube/jet extension: derives stability of flux-concentration paths, alternative paths, and bottleneck cascades in throat networks.

  4. Throughput ceiling extension: locks throughput definitions at the unit-cell level and derives scaling of the ceiling.

  5. Scale-up (macroscopic application) extension: locks \(\mu\)–\(m\)–\(M\) scale-conversion conventions and regimes, and derives macroscopic proxies.

2.3.5 0.2.5 Gate types that the optional route must pass (declaration)

The optional route presupposes PASS of all required-route Gates, and additionally requires PASS of the following Gate types.

  1. G-ANISO (anisotropy Gate): decides lock integrity of rotation-drive input, statistical stability of direction distributions, and per-direction reproducibility.

  2. G-SOC (SOC Gate): decides fixation of event definitions, existence of scale windows, and robustness of distribution diagnostics.

  3. G-PATH (path/bottleneck Gate): decides minimum-cut/alternative-path sensitivity, bottleneck-cascade stability, and invariance under throat-definition changes.

  4. G-CAP (capacity/throughput Gate): decides uniqueness of throughput definition, acyclicity of the ceiling derivation, and regime compatibility of the scaling.

  5. G-UP (scale-up Gate): decides lock integrity of scale-conversion conventions, meaning of proxies (including non-use ranges), and failure conditions of extrapolation.

2.4 0.3 Pipeline: “Input(LOCK)\(\rightarrow\)derive\(\rightarrow\)verify(Gate)” + file/registry overview

2.4.1 0.3.1 ASCII

All conclusions in this treatise follow the following one-way pipeline. The pipeline proceeds only in the order Input (LOCK) \(\rightarrow\) Derive (derived) \(\rightarrow\) Verify (Gate) \(\rightarrow\) Seal (snapshot).

+----------------------------------------------------------------------+
|  (A) Input (LOCK)                                                     |
|      - canon_lock : axioms/definitions/symbols/units/canonical numbers |
|      - realization_lock : length/time/energy fixed by unit realization |
|      - gate_lock : Gate definitions/thresholds/decision forms/cross-ch.|
|      - protocol_lock : code/input/seed/environment/log schema          |
+---------------+------------------------------------------------------+
                | (LOCK is read only from a single source: SSOT)
                v
+----------------------------------------------------------------------+
|  (B) Derive (derived)                                                 |
|      - derived_claims : propositions/theorems/numbers/ratio invariants |
|      - derived_tables : tables (including intermediate calculations)   |
|      - derived_figures : figures (scripts + parameters included)       |
|      - dependency_dag : LOCK->derived dependency graph                  |
+---------------+------------------------------------------------------+
                | (Derivation uses only LOCK items + permitted transforms)
                v
+----------------------------------------------------------------------+
|  (C) Verify (Gate)                                                    |
|      - gate_stack : stack of Gates (ordering included)                 |
|      - gate_inputs : inputs/data/logs used for verification            |
|      - gate_outputs : PASS/FAIL/INCONCLUSIVE + evidence metrics        |
|      - cross_checks : cross-channel consistency (RCROSS, etc.)          |
+---------------+------------------------------------------------------+
                | (Only PASS results acquire conclusion admissibility)
                v
+----------------------------------------------------------------------+
|  (D) Seal (registry_snapshot)                                         |
|      - registry_snapshot : frozen copy of registries used in release   |
|      - manifest : file list/roles/version/hash references              |
|      - checksums : per-file hashes (sha256, etc.)                      |
|      - release_tag : release identifier (version/date/change summary)  |
+----------------------------------------------------------------------+

2.4.2 0.3.2 SSOT

The registry is the file set that physically implements “where a definition is fixed” across the entire document. The registry is the single source of truth (SSOT): derivation code and verification code are allowed only to read it, never to modify it. Re-defining the same item outside the registry is forbidden.

The registry is composed of the following four types.

  1. canon_lock: axioms/definitions/symbols/units, canonical numeric inputs, diameter/radius meaning, and object definitions (cell/core/shell, etc.).

  2. realization_lock: length/time/energy/mass scales fixed by operational anchors (unit realization) and their derived quantities.

  3. gate_lock: Gate types, decision expressions, thresholds, tolerances, cross-channel layouts, and PASS/FAIL/INCONCLUSIVE output conventions.

  4. protocol_lock: execution environment, seed policy, input-file formats, log schemas, figure/table generation conventions, and naming rules.

2.4.3 0.3.3 Artifact package tree (overview)

The reproducibility package has the following top-level tree. The role of each directory is fixed, and creating duplicate files of the same role in a different location is forbidden.

bundle_root/
  registry/
    canon_lock.(yaml|json|toml)
    realization_lock.(yaml|json|toml)
    gate_lock.(yaml|json|toml)
    protocol_lock.(yaml|json|toml)
    symbols_table.(csv|tex)
    units_table.(csv|tex)
  derived/
    claims.(tex|json)
    tables/
      *.csv
      *.tex
    figures/
      *.pdf
    dag/
      dependency_dag.(json|dot|pdf)
  gate/
    gate_stack.(yaml|json)
    reports/
      gate_report_*.json
      gate_report_*.tex
    logs/
      *.log
      *.csv
  scripts/
    build_derived.(py|sh)
    run_gates.(py|sh)
    make_figures.(py|sh)
    utils/
  data/
    raw/
    processed/
  snapshot/
    registry_snapshot/
      (complete frozen copy of registry/)
    manifest.(json|yaml|csv)
    checksums.(txt|json)
    release_tag.(txt|json)

In this tree, snapshot/registry_snapshot/ is the frozen copy of the registry used in the release, preserving provenance of past conclusions even if registries are later versioned.

2.4.4 0.3.4 Overview of the manifest (file-list specification)

The manifest is the document that fixes, as a list, “all files included in the package.” The manifest has the following fields (the format is fixed to one of JSON/YAML/CSV; field names are kept identical).

  1. path: relative path from the bundle root.

  2. role: functional category of the file, e.g., registry, derived, gate_report, script, data_raw, data_processed, figure, table.

  3. content_type: text/tex, application/pdf, text/csv, etc.

  4. producer: producer identifier, e.g., manual, script:make_figures, script:run_gates.

  5. depends_on: list of dependency input paths (empty list if none).

  6. lock_version: registry-version identifiers referenced by the file (e.g., canon_lock_id, realization_lock_id).

  7. hash_ref: key referencing an entry in checksums (e.g., a sha256 value or checksum index).

  8. bytes: file size in bytes.

The manifest fixes “what is included” and serves as a reference point to immediately detect omission/duplication/substitution.

2.4.5 0.3.5 Overview of checksums (content-identity specification)

Checksums are the hash list used to verify the content identity of all files in the bundle. The checksum rules are fixed as follows.

  1. Algorithm fixed: the default algorithm is sha256. Additional algorithms may be listed, but sha256 must be present.

  2. Target fixed: all files under bundle_root/ are included. If exclusions are necessary, an exclusion-list file (checksum_exclusions) is created and itself included in checksums.

  3. Notation fixed: either “HASH<space>PATH” per line or a JSON key-value format.

  4. Linking fixed: each file references a checksum entry through the manifest field hash_ref; i.e., the manifest and checksums cross-reference each other.

Checksums decide not “same file name” but “same content,” sealing integrity of Gate reports and derived results.

2.4.6 0.3.6 Overview of registry_snapshot (frozen evidence)

The registry_snapshot is the directory that preserves, as-is, the registries referenced by a specific release. It serves the following purposes.

  1. Reproducibility: even if registries are updated later, the evidence of past conclusions (axioms/definitions/thresholds/operational anchors) can be restored identically.

  2. Dependency fixation: which LOCK version a derived result depended on is physically fixed (in combination with the lock_version fields).

  3. Branch management: versioning occurs only by creating new registries (never by editing old ones), preventing mixing of old-version and new-version conclusions.

The registry_snapshot resides under snapshot/registry_snapshot/ and is sealed by inclusion in the snapshot manifest and checksums.

2.4.7 0.3.7 Minimal schema (common fields for registry files)

Even if registry file formats differ, they share the following common fields.

  1. lock_id: unique identifier of the registry.

  2. created: creation time (string).

  3. scope: regime/protocol identifier to which the registry applies (default global).

  4. items: list of locked items (each includes name, unit, value or definition, description, and source path).

  5. invariants: invariant conditions enforced by the registry (e.g., diameter/radius meaning, unit dimensions, non-overloading rules).

  6. change_log: change history (appended only on version-up).

The common fields make “what the file locks” machine-readable and fix the input interface of derivation/verification scripts.

3 1. Governance: No-Tuning / LOCK / Gate

Declaration of global top-level rules

All definitions, all derivations, all numerical results, all comparisons, and all extensions in this document obey three global top-level rules: No-Tuning, LOCK, and Gate. A global top-level rule means that no section in the document is allowed to treat these rules as exceptions or ignore them partially. These rules are the minimal conditions for structural consistency and conclusion admissibility. A violation of any of the three rules is treated as an event that immediately revokes admissibility of the result produced in the violating sentence (or section).

Declaration of No-Tuning (ban on post-hoc adjustment)

No-Tuning is the global rule forbidding “changing inputs after seeing the result so that the conclusion matches a desired target.” No-Tuning forbids the following four categories.

  1. Post-hoc change of definitions: changing symbol meanings (e.g., diameter/radius), object definitions (e.g., cell/core/shell), unit interpretations, or inclusion/exclusion boundaries after seeing results.

  2. Post-hoc change of input values: changing canonical inputs, operational anchors, constants, thresholds, tolerances, or protocol parameters after seeing results.

  3. Selection bias: presenting only runs/settings/data that strengthen the conclusion. If selection is necessary, the selection rule itself must be pre-registered and locked as a LOCK item.

  4. Post-hoc change of verdict rules: changing Gate decision expressions, thresholds, cross-channel layout, or PASS/FAIL conditions after seeing results.

No-Tuning is not “no change ever” but “no post-hoc change inside the same version.” Any change is allowed only as a new version. If a change occurs, all artifacts produced after the change belong to a new LOCK identifier, while conclusions produced before the change remain assigned to the previous LOCK identifier. This structurally blocks retroactive reinterpretation of evidence for past conclusions.

Declaration of LOCK (canonical freezing and single source)

LOCK is the set of items that do not change within the same version once fixed. LOCK is the canonical baseline referenced by all derivations and all verifications, and it obeys the following principles.

  1. Single source of truth (SSOT): LOCK items are defined only at a specific location (or registry entry) inside the document. Re-defining the same item in another section is forbidden.

  2. Identifier-based attribution: every derived result and every Gate verdict must carry the identifiers of the referenced LOCK items. A conclusion is attributed not to “content” alone but to “content + LOCK identifiers.”

  3. Version-up only: changing a LOCK item occurs only by creating a new LOCK, not by editing an existing one. New LOCK items have new identifiers and do not change evidence of existing conclusions.

  4. Category separation: LOCK is stored in separated categories according to its role; items of different roles are not locked together.

At minimum, this document separates LOCK into the following four categories.

The purpose of LOCK is not only “to freeze values,” but to fix exactly one place where each definition lives across the whole document. A derivation without LOCK loses provenance, and a verification without LOCK loses decision legitimacy.

3.1 1.4 Declaration of Gate (admissibility verdict)

A Gate is the verdict device that grants or revokes conclusion admissibility of a derived result. Gates obey the following rules.

  1. Output format fixed: the Gate output is expressed only as one of PASS, FAIL, or INCONCLUSIVE.

  2. Pre-registration: the decision expression, thresholds, tolerances, cross-channel layout, and log format must be fixed in gate_lock before results are produced.

  3. Verdict precedence: results with FAIL or INCONCLUSIVE do not have conclusion admissibility and cannot be used as premises later.

  4. Stack allowed: a conclusion may need to pass a stack of multiple Gates; the stack order and composition must be pre-registered.

  5. Logs required: a Gate verdict must record logs of inputs, computation, intermediate metrics, and outputs; logs must follow the format fixed by protocol_lock.

Gate does not perform justification “because it matches.” Gate decides only whether pre-registered conditions are satisfied. Conclusion admissibility is granted only by PASS; interpretation content is constrained by LOCK and the derivation scope.

3.2 1.5 Coupled structure of the three rules (global precedence)

No-Tuning, LOCK, and Gate are not independent; they form a coupled structure.

  1. No-Tuning governs how LOCK and Gate are created and changed. It forbids changing LOCK or Gate after seeing results.

  2. LOCK governs the canonicity of inputs to derivations and Gates. LOCK does not change within the same version, and every derivation and Gate references LOCK.

  3. Gate governs conclusion admissibility of derived results. PASS is necessary for admissibility; FAIL/INCONCLUSIVE forbid use as evidence.

Therefore, the following hold globally.

3.3 1.1 No-Tuning rules and prohibited actions

3.3.1 1.1.1 Definition of No-Tuning

No-Tuning is the global rule forbidding “after seeing outputs, retroactively adjusting inputs/definitions/procedures/verdict criteria so that the outputs become the desired shape.” Here “outputs” include numerical values, tables, figures, logs, statistics, derived theorems, or Gate PASS/FAIL verdicts. The scope of No-Tuning is fixed to the following four layers.

  1. Definition layer: symbol meanings (e.g., diameter/radius), object definitions (cell/core/shell), unit meanings, inclusion/exclusion criteria, averaging/projection conventions.

  2. Input layer (LOCK): canonical numeric inputs, operational anchors (unit realization), critical thresholds, tolerances, protocol parameters, seed policy.

  3. Derivation layer: closure choices, algorithm choices, boundary/initial condition choices, selection of calculation paths (intermediate steps).

  4. Verdict layer (Gate): decision expressions, thresholds, cross-channel layout, PASS/FAIL rules, log schema.

No-Tuning does not forbid change itself; it forbids post-hoc change within the same version (same LOCK identifier set). If a change is necessary, it is allowed only as a version-up (new LOCK identifiers).

3.3.2 1.1.2 List of forbidden actions

Forbidden actions judged as No-Tuning violations are categorized as follows. If any category applies, the artifact immediately loses conclusion admissibility.

3.3.3 (A) Definition tuning

  1. After seeing a result, changing the meaning of diameter/radius or cell geometry (cube/sphere, etc.) so that the same number is reinterpreted as a different geometric quantity.

  2. After seeing a result, using the same symbol (e.g., \(\delta\), \(\Phi\)) with different meanings in different sections, or swapping meanings so that the derivation chain is effectively altered.

  3. After seeing a result, changing inclusion/exclusion criteria (e.g., which paths are counted as throats, which contacts are counted as effective contacts) so that numbers move toward a desired target.

  4. After seeing a result, changing averaging/projection/rectification conventions (angular averaging, amplitude squared, axis choice, etc.) so that rectification constants or event rates change.

3.3.4 (B) Value tuning

  1. After seeing a result, changing canonical inputs (anchor length, reference radius, rectification constants, etc.) to match a target number.

  2. After seeing a result, changing operational anchors (reference baselines/values fixed in unit realization) or reinterpreting the same anchor so that length/time/energy scales change.

  3. After seeing a result, changing critical thresholds or tolerances (e.g., shifting a threshold to turn FAIL into PASS).

  4. After seeing a result, changing the seed or sample count, repeatedly re-running “until PASS appears,” and keeping only the favorable run.

3.3.5 (C) Closure/model tuning

  1. After seeing a result, swapping closures or changing internal closure choices (e.g., representative path selections) so that numbers move closer to a target.

  2. After seeing a result, changing boundary/initial conditions so that conclusions change, yet describing it as “verification of the same conclusion.”

  3. After seeing a result, swapping algorithms (e.g., throat estimation, graph construction, relaxation rules) without declaring a LOCK version-up.

3.3.6 (D) Gate tuning

  1. After seeing a result, changing Gate decision expressions, thresholds, cross-channel layouts, or PASS/FAIL rules.

  2. Relaxing Gates when a result is FAIL, and strengthening Gates when a result is PASS, so that conclusions are kept only in a desired direction.

  3. Swapping metrics used in Gate decisions or changing metric definitions while still describing it as the same Gate without a gate_lock version-up.

3.3.7 (E) Selection/reporting tuning

  1. Selecting and presenting only cases where PASS occurred among multiple data/runs/samples, omitting FAIL or INCONCLUSIVE.

  2. Choosing a “representative” run or removing “outliers” after seeing results without a pre-registered selection rule.

  3. Removing negative results obtained under the same conditions from reports/logs by labeling them “unnecessary.”

3.3.8 (F) Protocol/code tuning

  1. After seeing a result, modifying code (algorithms/constants/definitions) without updating the identifiers that bind the modification to a LOCK version.

  2. Changing the execution environment (library versions, compile options, RNG implementation, etc.) without recording it in protocol_lock.

  3. Changing log formats or recorded fields so that unfavorable metrics are not recorded.

3.3.9 (G) External-justification tuning

  1. Using external standard frameworks/numbers as justification and retroactively adjusting internal definitions/inputs/closures to force agreement.

  2. Adopting justification sentences as conclusions, such as “because it agrees, the internal axiom is justified” (agreement is recorded only as a Gate item).

3.3.10 1.1.3 Format of allowed changes (exceptions): version-up only

There is no rule that treats No-Tuning as an exception inside the same version. Allowed changes exist only in the form of a version-up. A version-up holds only when all of the following conditions are satisfied.

  1. Issue a new LOCK identifier: create a new lock_id for the registry (canon_lock/realization_lock/gate_lock/protocol_lock) that contains the changed item.

  2. Fix a change log: record the reason for change (what changed and why), the before/after items, and affected derivation/verification items as change_log.

  3. Full re-derivation and re-verification: conclusions after the change must be regenerated from scratch under the new registry version and re-judged by the full required Gate stack.

  4. Preserve past conclusions: artifacts produced before the change remain assigned to the old lock_id and are not retroactively edited.

  5. Seal snapshots: seal registry_snapshot/manifest/checksums as a new release so that evidence of post-change results is physically frozen.

Therefore, “allowed exceptions” are not exceptions inside the same version, but only procedures for creating a new version.

3.3.11 1.1.4 Violation verdict and FAIL labels

A No-Tuning violation is recorded in the Gate system as a FAIL label. FAIL labels decompose the violation type, and multiple labels may be attached to one artifact. The label system is fixed as follows.

Label Meaning (violation type)
FAIL-NT-DEF Definition/meaning change (diameter/radius, object definitions, symbol-meaning swap, etc.)
FAIL-NT-VAL Input-value change (canonical inputs, operational anchors, post-hoc threshold/tolerance, post-hoc seeds)
FAIL-NT-CLS Post-hoc closure/model/algorithm swap (without version-up)
FAIL-NT-GATE Post-hoc change of Gate decision expression/threshold/rules
FAIL-NT-SEL Selection bias (keep only favorable results, omit unfavorable results, post-hoc outlier removal)
FAIL-NT-PROT Protocol/code/environment change unrecorded or identifier mismatch
FAIL-NT-LOG Log/schema modification to delete or avoid recording unfavorable metrics
FAIL-NT-RETRO Attempt to retroactively reinterpret or edit past conclusions

If FAIL-NT-* is assigned, the artifact immediately loses conclusion admissibility. Furthermore, all derived artifacts that use the failed artifact as a premise lose admissibility by propagation along the dependency graph.

3.3.12 1.1.5 Handling rules when a violation occurs (propagation of FAIL labels)

When a No-Tuning violation is confirmed, handling rules are fixed as follows.

  1. Quarantine: failed artifacts and their dependent derivatives are removed from the “candidate conclusion” list and quarantined as FAIL (state change; not deletion).

  2. Generate a FAIL report: create a fail_report that includes violation type (FAIL labels), relevant registry identifiers, relevant artifact paths, timestamp, and evidence of mismatch.

  3. Forbid evidence use: FAIL artifacts cannot be used as premises/evidence later; sentences premised on FAIL artifacts cannot be promoted as conclusion sentences.

  4. Constrained recovery paths: to restore admissibility, one must either (i) fully revert to the pre-violation state, or (ii) promote the change as a version-up and perform full re-derivation and re-verification under a new lock_id. Partial modifications inside the same version are not recognized as recovery.

  5. Maintain sealing consistency: the FAIL state and recovery attempts must appear as a continuous release record in manifest/checksums/registry_snapshot. Deleting or tampering with FAIL records is treated as an additional violation (FAIL-NT-LOG and FAIL-NT-RETRO).

3.4 1.2 Three LOCK types and SSOT

3.4.1 1.2.1 Definition of LOCK and SSOT

LOCK is the convention that fixes baseline items once so that they do not change within the same version. Its purpose is (i) to fix exactly one location of evidence, (ii) to make dependencies traceable, and (iii) to structurally block post-hoc adjustments (No-Tuning). SSOT (Single Source of Truth) means “the definition and value of the same item exist at exactly one location (one registry entry) across the whole document.” If SSOT holds, each section only references registry items instead of redefining them. If SSOT breaks, the same term is used with different meanings, or the same value is interpreted under different definitions, and derivation and verification cannot simultaneously hold.

LOCK is separated into three types by role.

  1. canon_lock: locks axioms/definitions/symbols/units/canonical inputs.

  2. realization_lock: locks physical scales and derived quantities fixed by unit realization (operational anchors).

  3. analysis_lock: locks analysis procedures used in derivation and verification (closures, estimators, algorithms, verdict composition, log schemas).

The three LOCKs have different roles and must not be mixed. For example, adjusting canonical definitions via unit realization, or treating an analysis procedure as if it were a canonical definition, is treated as a violation of the LOCK structure.

3.4.2 1.2.2 Definition of canon_lock (canonical freezing)

The canon_lock fixes items constituting the canonical baseline of the document. It contains only “fundamental units/meanings” and “baselines that must not move in later derivations.” canon_lock includes:

  1. Axioms: properties of VP, infinite rigidity (Stone), full packing, jamming-regime separation, etc.

  2. Definitions: objects (cell/core/shell/throat/path), operational definitions of event and event rate, diameter/radius meanings, and the non-overloading rule.

  3. Symbols/Units: unit dimensions of each symbol, dimensionless notation conventions, and symbol-collision rules.

  4. Canon inputs: numeric inputs adopted as canonical (reference radii, anchor lengths, rectification constants and where they are defined), together with their meanings.

  5. Invariants: invariant conditions that cannot be violated (fixed diameter/radius meaning, fixed inclusion/exclusion criteria of definitions, unit-dimension consistency).

Items in canon_lock are not derivation targets: they are not re-estimated or re-calibrated elsewhere in the document. canon_lock is an input to later derivations, and later derivations must output results without changing canon_lock.

3.4.3 1.2.3 Definition of realization_lock (unit-realization freezing)

The realization_lock contains items obtained by realizing a dimensionless (or internal-unit) world into physical units through operational anchors. It locks realization results, not definitions. realization_lock includes:

  1. Operational anchors: list of anchors used for realization and how anchors are fixed (input formats, measurement/record formats, channel layouts).

  2. Primary scales: first outputs of realization such as the length scale \(a\) and time scale \(\Delta t\), with identifiers of where/how they are computed.

  3. Derived scales: quantities derived from \(a\) and \(\Delta t\) (e.g., unit energy \(U_{\mathrm{lat}}\)) and identifiers of where they are used.

  4. Realization invariants: meaning locks and cross-consistency conditions preventing reinterpretation of the same \(a\) under different meanings.

  5. Cross-consistency spec: if consistency across independent channels is required, include channel lists and identifiers of the consistency decision configuration (decision expressions/thresholds themselves are fixed in analysis_lock).

realization_lock is not a tuning tool. It is a sealing of outputs produced by fixed operational anchors and fixed analysis procedures. Changing realization_lock implies changing anchors or analysis procedures and must be treated only as a version-up.

3.4.4 1.2.4 Definition of analysis_lock (freezing derivation/verification procedures)

The analysis_lock fixes procedural choices used in derivations and verifications. It locks the points where post-hoc tuning is easiest (closure choice, algorithm choice, verdict composition, log schema), implementing No-Tuning structurally. analysis_lock includes:

  1. Closures: definitions and selection rules when multiple closure candidates exist; include inputs/outputs/failure modes.

  2. Estimators/Metrics: formulas/procedures/normalizations/averaging conventions for throats, paths, distributions, invariants.

  3. Algorithms: steps/parameters for graph construction, relaxation, search, thresholding, sampling, bootstrap/resampling (if used).

  4. Gate-stack composition: which conclusions require which Gate stack (composition and order).

  5. Thresholds/Tolerances: fixed thresholds and tolerances deciding PASS/FAIL; cannot move post-hoc (movement is version-up only).

  6. Log schema: fields recorded for inputs, intermediate metrics, outputs, exceptions, fail labels, and environment info.

analysis_lock seals “how the calculation was performed.” Even under the same canon_lock and realization_lock, if analysis_lock differs then derived results and Gate verdicts can differ. Therefore the analysis_lock identifier must appear in every conclusion sentence.

3.4.5 1.2.5 SSOT implementation rules (registry unification)

SSOT is implemented not as a writing rule but as a file/registry structure. The SSOT rules are fixed as follows.

  1. Single location of definitions: items of canon_lock, realization_lock, and analysis_lock are defined only in their registry files; the main text only references item names.

  2. Item identifiability: every item has (i) name, (ii) unit, (iii) meaning (including diameter/radius), (iv) value or definition, (v) scope, and (vi) change history.

  3. Dependency direction: realization_lock and analysis_lock may reference canon_lock items, but the reverse direction (promoting realized outputs into canon inputs) is forbidden.

  4. No duplication: duplicates of the same item are not created in other files; references only are allowed, consisting of item name and lock_id.

SSOT violations are recorded as “definition collision” or “value collision,” and the artifact loses admissibility.

3.4.6 1.2.6 Change procedure (versioning and re-verification)

LOCK changes are allowed only as version-ups, not as edits inside the same version. The version-up procedure is fixed as follows.

  1. Specify the change target: declare whether the change belongs to canon_lock/realization_lock/analysis_lock, and identify the item name and scope.

  2. Create a new version: issue a new lock_id for the registry where the change occurs; the old lock_id is preserved and not edited.

  3. Fix a change log: record before/after items, change rationale, impact scope (dependent conclusions), expected failure modes as change_log.

  4. Update the dependency graph: mark all derived results and verification procedures that reference the changed item as re-computation targets.

  5. Full re-derivation: regenerate all relevant derived results from scratch under the new lock_id combination (including intermediates).

  6. Full re-verification: re-run the full required Gate stack for regenerated results to re-judge PASS/FAIL/INCONCLUSIVE.

  7. Seal snapshots: create and seal the frozen copy of the used registries (three LOCKs), the manifest, and the full checksums as registry_snapshot.

Conclusions produced before the version-up remain assigned to the old lock_id combination, and conclusions produced after the version-up are assigned to the new combination. Mixing results across different lock_id combinations into a single conclusion is forbidden.

3.5 1.3 Gate & PASS.rules

3.5.1 1.3.1 Definition of Gate and verdict output

Gate is the verdict device that grants or revokes conclusion admissibility for derived artifacts. Gate does not describe an artifact as “true/false”; it decides only whether pre-registered conditions are satisfied. Gate output is fixed to:

If the Gate output is FAIL or INCONCLUSIVE, the artifact does not have admissibility, and derived conclusions premised on it cannot have admissibility (dependency propagation). Admissibility is granted only by PASS.

3.5.1.1 Example (NON-LOCK calibration)

As a concrete example of how an external “inversion” claim can be structured without tuning any VP LOCK inputs, Appendix K reconstructs the mean free path of air from macroscopic acoustic and transport data. This appendix serves only as a methodological calibration of the reconstruction pipeline; it is not used to set any VP constants.

3.5.2 1.3.2 Classification of Gates (taxonomy and purpose)

Gates are classified by “what they decide.” The classification is fixed in analysis_lock; each Gate carries its inputs, decision expression, thresholds, log schema, and failure labels. Gate taxonomy is composed along three axes (target axis, function axis, stack axis).

3.5.3 (A) Target axis: what is decided

  1. Semantic Gates: decide consistency of symbol meanings, unit dimensions, diameter/radius meanings, and object definitions (cell/core/shell/throat/path).

  2. Lock-integrity Gates: decide consistency of referenced lock_id values, existence of registry snapshots, and absence of post-hoc changes.

  3. Regime Gates: decide whether the scope (dimension/boundary/initial/scale window) satisfies pre-registered regime conditions.

  4. Structure Gates: decide preservation of discrete-structure invariants (integer decompositions, cancellation rules, symmetries, sum/difference invariants).

  5. Numerical Gates: decide convergence, sensitivity, repetition stability, log completeness, and reproducibility of computation paths.

  6. Cross-consistency Gates: decide whether the same conclusion is maintained across independent channels/baselines/input combinations.

  7. Reproducibility Gates: decide whether the same package can be re-run with the same conventions to reach the same verdict.

  8. Anti-tuning Gates: decide whether post-hoc tuning occurred (definition/value/procedure/threshold changes, selection bias, log tampering, etc.).

3.5.4 (B) Function axis: how admissibility is decided

  1. Qualification Gates: Gates that allow/forbid creation of a conclusion sentence itself. If a qualification Gate does not PASS, the conclusion sentence cannot be generated.

  2. Limitation Gates: Gates that fix the scope/limitations of a conclusion. PASS fixes a scope; FAIL shrinks the scope or revokes admissibility.

  3. Decomposition Gates: Gates that record where a result broke down (semantic/regime/structure/numerical/cross-consistency/reproducibility/No-Tuning) as labels.

A single Gate may serve multiple functions, but the function role(s) must be declared in gate_lock.

3.5.5 (C) Stack axis: Gates operate as stacks

A conclusion must pass a stack of Gates rather than a single Gate. A stack includes “Gates that must pass first” and “conditionally executed Gates.” Stack order and conditions are pre-registered in analysis_lock. The purpose of the stack is:

  1. to prevent passing numerical Gates while semantic definitions are unstable,

  2. to prevent packaging regime-violating results as cross-consistency,

  3. to prevent describing results without reproducibility/snapshots as conclusions,

  4. to block No-Tuning violations early rather than only at the end.

3.5.6 1.3.3 Gate ID system (standard Gate list and roles)

Gates are called by identifiers (gate_id), and a gate_id encodes taxonomy and purpose. The following list is the standard Gate namespace used throughout this document (additional Gates extend the same convention).

3.5.7 (A) Global base Gates (common to all conclusions)

3.5.8 (B) Structure/rectification/procedure Gates (selected by claim type)

3.5.9 (C) Extension-regime Gates (used only in optional routes)

3.5.10 1.3.4 Standard fields for Gate records (log schema)

Every Gate verdict is recorded as a log with standard fields. The log format (JSON/YAML/CSV) is fixed in protocol_lock, while field meanings are fixed as follows.

  1. gate_id: Gate identifier.

  2. gate_version: Gate-definition version (including the analysis_lock identifier).

  3. lock_refs: referenced lock_id list (canon/realization/analysis).

  4. scope: applied scope (regime ID or condition expression).

  5. inputs: inputs used in the Gate decision (file paths, parameters, summaries).

  6. metrics: decision metrics (arrays/summary statistics/intermediate artifacts).

  7. thresholds: thresholds/tolerances (reference to pre-registered values).

  8. result: PASS/FAIL/INCONCLUSIVE.

  9. fail_labels: cause labels for FAIL/INCONCLUSIVE (multiple allowed).

  10. artifacts: list of generated artifacts (reports/figures/tables/logs) paths.

  11. timestamp: execution timestamp.

Gate records must be sealed by inclusion in the snapshot (manifest/checksums/registry_snapshot). Unsealed Gate records do not grant conclusion admissibility.

3.5.11 1.3.5 Definition of PASS.rules (sentence-admissibility rules)

PASS.rules is the rule set that fixes “which Gate combination must PASS in order to allow which sentence format as a conclusion.” PASS.rules is included in analysis_lock and cannot be modified after seeing results. PASS.rules has two layers.

  1. Claim-type layer: classifies conclusions by type and fixes the required Gate stack for each type.

  2. Sentence-template layer: fixes allowed conclusion sentence formats and forbidden sentence patterns.

The purpose of PASS.rules is simultaneously to block (i) unpassed results flowing into sentences and (ii) passed results inflating into over-interpretation.

3.5.12 1.3.6 Standard classification of claim types

Claim types used in PASS.rules are fixed as follows. Each type defines the range of allowed sentences; statements outside that range are forbidden.

  1. CT-DEF: declaration of definitions/axioms/conventions (LOCK items). Not a derivation/Gate target; attributed only by lock_id.

  2. CT-DER-FORM: derived form (equation structure, relations, invariant forms). Declares relations without numerical substitution.

  3. CT-DER-NUM: derived numbers (single values or a canonical list). Declared with units/meaning (including diameter/radius).

  4. CT-DER-RANGE: derived ranges (intervals, bounds, conditional ranges) with regime conditions.

  5. CT-STR: structural conclusions (integer decomposition, cancellation rules, symmetries, coordinate structures) with structural invariants.

  6. CT-REAL: unit-realization conclusions (derive \(a\), \(\Delta t\), etc from operational anchors) including cross-consistency Gate.

  7. CT-XCROSS: cross-consistency conclusions (consistency across independent channels) with channel layouts and thresholds.

  8. CT-REP: reproducibility conclusions (re-run the same package to reach the same verdict) with reproducibility logs and checksums.

  9. CT-LIM: limitation/non-scope conclusions (where it applies and where it breaks) with FAIL/INCONCLUSIVE cause labels.

3.5.13 1.3.7 Standard sentence templates conditioned on Gate PASS

The table below fixes, for each claim type, the required Gate stack and the allowed conclusion-sentence format. In the table, “required” means necessary to generate a conclusion sentence, and “conditional” means required only for that type.

Claim type Required Gates (PASS) Allowed conclusion sentence format (summary template)
CT-DEF (no Gate required; attributed by lock_id) “In [LOCK:{canon/realization/analysis}_lock_id], the following is defined: {definition/axiom/convention}.”
CT-DER-FORM required: G-SYM, G-LOCK, G-REG, G-NT “From [LOCK:{...}] and [DER:{derivation_id}], the relation {equation} is derived.”
CT-DER-NUM required: G-SYM, G-LOCK, G-REG, G-NT; conditional: G-NUM “By [LOCK:{...}][DER:{...}][GATE:{...}], {quantity} is fixed as {value}{unit}.”
CT-DER-RANGE required: G-SYM, G-LOCK, G-REG, G-NT; conditional: G-NUM “Under [REGIME:{condition}], {quantity}\(\in\)[{lower},{upper}] and the verdict passes [GATE:{...}].”
CT-STR required: G-SYM, G-LOCK, G-REG, G-NT; conditional: G-STR “[STRUCT:{structure_id}] preserves invariants {list} and is qualified by PASS of [GATE:G-STR].”
CT-REAL required: G-SYM, G-LOCK, G-REG, G-NT; conditional: G-RCROSS, G-NUM “From operational anchor {anchor_id}, {a, \(\Delta t\), \(\ldots\)} are obtained; cross-consistency is judged by [GATE:G-RCROSS].”
CT-XCROSS required: G-SYM, G-LOCK, G-REG, G-NT; conditional: G-RCROSS “Channels {A,B,\(\ldots\)} are consistent for {target}; the consistency threshold {thr} is fixed and passes [GATE:G-RCROSS].”
CT-REP required: G-SYM, G-LOCK, G-REG, G-NT; conditional: G-REP “Re-running with the same package (Manifest+Checksums+Registry_Snapshot) reproduces the same verdict, qualified by PASS of [GATE:G-REP].”
CT-LIM required: G-SYM, G-LOCK, G-NT “For {target}, under [REGIME:{condition}] the verdict is FAIL/INCONCLUSIVE; cause labels are {FAIL_LABELS}.”

3.5.14 1.3.8 Forbidden conclusion-sentence patterns (short)

PASS.rules fixes forbidden patterns as well as allowed templates. Forbidden patterns are fixed as follows.

  1. No-Gate conclusion forbidden: numerical/law statements without Gate identifiers or verdict status (PASS/FAIL/INCONCLUSIVE).

  2. No-regime universalization forbidden: scope expansion expressions such as “always/all/universal” without regime conditions.

  3. Post-hoc justification forbidden: verdict-neutralizing expressions such as “because it agrees, it is true” or “it fails Gate but is correct by interpretation.”

  4. No-LOCK attribution forbidden: re-stating definitions/values/thresholds/procedures without lock_id attribution.

3.5.15 1.3.9 Standard PASS.rules file template (machine-readable)

PASS.rules is included in analysis_lock in a machine-readable form. Below is a standard template (key names and structure are fixed).

pass_rules:
  - rule_id: PR-CT-DER-NUM-001
    claim_type: CT-DER-NUM
    claim_id: (internal claim identifier; e.g., CL-13-05-mp)
    scope: (regime id or condition expression; e.g., global | regime:R1 | condition:...)
    requires:
      gates_pass:
        - G-SYM
        - G-LOCK
        - G-REG
        - G-NT
        - (optional) G-NUM
      locks_present:
        - canon_lock
        - realization_lock
        - analysis_lock
      snapshot_required: true
      manifest_required: true
      checksums_required: true
    allows_sentences:
      - template_id: ST-CT-DER-NUM-01
        pattern: "[LOCK:{canon_id,real_id,ana_id}][DER:{der_id}][GATE:{gate_stack_id}] {Q} = {value} {unit}."
        requires_fields:
          - Q
          - value
          - unit
          - canon_id
          - real_id
          - ana_id
          - der_id
          - gate_stack_id
    forbids_sentences:
      - "Sentence asserting a numeric conclusion without a Gate tag"
      - "Sentence declaring universal scope without regime conditions"
      - "Post-hoc justification or verdict-neutralization sentence"
    on_fail:
      verdict: "revoked"
      label_prefix: "FAIL-PASSRULE"
      propagation: "propagate along dependency_dag"

In the template above, requires.gates_pass is a necessary condition to generate a conclusion sentence, and allows_sentences fixes the admissible sentence format. The snapshot/manifest/checksums requirements fix “evidence sealing” as a necessary condition for conclusion admissibility.

4 2. Notation, Terminology, and Scale Hierarchy

Purpose and top-level principles

This chapter fixes, under a single registry system, the symbols, units, and scales (length/time/mass/energy/force, etc.) used across the entire document, and structurally prevents the same item from being defined twice or assigned conflicting meanings across different sections. The conventions of this chapter act as the highest-precedence rules for the whole document. That is, any derivation, number, table, figure, or log that violates the symbol/unit/scale registry conventions fixed here cannot qualify as a conclusion.

Priority of symbol–unit–scale registries (conflict resolution order)

When a symbol/unit/scale conflict occurs, there is no rule that “patches” the conflict by interpretation. Conflicts are judged by Gate; if the verdict is FAIL or INCONCLUSIVE, the corresponding artifact loses admissibility as a conclusion. The precedence order used for conflict judgment is fixed as follows.

  1. Priority 1: Symbol Registry The meaning of a symbol has the highest priority. Here, meaning includes (i) the entity/object the symbol refers to (cell/core/shell/throat/path, etc.), (ii) the geometric meaning (diameter/radius, center/surface, cube/sphere, etc.), (iii) the measurement convention to which the numeric value is bound (which definition of length it belongs to), and (iv) the admissible operations (sum/product/integration, etc.). Assigning a unit or a scale value before the symbol meaning is locked is forbidden.

  2. Priority 2: Unit Registry Units are subordinate to symbol meaning. The unit registry fixes, for each symbol, the dimension (length/time/mass/dimensionless, etc.), the notation system (SI vs internal dimensionless, including mixing-prohibition rules), and the conversion policy. If a unit conflicts with the symbol meaning (e.g., assigning time to a symbol defined as length), the unit is automatically invalid and the conflict is judged as FAIL by Gate.

  3. Priority 3: Scale Registry The scale registry has meaning only after symbol and unit are locked. Scales are fixed by separating (i) canonical scales and (ii) realized scales. Canonical scales lock definitions (what the scale is), while realized scales lock the numerical values as sealed by operational anchors and procedures. No rule permits moving the numerical value of a scale to match a target outcome.

Therefore, changing a symbol meaning (priority-1), changing a unit (priority-2), or changing a scale value (priority-3) because “the number does not match” is forbidden within the same version. Change exists only via versioning (a new LOCK).

Single source of truth (SSOT) and separated storage of registries

Symbols/units/scales are stored in separate registry files, and each registry functions as SSOT. The SSOT implementation rules are fixed as follows.

  1. The meaning of a symbol exists only in the symbol registry.

  2. The unit dimension and notation rules of a symbol exist only in the unit registry.

  3. The definition and numeric attribution of a scale exist only in the scale registry.

  4. Body sections do not redefine registry items; they refer to them by item name and lock_id.

  5. Duplicated definitions/units/numbers outside registries are forbidden; upon detection, they are judged as conflicts.

Separated storage is a structural device to prevent “definitions (meaning)”, “units (dimensions)”, and “numbers (scales)” from retroactively adjusting each other.

Standard fields of the symbol registry (sealing meaning)

The symbol registry contains the following standard fields for each symbol. The presence of these fields is mandatory; if a field is missing, the corresponding symbol is judged unusable.

  1. symbol: the symbol string (e.g., a, D_anch, r_p, _rot, etc.).

  2. entity: the entity/object the symbol refers to (e.g., cell length, core radius, shell coordinates, critical throat thickness, etc.).

  3. geometry_meaning: geometric meaning (diameter/radius, center–surface, cube/sphere, axis/plane, etc.).

  4. definition: a definition sentence or equation (optionally with an identifier).

  5. allowed_operations: admissible operations and combinations (e.g., whether integration is allowed, what averaging convention applies, whether absolute value/square is allowed).

  6. scope: scope of applicability (a regime identifier or a condition). Use outside scope is forbidden.

  7. notes: conflict-prevention notes (distinguishing similarly spelled symbols, prohibiting reinterpretations, etc.).

The symbol registry forbids “the same notation used with different meanings.” A single symbol cannot carry multiple meanings; if needed, symbols must be split and each meaning locked as an independent entry.

4.1 2.5 Standard fields of the unit registry (sealing dimensions)

The unit registry contains the following standard fields for each symbol.

  1. symbol: linked 1:1 to the symbol field in the symbol registry.

  2. dimension: dimension (length/time/mass/dimensionless, etc.) or composite dimensions if required.

  3. unit_system: notation system (e.g., SI vs internal dimensionless), together with whether mixing is allowed.

  4. unit_name: unit notation (e.g., m, s, fm, pm, etc.). The notation itself is locked.

  5. conversion_policy: conversion policy (allowed/disallowed; where conversions may be performed).

  6. consistency_checks: rules for dimensional-consistency checks (identifiers or checklist items).

The unit registry is valid only under the “meaning-first” precedence. Therefore, the unit registry does not modify symbol meaning, and unit conflicts are not resolved by unit reinterpretation.

4.2 2.6 Standard fields of the scale registry (separating canonical vs realized)

The scale registry records scales by separating canonical and realized entries. The standard fields are as follows.

  1. scale_id: scale identifier (e.g., S-L-a, S-T-dt, S-L-Danch, S-L-rp, S-L-lrot, etc.).

  2. symbol: the symbol to which the scale is bound (linked to a symbol-registry entry).

  3. kind: one of canonical, realized, or derived.

  4. definition: the definition equation (canonical) or the output equation (realized/derived).

  5. value: the numerical value (mandatory only for realized/derived). For canonical, this field is left empty.

  6. unit_ref: unit-registry reference (binding to a unit dimension).

  7. anchor_ref: operational-anchor reference (mandatory only for realized). It identifies which anchor and which procedure sealed the value.

  8. dependencies: a list of dependencies (other scales, anchors, procedures). Circular dependency is forbidden.

  9. scope: scope of applicability (a regime identifier or a condition).

In the scale registry, realized is not “a measurement” but “a sealed result value fixed by an operational anchor and procedure,” and derived is a computed result obtained from locked scales. A derived entry does not modify or replace a canonical entry.

4.3 2.7 Conflict verdict and versioning procedure (no interpretation)

Symbol/unit/scale conflicts are not repaired by interpretation, and are handled by the following rules.

  1. Conflict detection: multiple meanings for the same symbol, unit-dimension mismatch, mismatch between scale definition and numeric attribution, ambiguous diameter/radius meaning, mixed cube/sphere interpretations, etc.

  2. Verdict: conflicts are judged only as FAIL or INCONCLUSIVE by Gate, accompanied by conflict-type labels.

  3. Isolation: any artifact containing the conflict and any artifacts derived from it are removed from the pool of admissible conclusions (this is a status transition, not deletion).

  4. Versioning: to resolve a conflict, a new lock_id must be issued for the relevant registry (canon/realization/analysis). Under the new lock_id, (i) fix the conflicting item, (ii) update dependencies, (iii) re-derive all dependent results, (iv) re-judge all required Gates, and (v) seal a snapshot.

  5. No mixing: building one conclusion by mixing symbol/unit/scale entries from different lock_id combinations is forbidden.

Therefore, sentences of the form “the same value can be read with another meaning” do not qualify as conclusions in this document. If the meaning is not uniquely locked, the value is not a conclusion.

4.4 2.1 Symbol / unit / object standards

4.4.1 2.1.1 Scope of the standard

This subsection fixes the global standards for (i) symbols, (ii) units, (iii) objects, and (iv) scales used throughout the document. The purpose is to structurally prevent: the same spelling being used with different meanings; the same meaning splitting into multiple notations; unclear unit dimensions; and mixed geometric meanings such as diameter vs radius. The standards fixed here are not redefined later; instead of restating them, later sections refer only to the corresponding registry entries (item name and identifier).

4.4.2 2.1.2 Symbol standard (notation rules)

Symbols are fixed across the whole document by the following conventions.

4.4.3 (A) Fonts and entity types

  1. Scalars are written in italic: \(a, r_0, r_p, \ell_{\mathrm{rot}}, \Delta t\).

  2. Vectors are written in bold italic: \(\mathbf{x}, \mathbf{v}, \mathbf{n}\).

  3. Matrices/tensors are written either as bold capitals or by a double-struck rule: \(\mathbf{M}\) or \(\mathbb{T}\). In this document, bold capitals are the default.

  4. Sets are written in calligraphic style: \(\mathcal{V}\) (VP set), \(\mathcal{E}\) (event set), \(\mathcal{G}\) (graph).

  5. Functions are written in roman: \(\mathrm{Gate}(\cdot)\), \(\mathrm{Rect}(\cdot)\), \(\mathrm{Hash}(\cdot)\).

4.4.4 (B) Indices and subscripts

  1. Entity indices are written as integer subscripts: \(i, j, k\).

  2. Component indices are written as Greek letters or coordinate subscripts: \(\alpha, \beta\) or \(x,y,z\).

  3. Object-label subscripts are written as roman abbreviations: \(r_p\) (proton core radius), \(\ell_{\mathrm{rot}}\) (rotation-driven length).

  4. Regime/scope subscripts are written in roman: \(a_{\mathrm{sim}}\) (internal unit), \(a_{\mathrm{real}}\) (after unit realization).

  5. Averaging/aggregation notation is fixed to an overbar: \(\overline{X}\) denotes only a result produced by a pre-registered averaging operator. The averaging operator is locked in analysis_lock.

4.4.5 (C) Reserved Symbols

To prevent symbol conflicts, the following reservation rules are fixed.

  1. Canonical input scales are written as \(D_{\mathrm{anch}}, r_p, \ell_{\mathrm{rot}}\), etc., and symbols used for canonical inputs are not reused with other meanings.

  2. Unit-realization scales use \(a, \Delta t\) as base symbols, and \(a\) and \(\Delta t\) are not reused with other meanings (e.g., area or acceleration).

  3. Diameter/radius meaning is locked in the symbol field geometry_meaning; a single symbol cannot mean both diameter and radius.

  4. Dimensionless numbers refer only to entries whose unit field specifies dimensionless, and dimensionless notation does not permit unit mixing.

4.4.6 2.1.3 Unit standard (dimension and notation rules)

Units are assigned only after symbol meaning has been locked. The unit standards are fixed as follows.

4.4.7 (A) Separation of unit systems

  1. Canonical units: used at the stage where meaning is locked (not values). Canonical units may be recorded as internal dimensionless or internal length/time units; in that case the unit registry locks unit_system=internal.

  2. Realized units: used after numeric values are fixed via operational anchors. Realized units are locked as unit_system=SI, and notation m, s with sub-prefix forms is allowed.

  3. No mixing: internal and SI units are not mixed within a single equation or a single table. If mixing is required, the conversion step itself must be separated as an independent derivation item and the input/output units of the conversion must be locked.

4.4.8 (B) Mandatory dimension field

Every symbol must have a mandatory dimension field in the unit registry.

  1. Length dimension: L

  2. Time dimension: T

  3. Mass dimension: M

  4. Energy dimension: E

  5. Force dimension: F

  6. Dimensionless: 1

If the dimension field is empty or conflicting, the symbol is judged unusable.

4.4.9 (C) Prefix notation and fixed scientific notation

In realized units, length and time may use prefixes, but the notation rules are fixed.

  1. Only prefix forms approved in the unit registry may be used.

  2. Scientific notation is fixed to the form \(x\times 10^n\).

  3. Significant-digit and rounding rules follow the rule locked in analysis_lock. Post-hoc changes are forbidden; changes are allowed only by versioning.

4.4.10 2.1.4 Object standard (object dictionary and meaning lock)

An object specifies “what a symbol refers to” and acts as the highest-level criterion. Objects are fixed in an Object Registry with the following conventions.

4.4.11 (A) Object ID and minimal fields

Every object has an object ID and the following minimal fields.

  1. object_id: e.g., OBJ-VP, OBJ-CELL, OBJ-CORE, OBJ-SHELL, OBJ-THROAT, OBJ-PATH, OBJ-EVENT.

  2. name: a human-readable object name.

  3. definition: the object definition (including defining equations if needed).

  4. geometry_meaning: the geometric meaning of the length symbols carried by the object (diameter/radius/gap/thickness/lattice constant, etc.).

  5. state_fields: state variables of the object (if any), locked as a list.

  6. allowed_maps: admissible mappings to other objects/observables (only allowed transforms are listed).

  7. scope: regime scope.

4.4.12 (B) Standard definitions of core objects (minimal declarations)

Core objects that must appear in this document are fixed as follows.

  1. Volume particle (VP), OBJ-VP: the fundamental unit composing space. VP properties and admissible rules are locked in the axioms/definitions of canon_lock.

  2. Cell, OBJ-CELL: the reference domain used to bundle a set of VP. Cell geometry (e.g., cube cell, spherical visualization cell) is locked by the geometry_meaning field; a single cell symbol cannot simultaneously refer to different geometries.

  3. Core, OBJ-CORE: the object defining the central structure. The radius/diameter meaning of the core length symbol must be locked.

  4. Shell, OBJ-SHELL: the structural object outside the core. Shell-coordinate sets, cancellation rules, survival vectors, etc., are locked together with procedures in analysis_lock.

  5. Throat, OBJ-THROAT: a microscopic bottleneck element in global connectivity. A throat thickness/gap/threshold symbol requires geometry_meaning, and the critical-throat estimator is locked in analysis_lock.

  6. Path, OBJ-PATH: a global transfer path made of connected throats. Path-selection rules and alternative-path rules are locked in analysis_lock.

  7. Event, OBJ-EVENT: the minimal observable/loggable event unit. Event definition is locked in canon_lock; event-rate estimation and aggregation are locked in analysis_lock.

4.4.13 2.1.5 Scale hierarchy for length/time/mass (hierarchy registry)

A scale hierarchy separates “what is an input,” “what is a derived result,” and “what is a realized output,” and locks the classification. Length/time/mass share a common structure.

4.4.14 (A) Hierarchy types

Each scale is classified as one of the following types.

  1. CAN-INPUT: canonical input. Meaning and value are locked in canon_lock.

  2. CAN-DERIVED: canonical derived. Derived from CAN-INPUT; derivation rules are locked in canon_lock or analysis_lock.

  3. REAL-PRIMARY: primary outputs of unit realization. Locked in realization_lock by operational anchors and realization rules.

  4. REAL-DERIVED: realized derived. Derived from REAL-PRIMARY and locked derivation rules; results are recorded in realization_lock or a derived registry.

  5. OBS-REF: observational reference. A reference obtained from an observation protocol; the meaning and protocol of the observation value are locked in analysis_lock.

4.4.15 (B) Standard entries for the length-scale hierarchy

The length-scale hierarchy contains the following standard entries. The existence of these items is enforced in the registry (whether a value is present depends on the type).

Item Type Geometric meaning Bound object / description
\(D_{\mathrm{anch}}\) CAN-INPUT diameter or length (locked) OBJ-CELL or canonical cell length (meaning must be locked)
\(r_0\) CAN-DERIVED radius (locked) fixed as \(r_0:=D_{\mathrm{anch}}/2\) (diameter–radius meaning lock)
\(r_p\) CAN-INPUT radius (locked) OBJ-CORE reference radius (meaning must be locked)
\(\ell_{\mathrm{rot}}\) CAN-INPUT or OBS-REF length (locked) OBJ-CELL/OBJ-THROAT rotation-drive input (adopted type is fixed by the lock)
\(a\) REAL-PRIMARY length (locked) OBJ-VP or the basic length unit inside the cell (unit realization output)
\(\delta_{\mathrm{gap}}\) REAL-DERIVED gap/thickness (locked) OBJ-THROAT critical-throat gap (estimator locked)

4.4.16 (C) Standard entries for the time-scale hierarchy

The time-scale hierarchy contains the following standard entries.

Item Type Meaning Bound object / description
\(\Delta t\) REAL-PRIMARY time tick (locked) OBJ-EVENT or a cell-based time unit (unit realization output)
\(T_{\mathrm{p}}\) REAL-DERIVED or CAN-DERIVED build time (locked) build time derived from canonical event rate and structural conventions
\(T_{\mathrm{n}}\) REAL-DERIVED or CAN-DERIVED auxiliary time (locked) an auxiliary time scale defined together with \(T_{\mathrm{p}}\) (definition location locked)

4.4.17 (D) Standard entries for the mass-scale hierarchy

Even when a number named “mass” appears, the mass-scale hierarchy fixes which internal definition that number comes from.

Item Type Meaning Bound object / description
\(m_p\) REAL-DERIVED mass scale (locked) OBJ-CORE mass scale (derivation rules + Gate required)
\(m_e\) REAL-DERIVED mass scale (locked) OBJ-EVENT or a shell-survival-structure-based mass scale (derivation rules + Gate required)
\(m_H\) REAL-DERIVED mass scale (locked) mass scale derived from the lattice unit energy and geometric resistance conventions (derivation rules + Gate required)

Mass-scale entries do not qualify as conclusions by value alone. A mass scale acquires admissibility only together with (i) meaning lock, (ii) locked derivation rules, and (iii) a Gate verdict.

4.4.18 2.1.6 Prohibition of symbol reuse (no overloading)

Symbol reuse (overloading) is a global prohibition rule in this document. “Overloading” refers to any case where at least one of the following holds.

  1. The same symbol is bound to different objects (OBJ-*).

  2. The same symbol has different geometric meanings (diameter/radius/gap/thickness/lattice constant, etc.).

  3. The same symbol has different dimensions (L, T, M, 1, etc.).

  4. The same symbol simultaneously has different hierarchy types (CAN-INPUT vs REAL-PRIMARY, etc.).

  5. The same symbol is subjected to different averaging/aggregation conventions (e.g., different averaging operators for \(\overline{X}\)).

A need for overloading is not accepted as a justification. If meanings differ, symbols must be split. Symbol splitting follows these fixed rules.

  1. Object-subscript split: if the same spelling is maintained, attach an object-label subscript (e.g., \(r_p\), \(r_0\), \(r_{\mathrm{cell}}\)).

  2. Regime-subscript split: if regimes differ, split with a regime subscript (e.g., \(a_{\mathrm{sim}}\), \(a_{\mathrm{real}}\)). However, even then the meaning and dimension must remain identical.

  3. Full split: if meaning and dimension differ, change the spelling itself and fix a purpose subscript (e.g., \(\delta_{\mathrm{rect}}\) vs \(\delta_{\mathrm{gap}}\)).

If symbol overloading is detected, the corresponding artifact is judged FAIL by the symbol Gate and loses admissibility as a conclusion. Overloading is not resolved by “interpretation” within the same version; if a registry entry must be modified, it is permitted only via versioning.

4.5 2.2 CANON inputs (canonical)

4.5.1 2.2.1 Definition of canonical inputs

Canonical inputs (CANON inputs) are the set of inputs treated as starting points of evidence throughout the document. A canonical input must satisfy all of the following properties simultaneously.

  1. Meaning lock: what the item means (object binding, geometric meaning such as diameter/radius, inclusion/exclusion criteria) is fixed uniquely.

  2. Unit lock: its dimension and unit notation are fixed. Items that require units have their units locked.

  3. Value lock: if a value is given, it is fixed (including accuracy/significant-digit rules).

  4. Scope lock: the applicable regime scope (global vs specific regimes) is fixed. It cannot be referenced outside scope.

  5. SSOT: canonical inputs exist only in a single location in the canon_lock registry. The body does not redefine them and refers only to item name and identifier.

Canonical inputs are not “targets to be derived.” They are derivation inputs, and replacing a canonical input by a derived result, or retroactively modifying a canonical input based on a derived result, is forbidden.

4.5.2 2.2.2 List and classification of canonical input items

In this document, the following five items are classified as canonical inputs.

  1. \(D_{\mathrm{anch}}\) : canonical anchor length (representative length of the anchor cell).

  2. \(r_p\) : canonical radius (defined as the proton core radius).

  3. \(\pi\) : the circle constant (dimensionless).

  4. \(\delta\) : a rectification constant (dimensionless).

  5. \(\ell_{\mathrm{rot}}\) : rotation-drive length (reference canonical input).

Because these items have different roles, canonical inputs are further classified as follows.

4.5.3 2.2.3 Meaning lock for each item (object binding and geometric meaning)

For canonical inputs, “meaning” must be locked before “value.” The object binding and geometric meaning of each item are fixed as follows.

4.5.4 (A) \(D_{\mathrm{anch}}\) : canonical anchor length

\(D_{\mathrm{anch}}\) is defined as the representative length of OBJ-CELL (cell) or a canonical geometric object treated as equivalent to the cell. The geometric meaning of \(D_{\mathrm{anch}}\) (diameter/radius/edge length) is fixed uniquely in the geometry_meaning field of canon_lock. Because \(D_{\mathrm{anch}}\) is the canonical anchor, no section may reinterpret its meaning and replace it by a different geometric quantity. If a radius-like derived quantity is needed, it is introduced only as a derived definition with a separate symbol, for example \[r_0 := \frac{D_{\mathrm{anch}}}{2}.\] Here, \(r_0\) is a derived symbol, and it is usable only after the meaning lock of \(D_{\mathrm{anch}}\) is satisfied.

4.5.5 (B) \(r_p\) : canonical radius (core radius)

\(r_p\) is defined as the radius of OBJ-CORE (core). \(r_p\) is locked as a radius and cannot be reinterpreted as a diameter. The value of \(r_p\) is locked as a canonical input value; the unit is locked as length. The unit notation is fixed to one of the registry-approved prefix forms (e.g., fm). \(r_p\) is used as an evidence starting point in core geometry and event-rate derivations; changing \(r_p\) or replacing it by a different meaning is forbidden within the same version.

4.5.6 (C) \(\pi\) : the circle constant

\(\pi\) is dimensionless and its value is globally fixed. Since \(\pi\) is a defined constant rather than a measurement, no section may estimate or adjust \(\pi\). \(\pi\) is used in rectification constants and geometric ratios, and every derivation that uses \(\pi\) refers to it only as a canonical constant.

4.5.7 (D) \(\delta\) : rectification constant

\(\delta\) is defined as a dimensionless rectification constant. The definition of \(\delta\) is fixed in a single location as \[\delta := \frac{1}{\pi^2}.\] This definition is locked in canon_lock; later sections do not re-derive \(\delta\). Recomputing \(\delta\) from a different averaging or projection convention in order to change its value is forbidden. The usage of \(\delta\) is restricted to the defined form; if another rectification constant is needed, it must be separated as a new symbol and locked as a separate entry.

4.5.8 (E) \(\ell_{\mathrm{rot}}\) : rotation-drive length (reference canonical input)

\(\ell_{\mathrm{rot}}\) is defined as a length input used in rotation-drive regimes. Its geometric meaning is locked as a diameter (no diameter/radius mixing). \(\ell_{\mathrm{rot}}\) is classified as follows.

Therefore, replacing \(\ell_{\mathrm{rot}}\) by \(D_{\mathrm{anch}}\), or redefining the meaning of \(D_{\mathrm{anch}}\) from \(\ell_{\mathrm{rot}}\), is forbidden. If promotion is required, the versioning procedure of 2.2.6 must be followed.

4.5.9 2.2.4 Value/unit lock (standard notation and significant digits)

Value and unit notation of canonical inputs are locked by the following rules.

  1. \(D_{\mathrm{anch}}\), \(r_p\), and \(\ell_{\mathrm{rot}}\) are locked as length dimension (L).

  2. \(\pi\) and \(\delta\) are locked as dimensionless (1).

  3. If a value is provided, it is fixed either in standard scientific notation (\(x\times 10^n\)) or in a fixed-point notation as specified in the registry.

  4. Significant-digit / rounding rules are fixed in analysis_lock under numeric_format, and do not change within the same version.

If a value for \(\ell_{\mathrm{rot}}\) is provided (e.g., \(\ell_{\mathrm{rot}}=\lrot\)), its diameter meaning and unit (e.g., pm) must be locked simultaneously. Recording a value without locking meaning is not allowed.

4.5.10 2.2.5 Reference rules for canonical inputs (usage rules in the body)

Canonical inputs are used in the body only under the following rules.

  1. The body does not redefine canonical inputs. It refers only to the canon_lock item names (e.g., D_anch, rp, pi, delta, l_rot) and lock_id.

  2. Every equation/table/figure/log that uses a canonical input must also reference the corresponding meaning lock information (geometry_meaning, entity, object_id).

  3. Any section that requires a canonical input must declare the regime scope together. For example, \(\ell_{\mathrm{rot}}\) may be referenced only in rotation-drive regimes; unconditional reference in global sections is forbidden.

  4. When \(\pi\) and \(\delta\) appear together in the same section, \(\delta\) must always be used by referencing the definition \(\delta:=1/\pi^2\), and may not be treated as an independently adjustable constant.

4.5.11 2.2.6 Change procedure (versioning and re-verification)

Changing a canonical input is not allowed within the same version. Changes are performed only via versioning, under the following fixed procedure.

  1. Specify the category of change: a canon_lock item change (canonical input change).

  2. Issue a new canon_lock lock_id. The previous lock_id is preserved and is not modified.

  3. Fix a change log that records, before/after, (i) meaning (entity, geometry_meaning, object_id), (ii) value, and (iii) unit.

  4. Mark all derived results that depend on the changed item as targets for regeneration in the dependency graph.

  5. Re-run the required Gate stacks and re-judge PASS/FAIL/INCONCLUSIVE for all affected results.

  6. Seal the new version with a registry snapshot, manifest, and checksums.

In particular, promoting \(\ell_{\mathrm{rot}}\) to a required input, or modifying the meaning of \(D_{\mathrm{anch}}\) by coupling it with \(\ell_{\mathrm{rot}}\), is a change of the canonical-input structure itself and therefore requires versioning and full re-verification.

4.6 2.3 REALIZATION inputs

4.6.1 2.3.1 Definition of REALIZATION inputs

REALIZATION inputs are the set of items fixed to realize an internally described world (dimensionless or internal length/time units) into external units (length/time). REALIZATION inputs are included in realization_lock and do not change within the same lock_id. REALIZATION inputs are not “canonical inputs”; they are treated as “realization outputs” or “reference anchors used for realization.” Therefore, REALIZATION inputs must simultaneously lock (i) meaning (what length/what time), (ii) unit, (iii) numeric value, and (iv) procedural attribution (which reference channel and which verdict rule sealed it).

4.6.2 2.3.2 Internal coordinates and realization map (length/time conversion)

Define internal coordinates as follows.

The realization map is fixed as \[x \;:=\; a\,\tilde{x}, \qquad t \;:=\; \Delta t\,\tilde{t}, \label{eq:realization_map_xt}\] where \(x\) is realized length (length dimension) and \(t\) is realized time (time dimension). Accordingly, internal velocity \(\tilde{v}\) and realized velocity \(v\) are related by \[\tilde{v} \;:=\; \frac{d\tilde{x}}{d\tilde{t}}, \qquad v \;:=\; \frac{dx}{dt} \;=\; \frac{a}{\Delta t}\,\tilde{v} \label{eq:realization_map_v}\] In this relation, \(a/\Delta t\) is the fundamental scaling factor with dimension (length/time), and fixing this factor is the core of realization.

4.6.3 2.3.3 Meaning lock for \(a\) (realized length scale)

\(a\) is the realized primary length scale. The meaning of \(a\) is fixed as follows.

  1. \(a\) is defined as the fundamental diameter of a volume particle (VP).

  2. The geometric meaning of \(a\) is locked as a diameter and is not reinterpreted as a radius.

  3. The dimension of \(a\) is locked as length (L), and the unit system is locked as SI.

  4. The numerical value of \(a\) is locked in realization_lock as \[a \;=\; \aVP. \label{eq:a_value_lock}\]

If a radius is needed, introduce a separate derived symbol \[r_a \;:=\; \frac{a}{2}. \label{eq:ra_derived}\] \(r_a\) is a derived quantity; it is not a device to alter the meaning of \(a\).

4.6.4 2.3.4 Meaning lock for \(\Delta t\) (realized time tick)

\(\Delta t\) is the realized primary time tick. Its meaning is fixed as follows.

  1. \(\Delta t\) is defined as the scaling factor that converts one unit of internal time \(\tilde{t}\) into realized time \(t\) (Eq. [eq:realization_map_xt]).

  2. The dimension of \(\Delta t\) is locked as time (T), and the unit system is locked as SI.

  3. The numerical value of \(\Delta t\) is locked in realization_lock as \[\Delta t \;=\; 1.86\times 10^{-21}\ \mathrm{s}. \label{eq:dt_value_lock}\]

\(\Delta t\) is not identified with an algorithmic time step used for numerical stability. If an algorithmic step exists, it is locked as a protocol parameter in analysis_lock and kept distinct from \(\Delta t\). \(\Delta t\) is used only as the fixed scaling factor of the realization map ([eq:realization_map_xt]).

4.6.5 2.3.5 Meaning lock for \(c_{\mathrm{ref}}\) (reference speed constant for realization)

\(c_{\mathrm{ref}}\) is a reference speed constant used during realization. Its meaning is fixed as follows.

  1. \(c_{\mathrm{ref}}\) is defined as a constant of an external reference channel used to fix the realized speed unit \(a/\Delta t\).

  2. \(c_{\mathrm{ref}}\) is locked with dimension (L/T) and unit notation m/s.

  3. \(c_{\mathrm{ref}}\) is recorded in realization_lock together with both a “value” and a “reference-channel identifier.” Locking only a value without the channel/procedure is not allowed.

  4. \(c_{\mathrm{ref}}\) is not identified with an internally derived propagation indicator (e.g., a particular value of internal speed \(\tilde{v}\)). The internal propagation indicator is a derived result; \(c_{\mathrm{ref}}\) is a realization reference. Their connection is made only via the realization scaling factor \(a/\Delta t\) in Eq. [eq:realization_map_v].

Therefore, if \(c_{\mathrm{ref}}\) is adopted as a realization input, the realized speed scale is fixed as \[\frac{a}{\Delta t} \quad\text{(realized speed unit)} \label{eq:a_over_dt}\] This fixing is valid only when the meaning locks of \(a\), \(\Delta t\), and \(c_{\mathrm{ref}}\) all hold simultaneously.

4.6.6 2.3.6 Lock rules for REALIZATION inputs (nature of realized values)

REALIZATION inputs (\(a\), \(\Delta t\), \(c_{\mathrm{ref}}\)) are locked by the following rules.

  1. Simultaneous meaning–unit–value lock: for all three items, (i) object binding, (ii) geometric meaning (diameter/radius, etc.), (iii) dimension and unit notation, and (iv) numerical value must be locked simultaneously.

  2. SSOT: realized values exist only in a single location in realization_lock. The body refers only to item name and lock_id.

  3. Procedural attribution: \(a\), \(\Delta t\), and \(c_{\mathrm{ref}}\) must carry which reference channel / which cross-consistency / which thresholds sealed them. This procedural information cross-references the Gate composition and protocol conventions in analysis_lock.

  4. No reinterpretation: reinterpreting \(a\) as a radius, replacing \(\Delta t\) by an algorithmic step, or retroactively identifying \(c_{\mathrm{ref}}\) with an internal derived indicator is forbidden within the same version.

If any of the above rules is violated, the realization map ([eq:realization_map_xt]) collapses; the corresponding artifact loses admissibility as a conclusion.

4.6.7 2.3.7 Change procedure for realized values (versioning and full re-judgment)

Changing realized values (\(a\), \(\Delta t\), \(c_{\mathrm{ref}}\)) is not allowed within the same version. Changes are performed only via versioning, under the following fixed procedure.

  1. Specify what changes: identify whether the change is in \(a\)/\(\Delta t\)/\(c_{\mathrm{ref}}\), and whether it changes (i) meaning, (ii) unit, (iii) value, and/or (iv) reference channel/procedure.

  2. Create a new realization_lock: issue a new realization_lock_id. The previous realization_lock_id is preserved and not modified.

  3. Fix a change_log: lock the before/after items (meaning/unit/value/channel identifier), the reason, and the list of affected derived quantities.

  4. Update dependency graph: mark all derived quantities that depend on \(a\) or \(\Delta t\) (length/time/velocity scaling factors, energy scale, mass scale, force scale, etc.) as regeneration targets.

  5. Full re-derivation: regenerate all related derived artifacts from scratch (including intermediate artifacts) using the changed realized values.

  6. Full re-judgment: re-run Gate stacks from scratch, including cross-consistency, lock integrity, reproducibility, and No-Tuning violations, and re-judge PASS/FAIL/INCONCLUSIVE.

  7. Sealing: seal the new version by generating and freezing registry_snapshot, manifest, and checksums.

Conclusions prior to versioning belong to the previous realization_lock_id combination; conclusions after versioning belong to the new combination. Mixing results across different realization_lock_id values in one conclusion sentence is forbidden.

4.7 2.4 Preventing confusion between diameter/radius/cell geometry

4.7.1 2.4.1 Purpose

This subsection locks (i) standard definitions of diameter/radius/cell geometry, (ii) standard notation rules, (iii) allowed derived-conversion rules, and (iv) an immediate-FAIL policy for ambiguity/conflict detection, in order to structurally prevent a length symbol from being used interchangeably as a diameter, a radius, or a cell representative length (edge/diameter, etc.).

4.7.2 2.4.2 Standard definitions: radius and diameter

In this document, “radius” and “diameter” are fixed by the following definitions. Definitions are locked in the geometry_meaning field, and a single symbol cannot carry both meanings.

4.7.3 (A) radius

A radius is defined as the distance from a chosen center (or reference point) to a chosen boundary (or reference boundary surface). Radius symbols follow these conventions.

  1. Use \(r\) in the symbol name: \(r_0, r_p, r_a\), etc.

  2. Lock geometry_meaning as radius.

  3. A radius is not defined via the conventional relation “half of a diameter.” It is defined directly together with the object definition; its relation to diameter is connected only via a separate derived definition.

4.7.4 (B) diameter

A diameter is defined as the distance between two opposite boundaries (or reference boundary surfaces) of the same object. Diameter symbols follow these conventions.

  1. Use \(D\) in the symbol name: \(D_{\mathrm{anch}}, D_a\), etc.

  2. Lock geometry_meaning as diameter.

  3. A diameter is not defined via the conventional relation “twice a radius.” It is defined directly together with the object definition; its relation to radius is connected only via a separate derived definition.

4.7.5 (C) Radius–diameter linkage (derived definitions only)

The linkage between radius and diameter is permitted only as derived definitions as follows. \[r \;:=\; \frac{D}{2}, \qquad D \;:=\; 2r. \label{eq:rd_relation}\] Equation [eq:rd_relation] does not mean “converting symbol meaning” but “deriving a new symbol.” That is, if \(D\) is locked as a diameter in a section, then when a radius is needed one must introduce a new symbol \(r:=D/2\) and use it; within that section, \(D\) must not be used as if it were a radius. Likewise, if \(r\) is locked as a radius, then when a diameter is needed one must introduce a new symbol \(D:=2r\); within that section, \(r\) must not be used as if it were a diameter.

4.7.6 2.4.3 Standard definition: cell and cell geometry

In this document, a “cell” is the reference domain that bundles a VP set and is defined as an object. A cell must lock “which geometry it uses” together with the cell definition; a cell length/cell radius/cell diameter is unusable if cell geometry is not locked.

4.7.7 (A) Required fields of the cell object

The cell object OBJ-CELL must include the following mandatory fields.

  1. cell_id: cell identifier.

  2. cell_geometry: cell-geometry type.

  3. cell_length_symbol: the symbol used as the cell representative length.

  4. geometry_meaning: geometric meaning of the representative length (edge length, diameter, radius, etc.).

  5. definition: definition of the cell and the representative length.

  6. scope: applicable regime.

4.7.8 (B) Standard list of cell geometry types (enumeration)

Cell geometry type is locked as one of the following enumerated values.

  1. CELL-CUBE: the cell is a cube, and the representative cell length is defined as an edge length.

  2. CELL-SPHERE-VIS: the cell is used as a sphere for visualization or auxiliary definition, and the representative cell length is defined as either a diameter or a radius (exactly one is chosen and locked).

  3. CELL-OTHER: if a geometry other than the above two is used, the geometry definition (boundary, representative length, measurement convention) must be fully specified and locked with additional fields.

CELL-OTHER is not an “ambiguous free parameter.” If CELL-OTHER is used, the cell-geometry definition must be complete in the document, and the meaning and conversion relations of the representative length must all be locked.

4.7.9 (C) CELL-CUBE

For a cube cell, the representative length is defined as an edge length.

  1. The representative-length symbol of a cube cell may be written as \(L_{\mathrm{cell}}\) or \(D_{\mathrm{anch}}\), etc., but regardless of the spelling the geometry_meaning must be locked as edge.

  2. To introduce a “radius” or “diameter” for a cube cell, it must be introduced only as a derived symbol. For example, to connect a cube diagonal or an equivalent visualization sphere, the connection rule must be fixed either in analysis_lock (procedure) or canon_lock (definition), and the derived symbol must be created separately.

Interpreting a cube cell representative length as a radius or a diameter is forbidden.

4.7.10 (D) CELL-SPHERE-VIS

For a spherical cell, the representative length is defined as exactly one of diameter or radius; the two are not used simultaneously.

  1. If the representative length is defined as a diameter: use symbol \(D_{\mathrm{cell}}\) and lock geometry_meaning=diameter.

  2. If the representative length is defined as a radius: use symbol \(r_{\mathrm{cell}}\) and lock geometry_meaning=radius.

  3. If both diameter and radius are needed, introduce a separate symbol connected by the derived definition (Eq. [eq:rd_relation]).

CELL-SPHERE-VIS is a choice of cell geometry; using one cell representative length “as if” it were both CELL-CUBE and CELL-SPHERE-VIS within the same section is forbidden.

4.7.11 2.4.4 Confusion-prevention rules (lock rules)

This subsection locks confusion-prevention rules as follows. Violations are judged immediately as FAIL.

4.7.12 (R1) Mandatory meaning lock

Before a length symbol is used, it must simultaneously carry the following three fields.

  1. object_id (which object’s length)

  2. geometry_meaning (diameter/radius/edge/gap/thickness, etc.)

  3. dimension (length dimension)

If any of the three is missing, the symbol is unusable and any equation/table/figure/log that contains it is judged immediately as FAIL.

4.7.13 (R2) Single-meaning principle for the same symbol (no overloading)

The same symbol (same spelling) may carry only a single triple \((\texttt{object\_id},\ \texttt{geometry\_meaning},\ \texttt{dimension})\) across the whole document. Once the same spelling appears with a different triple, the conflict is not repaired by interpretation and is immediately FAIL.

4.7.14 (R3) Explicit derived conversion principle (no implicit conversion)

Diameter\(\leftrightarrow\)radius conversion, cube\(\leftrightarrow\)sphere mapping, and representative-length\(\leftrightarrow\)derived-length conversion are permitted only through explicit derived symbols.

  1. “Using \(D\) as if it were \(r\)” or “using \(r\) as if it were \(D\)” is forbidden.

  2. “Changing the cell representative length into diameter/radius/edge depending on context” is forbidden.

  3. If a conversion is needed, a new symbol must be introduced, and the new symbol must be registered in the registry together with the conversion equation.

4.7.15 (R4) No cross-mixing of diameter/radius/cell geometry

Within the same section, the following mixes are forbidden.

  1. After locking a cell representative length as edge, using the same symbol as diameter or radius.

  2. In a cell locked as CELL-CUBE, assigning a CELL-SPHERE-VIS diameter/radius definition to the same symbol without an explicit derived definition.

  3. Treating different cell geometry types as if they were the “same cell” and composing them within one derivation chain.

4.7.16 2.4.5 Immediate FAIL policy upon confusion detection (Gate verdict)

Detection of confusion (ambiguity/conflict) is performed by Gate; upon detection the verdict is immediately FAIL. The verdict is independent of interpretation and depends only on the integrity of definitions/locks.

4.7.17 (A) Immediate FAIL conditions

If any of the following holds, the verdict is immediately FAIL.

  1. RD-AMB (radius/diameter ambiguity): the registry does not determine whether a symbol is radius or diameter (missing or multiple geometry_meaning).

  2. RD-CONFLICT (radius/diameter conflict): the same symbol is used as radius in one place and as diameter in another.

  3. CELL-AMB (cell-geometry ambiguity): the cell geometry type (CELL-CUBE/CELL-SPHERE-VIS/CELL-OTHER) is not locked.

  4. CELL-CONFLICT (cell-geometry conflict): the same cell representative-length symbol is used as a representative length under different cell geometry types.

  5. IMPLICIT-CONV (implicit conversion): \(D\leftrightarrow r\) conversion or cube\(\leftrightarrow\)sphere mapping is performed implicitly in an equation/table/figure/log without a derived symbol.

  6. DIM-MISMATCH (dimension mismatch): a symbol locked as length is used with a different dimension, or unit notation changes without an explicit conversion step.

4.7.18 (B) FAIL labels (standard labels)

The FAIL labels for confusion verdicts are fixed as follows (multiple labels are allowed).

Label Meaning
FAIL-GEO-RD-AMB radius/diameter meaning is not locked (ambiguous)
FAIL-GEO-RD-CONF the same symbol is used conflictingly as radius/diameter
FAIL-GEO-CELL-AMB cell geometry type is not locked (ambiguous)
FAIL-GEO-CELL-CONF cell geometry type or representative-length meaning conflicts
FAIL-GEO-IMPL implicit conversion performed without derived symbols
FAIL-GEO-DIM unit/dimension mismatch or missing conversion step

4.7.19 (C) Propagation rule for immediate FAIL

If FAIL occurs, the following immediately hold.

  1. The artifact containing the failing equation/table/figure/log loses admissibility as a conclusion.

  2. In the dependency graph, all derived artifacts that use the failing artifact as input lose admissibility in a cascading manner.

  3. FAIL is not resolved by “editing text”; resolution exists only via versioning (registry modification followed by full re-derivation and re-judgment).

4.7.20 2.4.6 The only path to resolve confusion (versioning)

The procedure for resolving confusion is fixed as follows.

  1. Identify the location of the conflicting items (symbol meaning / cell geometry / unit / derived definition) in the registry.

  2. Issue a new lock_id for the relevant registry (canon_lock or analysis_lock or realization_lock).

  3. Modify items so as to remove the conflict (symbol splitting, enforcing a single geometry_meaning, enforcing a single cell geometry type, adding derived symbols, etc.).

  4. Regenerate all related derivations under the modified lock_id combination and re-judge all relevant Gates.

  5. Seal the new version with a registry snapshot, manifest, and checksums.

Interpretation or sentence editing within the same version is not accepted as confusion resolution. Confusion is a definition/lock problem; without changing definitions/locks, confusion remains.

5 3. Axioms and Primitives (Volume Particle / Lattice / Quantum Cell)

Chapter declaration: a single backbone starting from primitives

This chapter fixes, for the entire document, the minimal building blocks of the world as primitives. The primitives are (i) the Volume Particle (VP), (ii) the Stone regime, and (iii) the Cell (Anchor Cell). Their relationship is fixed as a single backbone as follows. \[\text{VP axiom set} \;\Longrightarrow\; \text{Allowed configurations (full packing / non-overlap / contact constraints)} \;\Longrightarrow\; \text{Lattice/graph (definition of adjacency)} \;\Longrightarrow\; \text{Cell definition (aggregation / counting / coordinate system)} \;\Longrightarrow\; \text{Subsequent derivations (rectification / events / realization / mass / force)}. \label{eq:primitive_chain}\] In Eq. [eq:primitive_chain], items to the right depend on items to the left. Retroactive interpretation in the opposite direction (changing a left-hand definition because a right-hand result is inconvenient) is not allowed.

VP axiom set (fixing a minimal set of axioms)

The Volume Particle (VP) is the basic unit constituting space. The properties of VP are fixed by the following axiom set. Each axiom has a distinct meaning; redundant statements across axioms are forbidden.

5.0.1 (VP-A1) Stone axiom: infinite rigidity (incompressible) and identity

Every VP has the “Stone” property. The Stone property is fixed as follows.

  1. Incompressible: the internal volume of a VP does not change. The internal volume of the same VP does not decrease or increase due to state change or configuration change.

  2. Non-penetration: two different VPs cannot occupy the same spatial region simultaneously. In other words, occupied regions of VPs do not overlap.

  3. Identity: VPs are treated as identical basic units; no intrinsic “species” differences are introduced at the axiomatic level. Differences are expressed only by configuration, contact relations, and local state variables (defined later).

This axiom does not introduce “rigidity” as an additional dynamical law; it is introduced as a constraint that restricts the set of admissible configurations. In short, the Stone axiom fixes the domain of “possible configurations” first.

5.0.2 (VP-A2) Full-packing axiom: space is fully occupied

Space is fully occupied by VPs. The full-packing axiom is fixed as follows.

  1. Full occupation: inside the region of interest (a cell or a domain), space is represented by the union of VP-occupied regions.

  2. Where defects live: an explicit “void” degree of freedom that violates full packing is not introduced as a primitive. Instead, defects, throats, gaps, and deficits appear only as local structural quantities defined as consequences of configuration and adjacency (later fixed as objects).

  3. Boundary treatment: a boundary does not mean “outside VP.” It is fixed as a procedural boundary introduced by domain selection (cell selection). The boundary type (closed/open/driven) is locked as part of the protocol.

5.0.3 (VP-A3) Local-rule axiom: contact-based configuration and local update

Changes of VP configurations are defined only by local rules. The local-rule axiom is fixed as follows.

  1. Contact-based: VP–VP interaction is expressed only through “contact” or “proximity.” Contact is reduced to adjacency; adjacency is recorded via a graph or lattice structure.

  2. Local update: changes (rearrangement, driving, relaxation) occur locally, and the update rule is protocol-locked. Updates are never defined in a way that exits the set of admissible configurations.

  3. Place for state variables: internal state variables carried by a VP (e.g., phase, orientation, local indicators) are only given a slot in this axiom. The concrete list and meaning of those state variables are defined and locked later.

5.0.4 (VP-A4) Adjacency axiom: lattice/graph as a primary object

A configuration is externalized as an adjacency structure. The adjacency axiom is fixed as follows.

  1. Adjacency graph: for a VP set \(\mathcal{V}\) inside a domain, define an edge set \(\mathcal{E}\) by an adjacency relation and construct a graph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\).

  2. Lattice/network: in this document, “lattice” does not mean a regular array; it is fixed to mean a network object whose adjacency graph provides connectivity/transfer inside the domain.

  3. Place for measurement: length-like quantities (distance, thickness, gap, etc.) are derived by combining the adjacency structure with the cell definition. They are not given as primitives that define adjacency itself.

5.0.5 (VP-N1) Effective stiffness depends on observation time: dynamic rigidity, jamming, and unjamming

The Stone axiom (VP-A1) locks a constraint: the internal volume of each VP does not change. However, experimentally reported “stiffness” (or stress response) is typically determined by whether a VP ensemble can rearrange within the time scale of observation. That is, the relevant observable is dynamic stiffness, not static stiffness, and the same configuration can appear “soft” or “hard” depending on observation time / driving rate.

To summarize this with minimal operational variables, introduce a relaxation time \(\tau_{\rm relax}\) and an observation time \(\tau_{\rm obs}\), and use the following dimensionless number as a regime label. \[\mathrm{De} \;\equiv\; \frac{\tau_{\rm relax}}{\tau_{\rm obs}}. \label{eq:deborah_like}\] Regime interpretation: If \(\mathrm{De}\ll 1\), the VP ensemble can rearrange within the observation window and distribute load, yielding a “fluid-like (soft)” response. If \(\mathrm{De}\gg 1\), rearrangement is suppressed and a rigid backbone persists, yielding a “solid-like (hard)” response. The jamming/unjamming regimes in this document may be used in a form that explicitly includes this time-scale dependence; this is also why protocol inputs such as \(\ell_{\mathrm{rot}}\) are recorded alongside specific conclusions.

5.0.5.1 Eggshell transition: a minimal state machine (conceptual)

In this document, “critical” does not mean a single number; it is defined as a regime transition. The response of a VP ensemble can be summarized by the following minimal sequence.

This sequence justifies the approach of not trying to compute stiffness microscopically to arbitrary precision, but instead locking the observed critical (yield/saturation) conditions together with the protocol (LOCK), and then performing derivations and Gate verdicts on top of that. Therefore, whenever critical scales such as \(c^2\), \(g_\star\), and \(g^\ast\) appear in a conclusion, one must always record the protocol (time window / driving rate / geometry) together; without that record, the result is treated as INCONCLUSIVE.

5.0.5.2 Minimal closure: unjamming trigger and self-healing dynamics

To turn the above conceptual sequence into verifiable statements, this document proposes the following minimal operational closure candidates.

First, the trigger of “eggshell” failure (unjamming) is fixed as: a protocol-defined demand \(\Psi_{\rm req}(t)\) exceeds a yield threshold \(\Psi_{\rm yield}\). \[\Psi_{\rm req}(t) > \Psi_{\rm yield} \;\Rightarrow\; \text{unjamming / channel open}. \label{eq:unjamming_trigger}\] Here \(\Psi_{\rm req}\) is locked in analysis_lock (together with type/dimension/unit) by exactly one definition among stress, energy density, curvature demand, event rate, etc., and \(\Psi_{\rm yield}\) is locked together under the same protocol (no post-hoc adjustment).

To align symbols with later saturation/yield descriptions (e.g., Appendix G), one can introduce the effective demand after yielding as follows. \[\Psi_{\rm eff}(t) := \min\bigl(\Psi_{\rm req}(t),\ \Psi_{\rm yield}\bigr) \label{eq:psi_eff_clamp}\] By definition, \(\Psi_{\rm eff}(t)\le \Psi_{\rm yield}\).

Second, introduce a macroscopic state variable \(\xi(t)\in[0,1]\) representing the “health” of the jamming network (\(\xi=1\): jammed, \(\xi=0\): fully unjammed). Close self-repair (re-jamming) with a first-order relaxation; the simplest form is the following piecewise dynamics. \[\dot \xi(t)= \begin{cases} -\dfrac{\xi(t)}{\tau_{\rm break}}, & \Psi_{\rm req}(t)>\Psi_{\rm yield},\\[6pt] +\dfrac{1-\xi(t)}{\tau_{\rm heal}}, & \Psi_{\rm req}(t)\le \Psi_{\rm yield}. \end{cases} \label{eq:g_dynamics_piecewise}\] Here \(\tau_{\rm break}\) and \(\tau_{\rm heal}\) are time scales depending on regime/environment/defect distribution and are locked in analysis_lock. If needed, an effective stiffness (or resistance) can be connected by a linear mixture such as \(K_{\rm eff}(t)=K_{\rm soft}+(K_{\rm jam}-K_{\rm soft})\,\xi(t)\), and the stiffness indicator \(\chi_{\mathrm{ST}}\) of Sec. 3.2 can be closed as a threshold function \(\chi_{\mathrm{ST}}=\mathbf{1}[\xi\ge \xi_{\rm th}]\) (the threshold \(\xi_{\rm th}\) is protocol-locked).

Third, to connect the “inflow/outflow (flux channel)” of stage [D] to the flux definition in Sec. 4.1, introduce the channel-open indicator \[\chi_{\rm open}(t):=\mathbf{1}[\Psi_{\rm req}(t)>\Psi_{\rm yield}] \label{eq:chi_open_def}\] and one can fix, as an operational cap for a cut flux \(J\) (Definition [eq:flux_def]), at least \[|J| \le c_{\mathrm{ref}}\,\chi_{\rm open}. \label{eq:flux_cap_cref}\] This is the minimal statement that treats “inflow at the speed of light” not as an identity (\(=\)) but as an upper bound (\(\le\)). Whether a nonzero flux can exist in the regime \(\chi_{\rm open}=0\) is judged separately by the protocol definition.

This closure is not a substitute for microscopic rules. It is a minimal model to describe dynamic rigidity (time-scale dependence) and the jamming–unjamming transition in an experimentally decidable form. The purpose of these equations is not “decimal places” but to separate and record (i) when a transition happens (Trigger) and (ii) how fast the system recovers (Time-scale) as measurable parameters.

Stone regime and its relation to the VP axioms

Stone is another name for the infinite rigidity introduced in VP-A1; in this document it is not an “additional entity.” Stone means the following.

  1. The regime in which the incompressibility constraint of VPs applies globally is called the “Stone regime.”

  2. In the Stone regime, “volume change” is not treated as a degree of freedom; derivations proceed only via configuration (adjacency) and local update rules.

  3. The scope of applicability (when the Stone regime is assumed) is locked as a regime declaration; extensions outside the Stone regime are introduced only as closure candidates in separate sections.

Therefore, “Stone” is a shorthand that refers to an axiom set; introducing Stone does not import a new evidence system.

Cell definition: a primary object for aggregation and coordinates

A Cell is a unit of aggregation introduced to describe VP configurations and also a reference for coordinates. A cell can be defined only within the constraints of VP-A1~VP-A4. The core of the cell definition is to lock the following three items.

  1. Object attribution: the cell is fixed as object OBJ-CELL; a cell represents a domain selection that groups a VP set.

  2. Cell geometry: lock the geometry type of the cell (e.g., cubic cell), and lock the meaning of the representative cell length (edge length / diameter / radius) as exactly one.

  3. Representative cell length: the representative cell length (e.g., \(D_{\mathrm{anch}}\) or an equivalent notation) is locked as a canonical input in canon_lock and is never reinterpreted in later sections.

A cell is not “a property of VP”; it is “a way of describing VP configurations.” That is, the cell exists only as a tool for describing a world constrained by axioms, not as a source of axioms.

5.1 3.5 Coupling the VP axioms with the cell definition: what is prior and what is derived?

The coupling priority between the VP axiom set and the cell definition is fixed as follows.

  1. Priority 1 (VP axioms): the set of admissible configurations (incompressible, non-penetration, full packing, local updates, adjacency) is fixed first.

  2. Priority 2 (cell definition): within the admissible set, the cell selects a domain and performs aggregation and coordinatization.

  3. Priority 3 (derived quantities): inside the cell, counting, distributions, paths, throats, and event aggregation are defined, and derivations of length/time/energy/mass follow.

Therefore, the following retroactive moves are forbidden.

Allowed change exists only by versioning (a new lock_id). After a version upgrade, one must rerun the entire chain VP axioms \(\rightarrow\) cell \(\rightarrow\) derived quantities \(\rightarrow\) Gate verdicts.

5.2 3.1 VP axioms (minimal assumptions)

5.2.1 3.1.1 [D]

In this section [A] denotes an axiom. An axiom is a starting point that is not further derived in the theory and is not modified within a single version. In this section [D] denotes a definition. A definition is a linguistic /formal convention that fixes the meaning of all subsequent statements; if a definition changes, the same symbol would point to a different target, therefore definitions are also not modified within a single version. This section locks infinite rigidity, full packing, and the local rule as [A] for the VP world, and locks the essential terms used by those axioms as [D]. We also list examples of assumptions that are easy to sneak in but are prohibited within the same version.

5.2.2 3.1.2 [D] Primitive objects and basic terms

5.2.2.1 [D-1] VP

A Volume Particle (VP) is the basic constituent unit of space. A VP is defined not as a “point” or a “coordinate” but as an object that has an occupied region. The set of VPs is denoted by \(\mathcal{V}\).

5.2.2.2 [D-2] configuration

Inside a domain (see [D-4]), each VP \(i\in\mathcal{V}\) has an occupied region \(\Omega_i\). A configuration is defined as the set \(\{\Omega_i\}_{i\in\mathcal{V}}\). Configurations are classified as “admissible” or “inadmissible”; the criterion is determined by [A].

5.2.2.3 [D-3] non-overlap

For two distinct VPs \(i\neq j\), define impenetrability (non-overlap) as the property that the intersection of occupied regions is empty: \[\Omega_i \cap \Omega_j = \varnothing \quad (i\neq j).\] Non-overlap is used only as a condition on admissible configurations, not as a force law or an equation of motion.

5.2.2.4 [D-4] boundary

A domain \(\mathcal{D}\) is a finite region of interest in which derivations and aggregations are performed. Domain selection is a procedural choice and does not change the properties of VPs. A boundary is the separation introduced by choosing a domain; the boundary type (closed/open/driven) is not an axiom but a protocol item.

5.2.2.5 [D-5] Full Packing

Define Full Packing as the property that the union of VP occupied regions fills the domain. In this section, “fills” is locked to mean that we introduce no new degrees of freedom for empty space. That is, void space inside the domain is not treated as an independent object; it is treated only as a structural quantity defined as a consequence of configuration and adjacency.

5.2.2.6 [D-6] adjacency

Two VPs \(i,j\) are said to be “in contact” when a pre-registered contact predicate (one of: distance-based, surface-based, etc.) is satisfied. The contact predicate itself is a definition item and may not be replaced after seeing results. Adjacency is the discrete record of contacts. Define the adjacency graph as \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), where \((i,j)\in\mathcal{E}\) means that \(i\) and \(j\) are adjacent.

5.2.2.7 [D-7] neighborhood

Define the local neighborhood of VP \(i\) by \[\mathcal{N}(i) := \{\, j\in\mathcal{V}\;|\;(i,j)\in\mathcal{E}\,\}.\] The neighborhood is the minimal unit that determines “what information is local.”

5.2.2.8 [D-8] global update

A local update is an operation that changes part of a configuration by depending only on a chosen VP \(i\) and its neighborhood \(\mathcal{N}(i)\). Denote it by an operator \(\mathcal{U}_i\). A global update is an update that can be expressed as a finite composition of local updates: \[\mathcal{U}_{\mathrm{global}} = \mathcal{U}_{i_K}\circ\cdots\circ\mathcal{U}_{i_2}\circ\mathcal{U}_{i_1}.\] The composition order and selection rule are fixed as a protocol and may not be arbitrarily swapped after seeing results.

5.2.3 3.1.3 [A] Fixing the VP axiom set (minimal assumptions)

5.2.4 [A-1] Infinite rigidity (Stone): volume invariance and non-overlap

The infinite-rigidity axiom is fixed by the following two statements.

  1. Volume invariance: the internal volume of every VP does not change. That is, in any admissible configuration the “volume measure” of each occupied region \(\Omega_i\) is preserved. This holds regardless of configuration changes, updates, or driving.

  2. Non-overlap: in every admissible configuration, VP occupied regions do not overlap ([D-3]). Non-overlap is not relaxed after seeing results; any configuration that violates it is judged inadmissible and cannot be used as an input for derivation/verification.

This axiom does not introduce “rigidity” as a dynamical law. Infinite rigidity is a constraint axiom that restricts the admissible configuration set; no later section interprets it as “tuning a rigidity value.”

5.2.5 [A-2] Full Packing: banning empty space as an independent degree of freedom

The full-packing axiom is fixed as follows.

  1. Inside the domain \(\mathcal{D}\), space is fully occupied by VPs ([D-5]). That is, we do not add “unoccupied space” as an independent object (or an independent field).

  2. Any structure that appears inside the domain, such as a gap, a deficit, or a throat, is defined not as “empty space itself” but only as a structural quantity derived from VP configuration and adjacency.

Therefore, the full-packing axiom forbids a mixed world of the form “space = VP + (additional medium).” If an additional medium is introduced, the meaning of full packing changes, hence it is not allowed within the same version.

5.2.6 [A-3] Local Rule: neighborhood-dependent updates

The local-rule axiom is fixed as follows.

  1. Locality: every change (rearrangement, relaxation, driving, transport) is expressed as a composition of local updates that depend only on the local neighborhood \(\mathcal{N}(i)\) ([D-8]).

  2. Admissibility preservation: a local update \(\mathcal{U}_i\) may not generate a configuration that violates [A-1] and [A-2]. That is, updates are defined only in a way that preserves non-overlap and full packing.

  3. Pre-registration of rules: the selection rule for local updates (which \(i\) is chosen), composition order, repetition condition, and termination condition are fixed as a protocol and may not be changed after seeing outcomes.

The local-rule axiom forbids a “one-shot global rule.” If a one-shot global rule is introduced, what counts as local information and what counts as global information can be retroactively adjusted, which structurally violates No-Tuning.

5.2.7 3.1.4 [D] Immediate consequences (admissible/inadmissible verdicts)

The axioms in this section immediately provide the following verdict criteria.

  1. Violating [A-1] (volume invariance or non-overlap) yields an inadmissible configuration.

  2. Violating [A-2] (adding empty space as an independent degree of freedom, or treating gap/throat/deficit as an independent object) yields an inadmissible description.

  3. Violating [A-3] (a global update that cannot be described by neighborhood-based local updates, or post-hoc modification of rules) yields an inadmissible procedure.

These verdicts are not softened by “interpretation.” The verdict is determined automatically by definitions and axioms; changing the meaning of axioms in order to dispute a verdict is forbidden within the same version.

5.2.8 3.1.5 Examples of prohibited extra assumptions (brief)

The following are examples of additional assumptions that are easy to insert on top of the minimal axiom set, but are prohibited within the same version. The list is only to clarify what is prohibited; it does not introduce new axioms.

  1. A universal interaction assumed as a function of distance only: adding a universal function \(f(d_{ij})\) for every VP pair \((i,j)\) and using it as the ground for all later derivations.

  2. Introducing a global continuous field: introducing a continuous field (scalar/vector, etc.) defined over the whole domain as a primary object, then treating the VP configuration as a secondary consequence.

  3. Axiomatizing equilibrium/optimization goals: adding a global principle of the form “a global objective is always minimized/maximized” and retroactively constructing local rules to fit that goal.

  4. Axiomatizing a probability distribution: fixing a specific distribution (e.g., a particular noise model or randomness) as a primary axiom for the initial condition or update process and justifying results as a consequence of that distribution.

  5. Automatically axiomatizing isotropy/homogeneity: adding “same in all directions” or “same at all positions” as an axiom and erasing anisotropy/defect structures that appear in actual configurations/adjacency.

  6. Implicitly axiomatizing cell geometry: promoting the choice of a particular cell geometry (cube/sphere, etc.) into an “axiom” without separately recording its influence on derived results.

If any of the above must be adopted, it cannot be silently merged into the axiom set. It must be (i) explicitly written as a definition or closure, (ii) scope-locked, and (iii) judged by Gates. “Quiet insertion” within the same version is prohibited.

5.3 3.2 Jamming lattice and Point-J

5.3.1 3.2.1 Fluidity as Failure Rate

In this theory, since VPs have infinite rigidity (Stone), we do not add “softness” or “viscosity” as new axioms. Observed “fluidity” is defined not as an intrinsic material property but as the statistics of events—collapse and recovery of the stiffness network (the jamming lattice). Thus we do not declare fluid/solid as separate phases (additional axioms); instead, on the same VP axioms we judge them via a stiffness failure rate.

5.3.1.1 Failure rate

Lock a protocol \(\mathcal{P}\) and an observation window \(W=[t_0,t_0+\Delta t]\) (LOCK). Let \(N_{\mathrm{total}}(W)\) be the total number of update/observation steps in \(W\), and at each step \(n=1,\ldots,N_{\mathrm{total}}(W)\) construct a jamming lattice \(\mathfrak{J}_n\). Using the stiffness indicator \(\chi_{\mathrm{ST}}(\mathfrak{J}_n)\) defined in 3.2.3.3, define the total number of “non-stiff (unjamming) events” by \[N_{\mathrm{unjam}}(W) :=\sum_{n=1}^{N_{\mathrm{total}}(W)} \mathbf{1}\!\left[\chi_{\mathrm{ST}}\!\left(\mathfrak{J}_n\right)=0\right] \label{eq:unjam_count_by_chiST}\] (the event index \(n\) and the construction convention for \(\mathfrak{J}_n\) must be part of \(\mathcal{P}\)). The fluidity index is then defined as \[\phi(\mathcal{P};W):=\frac{N_{\mathrm{unjam}}(W)}{N_{\mathrm{total}}(W)}\in[0,1] \label{eq:fluidity_phi_def}\]

[LOCK] The reporting schema for this definition of \(\phi\)—protocol / time window / judgement function—is sealed in 04_vp_whitepaper/LOCK/fluidity_phi_lock.json.

5.3.1.2 Interpretation: “it resists when struck, but flows when pushed”

\(\phi\) depends on the observation window and the driving rate. For a short observation time scale \(\tau_{\rm obs}\) as in high-speed impact, rearrangements are suppressed (Deborah-like [eq:deborah_like] gives \(\mathrm{De}\gg 1\)), \(\chi_{\mathrm{ST}}=1\) persists within \(W\), and a “solid-like” response with \(\phi\approx 0\) appears. Conversely, under slow driving, rearrangement/slip events accumulate (\(\mathrm{De}\ll 1\)), events with \(\chi_{\mathrm{ST}}=0\) repeat, \(\phi\) increases, and a “fluid-like” response appears. Moreover, under a large load, if the yield condition [eq:unjamming_trigger] is satisfied, \(\phi\) can surge, producing an unjamming (channel-opening) transition.

5.3.1.3 Reader map (NON-LOCK)

For convenience, a compact mapping between this failure-rate definition and the familiar solid/liquid/gas language is provided in Appendix L. Appendix L is explicitly interpretive (NON-LOCK): it introduces no new axioms and is never used as an input to any locked numerical derivation.

5.3.1.4 Purpose and premises

This section externalizes a VP configuration into an adjacency structure, distinguishes within a domain the regimes in which “global transmission (connectivity)” holds or fails, and defines their boundary as Point-J. The “stiff/non-stiff” used here is not grounded on external continuum notions such as elastic moduli. In this section, stiff/non-stiff is a structural regime defined only by (i) an adjacency graph, (ii) a set of domain boundaries, and (iii) pre-registered judgement conventions.

5.3.1.5 Reader note (4-3-1 dictionary)

For a compact state-language mapping (solid–liquid–gas vs. jammed–flowing–unjammed) tied to \(\phi\) and \(\chi_{\mathrm{ST}}\), see Appendix L.

5.3.2 3.2.2 Definition of the jamming lattice

5.3.2.1 3.2.2.1 Domain and boundary sets

Define the domain \(\mathcal{D}\) as a finite region composed of one cell or a union of cells. Inside the domain, designate “two opposing boundary sets.” \[\partial\mathcal{D}^{-},\ \partial\mathcal{D}^{+}\subset \partial\mathcal{D}, \qquad \partial\mathcal{D}^{-}\cap \partial\mathcal{D}^{+}=\varnothing. \label{eq:domain_boundaries}\] \(\partial\mathcal{D}^{-}\) and \(\partial\mathcal{D}^{+}\) are two subsets selected on the domain boundary \(\partial\mathcal{D}\) and are used only as a criterion to judge whether global transmission holds. The selection rule for the boundary sets (e.g., which face is taken as \(\partial\mathcal{D}^{-}\)) is fixed as a procedure locked in analysis_lock.

5.3.2.2 3.2.2.2 Contact predicate and contact graph

Let the VP set be \(\mathcal{V}\) and each VP be indexed by \(i\in\mathcal{V}\). Define a binary contact predicate \(C(i,j)\) for two VPs \(i,j\) by \[C(i,j)\in\{0,1\}, \qquad C(i,j)=1\ \Longleftrightarrow\ i \text{ and } j \text{ satisfy a pre-registered contact convention}. \label{eq:contact_predicate}\] The contact convention can be defined as (i) distance-based, (ii) surface-based, (iii) persistent-contact-after-relaxation-based, etc., but regardless of the choice, the convention itself must be locked as a definition and cannot be replaced after seeing results.

Define the contact graph (contact network) \(\mathcal{G}_c\) by \[\mathcal{G}_c := (\mathcal{V}, \mathcal{E}_c), \qquad \mathcal{E}_c := \{(i,j)\ |\ i\neq j,\ C(i,j)=1\}. \label{eq:contact_graph}\] The contact graph is a discrete externalization of the configuration; regime judgement below is performed based on \(\mathcal{G}_c\).

5.3.2.3 3.2.2.3 Node sets touching the domain boundaries

Define the VP node sets that touch the boundary sets \(\partial\mathcal{D}^{-}, \partial\mathcal{D}^{+}\) by \[\mathcal{V}^{-} := \{\, i\in\mathcal{V}\ |\ i \text{ touches } \partial\mathcal{D}^{-}\,\}, \qquad \mathcal{V}^{+} := \{\, i\in\mathcal{V}\ |\ i \text{ touches } \partial\mathcal{D}^{+}\,\}. \label{eq:boundary_nodes}\] The criterion for “touches” (e.g., whether the occupied region crosses the boundary, or whether the VP is within a threshold distance from the boundary) is fixed as a convention locked in analysis_lock. \(\mathcal{V}^{-}\) and \(\mathcal{V}^{+}\) are used as the source/target node sets for global transmission.

5.3.2.4 3.2.2.4 Jamming lattice

In this section the “jamming lattice” is defined as the bundle of the following four elements. \[\mathfrak{J} := \bigl(\mathcal{D},\ \mathcal{G}_c,\ \mathcal{V}^{-},\ \mathcal{V}^{+}\bigr). \label{eq:jamming_lattice}\] A jamming lattice \(\mathfrak{J}\) simultaneously includes (i) the domain, (ii) the contact graph, and (iii) the node sets touching the opposing boundaries. The stiff/non-stiff regimes and Point-J are defined only on \(\mathfrak{J}\).

5.3.3 3.2.3 Definition of stiff / non-stiff regimes

5.3.3.1 3.2.3.1 Global transmission and “spanning”

If there exists a path in the contact graph \(\mathcal{G}_c\) from some node in \(\mathcal{V}^{-}\) to some node in \(\mathcal{V}^{+}\), we define that “global transmission (spanning connection)” holds. Define the indicator by \[\chi_{\mathrm{span}}(\mathfrak{J}) := \begin{cases} 1, & \exists\ i\in\mathcal{V}^{-},\ \exists\ j\in\mathcal{V}^{+}\ \text{s.t.}\ i\leadsto j\ \text{in }\mathcal{G}_c,\\ 0, & \text{otherwise}. \end{cases} \label{eq:chi_span}\] Here \(i\leadsto j\) means that a path from \(i\) to \(j\) exists in \(\mathcal{G}_c\).

5.3.3.2 3.2.3.2 Bottleneck sensitivity (single cut) and the stiffness backbone

Spanning alone (\(\chi_{\mathrm{span}}=1\)) does not define “stiffness.” If spanning is maintained by only a single edge or a single node (a chain-like connection), it can collapse immediately by a local defect, and in this section we do not treat it as stiffness. Therefore we define a “bottleneck sensitivity” measure. Define the min-cut size between the two boundary node sets \(\mathcal{V}^{-},\mathcal{V}^{+}\) by \[\kappa_{\min}(\mathfrak{J}) := \min\bigl\{\, |\mathcal{C}|\ \big|\ \mathcal{C}\subseteq \mathcal{E}_c,\ \text{$\mathcal{V}^{-}$ and $\mathcal{V}^{+}$ are separated in }\mathcal{E}_c\setminus\mathcal{C}\,\bigr\}. \label{eq:kappa_min}\] \(\kappa_{\min}\) is a structural indicator of “how many edges must be cut to break global transmission.” The algorithm used to compute \(\kappa_{\min}\) (exact/approximate, weighted/unweighted, etc.) is locked in analysis_lock.

Define the stiffness backbone \(\mathcal{B}\) as a subgraph of the contact graph that satisfies the following conditions. \[\mathcal{B} := (\mathcal{V}_B, \mathcal{E}_B)\subseteq (\mathcal{V},\mathcal{E}_c) \label{eq:backbone_def}\]

  1. (Spanning) Within \(\mathcal{B}\), the sets \(\mathcal{V}^{-}\cap\mathcal{V}_B\) and \(\mathcal{V}^{+}\cap\mathcal{V}_B\) are connected by a path.

  2. (Bottleneck lower bound) The min-cut size defined on \(\mathcal{B}\) satisfies a pre-registered integer lower bound \(\kappa_{\mathrm{ST}}\): \[\kappa_{\min}(\mathcal{B}) \ge \kappa_{\mathrm{ST}}. \label{eq:kappa_threshold}\]

  3. (Maximality) Among the subgraphs satisfying the above two conditions, choose \(\mathcal{B}\) as the one that satisfies a pre-registered “maximality convention” (e.g., maximum number of edges, maximum number of nodes, or maximum score).

\(\kappa_{\mathrm{ST}}\) and the maximality convention are locked in analysis_lock. This section fixes only the definition and does not force a particular value (e.g., \(\kappa_{\mathrm{ST}}=2\)) as an axiom.

5.3.3.3 3.2.3.3 Definition of the stiff regime

Define the stiff-regime indicator by \[\chi_{\mathrm{ST}}(\mathfrak{J}) := \begin{cases} 1, & \exists\ \mathcal{B}\ \text{s.t. }\mathcal{B} \text{ satisfies \eqref{eq:backbone_def}--\eqref{eq:kappa_threshold}},\\ 0, & \text{otherwise}. \end{cases} \label{eq:chi_ST}\] That is, the stiff regime is defined as the set of jamming lattices with \(\chi_{\mathrm{ST}}(\mathfrak{J})=1\). The essence of the stiff regime is not only “existence of global transmission” but also “existence of a backbone that satisfies a bottleneck lower bound.”

5.3.3.4 3.2.3.4 Definition of the non-stiff regime

Define the non-stiff regime as the complement of the stiff regime: \[\chi_{\mathrm{ST}}(\mathfrak{J})=0 \quad\Longleftrightarrow\quad \text{non-stiff regime}. \label{eq:nonstiff_def}\] The non-stiff regime includes the following two cases.

  1. (Non-spanning) \(\chi_{\mathrm{span}}(\mathfrak{J})=0\): no global transmission path exists.

  2. (Bottleneck collapse) \(\chi_{\mathrm{span}}(\mathfrak{J})=1\) but no subgraph satisfies the bottleneck lower bound [eq:kappa_threshold]: global transmission exists but is structurally fragile, hence a “stiffness backbone” does not hold.

Therefore, the non-stiff regime is fixed to include both “no connection at all” and “a connection exists but without a bottleneck lower bound.”

5.3.4 3.2.4 Switch observables (definitions only)

In this section, “switch observables” are observables (or computables) defined to judge transitions between stiff and non-stiff regimes. We define three types.

5.3.4.1 3.2.4.1 Primary switch: regime indicator

The primary switch observable is the stiff-regime indicator \(\chi_{\mathrm{ST}}\) (Eq. [eq:chi_ST]). If \(\chi_{\mathrm{ST}}=0\) we classify as non-stiff; if \(\chi_{\mathrm{ST}}=1\) we classify as stiff. This classification is a definition and is not softened by numerical approximation or interpretation.

5.3.4.2 3.2.4.2 Secondary switch: min-cut size

The secondary switch observable is the min-cut size \(\kappa_{\min}\) (Eq. [eq:kappa_min]). \(\kappa_{\min}\) is an integer; a larger value means that boundary-to-boundary transmission is less sensitive to removing a single edge. This section does not add an extra physical interpretation of \(\kappa_{\min}\) and keeps it as a purely structural indicator.

5.3.4.3 3.2.4.3 Tertiary switch: backbone existence and size

The tertiary switch observable is the existence and size of the stiffness backbone \(\mathcal{B}\). \[N_B := |\mathcal{V}_B|, \qquad E_B := |\mathcal{E}_B|. \label{eq:backbone_size}\] Only when the backbone selection convention is locked do \(N_B\) and \(E_B\) become comparable observables. If the convention changes, the backbone can change even for the same configuration, hence the selection convention must be locked in analysis_lock.

5.3.5 3.2.5 Definition of Point-J

Define Point-J as “the boundary at which the stiffness switch transitions when a control parameter is varied.” To do so, define a control parameter \(u\) and a family of configurations \(\{\mathfrak{J}(u)\}\).

5.3.5.1 3.2.5.1 Control parameter \(u\) (example set)

The control parameter \(u\) is defined as an internal indicator that changes the structure of the jamming lattice \(\mathfrak{J}\) monotonically. Examples of admissible control parameters in this document are as follows (choose one or combine).

  1. Contact-density indicator: \[\bar{z} := \frac{2|\mathcal{E}_c|}{|\mathcal{V}|}, \label{eq:mean_degree}\] i.e., the mean degree (mean number of contacts).

  2. Critical-throat-based indicator: if throat objects are defined, use a throat-related indicator (e.g., a representative throat thickness, number of throats, bottleneck-chain indicator, etc.) as \(u\). The throat definition and estimator must be pre-registered in analysis_lock.

  3. Occupancy-based indicator: if the cell definition is locked, use a monotone indicator obtained from occupancy aggregation (counting) inside the domain as \(u\). The occupancy indicator definition (what is counted and which cell geometry is used) must be locked in canon_lock and analysis_lock.

This section does not force a specific choice of \(u\) as an axiom. The chosen \(u\) must be locked in analysis_lock and cannot be replaced after seeing results.

5.3.5.2 3.2.5.2 Point-J (transition point)

Given a family of jamming lattices \(\{\mathfrak{J}(u)\}\) parameterized by \(u\), define Point-J by \[u_J := \inf\{\, u\ |\ \chi_{\mathrm{ST}}(\mathfrak{J}(u))=1\,\}. \label{eq:pointJ_def}\] That is, we define the location of Point-J as the smallest \(u\) at which \(\chi_{\mathrm{ST}}\) transitions from \(0\) to \(1\). If in practice \(u\) is given only on a discrete sample grid (e.g., \(u_1<u_2<\cdots<u_K\)), Point-J is locked by the following discrete definition. \[u_J := u_{k^\ast}, \qquad k^\ast := \min\{\, k\ |\ \chi_{\mathrm{ST}}(\mathfrak{J}(u_k))=1\,\}. \label{eq:pointJ_discrete}\] How to judge “the first transition” (e.g., whether it is stably \(1\) over repeated runs, or exceeds a certain fraction, etc.) must be locked as a Gate convention in analysis_lock and may not be modified after seeing results.

5.4 3.3 Definition of the Quantum Cell (Anchor Cell)

5.4.1 3.3.1 Status of the Anchor Cell (canonical domain)

The Anchor Cell is the minimal domain fixed for aggregation and coordinatization of a VP configuration. The Anchor Cell definition holds only when the following three items are simultaneously locked (LOCK).

  1. Identification of the cell object (OBJ-CELL) and the cell geometry type (CELL-CUBE or CELL-SPHERE-VIS).

  2. The meaning of the representative cell length symbol \(D_{\mathrm{anch}}\) (diameter/radius/edge, etc.) and its unit dimension (length).

  3. The extent of the internal coordinate system (domain boundary) and inclusion/exclusion rules (including counting rules).

In this section, the canonical cell geometry is fixed as CELL-CUBE, and CELL-SPHERE-VIS is allowed only as a visualization mapping. A visualization mapping may not directly participate in a canonical derivation; to make it participate, a separate LOCK (version-up) and Gate judgement are required.

5.4.2 3.3.2 CELL-CUBE

Fix the geometry type of the canonical Anchor Cell as \[\mathrm{cell\_geometry} := \texttt{CELL-CUBE}. \label{eq:cell_geom_cube}\] Define the representative length \(D_{\mathrm{anch}}\) of the canonical Anchor Cell as the edge length of CELL-CUBE. That is, \[\mathrm{geometry\_meaning}(D_{\mathrm{anch}}) := \texttt{edge}. \label{eq:Danch_edge_lock}\] Define the canonical Anchor Cell domain \(\mathcal{D}_{\square}\) by \[\mathcal{D}_{\square} := \left\{\mathbf{x}\in\mathbb{R}^3\ \big|\ 0\le x< D_{\mathrm{anch}},\ 0\le y< D_{\mathrm{anch}},\ 0\le z< D_{\mathrm{anch}}\right\}. \label{eq:anchor_cube_domain}\] The canonical cell volume is fixed by the geometric definition \[V_{\square} := |\mathcal{D}_{\square}| = D_{\mathrm{anch}}^{3} \label{eq:Vcube}\] \(V_{\square}\) is a derived quantity of the canonical definition; if the meaning of \(D_{\mathrm{anch}}\) is not locked, \(V_{\square}\) cannot be used.

5.4.3 3.3.3 Fixing \(r_0 := D_{\mathrm{anch}}/2\) (half-length scale)

From the representative canonical cell length, fix the half-length scale \(r_0\) as the following derived definition. \[r_0 := \frac{D_{\mathrm{anch}}}{2}. \label{eq:r0_def}\] The meaning of \(r_0\) is locked as follows.

  1. \(r_0\) is half of the edge length of the canonical Anchor Cell; it does not automatically carry the geometric meaning of a radius.

  2. \(r_0\) is locked as a length-dimension quantity (L), and its unit notation is locked in the same family as the unit of \(D_{\mathrm{anch}}\).

  3. \(r_0\) is a derived length scale that can be freely used in canonical derivations, but it is never used as a substitute that changes the meaning of \(D_{\mathrm{anch}}\) (edge length).

Therefore, statements like “\(r_0\) is the radius of a sphere” hold only when a separate visualization mapping is locked (see 3.3.4 below). Using \(r_0\) as a sphere radius without such a lock is a definition conflict and is immediate FAIL.

5.4.4 3.3.4 CELL-SPHERE-VIS

CELL-SPHERE-VIS is a mapping rule that associates the canonical cell (CELL-CUBE) with a spherical domain only for visualization. The visualization cell does not replace the canonical cell; in canonical derivations the cell is always the cube domain of [eq:cell_geom_cube].

5.4.4.1 3.3.4.1 Locking the visualization mode (discrete choice)

Since the visualization mapping cannot be adjusted after seeing results, the visualization mode must be chosen as one element of the following discrete set and locked in analysis_lock. \[\mathrm{vis\_mode}\in\left\{\texttt{VIS-EQUAL-DIAMETER},\ \texttt{VIS-EQUAL-VOLUME}\right\}. \label{eq:vis_mode}\] The two modes generate different spherical domains and cannot be mixed within the same section/output.

5.4.4.2 3.3.4.2 Standard definition of the visualization sphere domain

Define the visualization sphere domain by \[\mathcal{D}_{\circ}(r_{\mathrm{vis}}) := \left\{\mathbf{x}\in\mathbb{R}^3\ \big|\ \|\mathbf{x}-\mathbf{x}_c\| < r_{\mathrm{vis}}\right\}, \label{eq:anchor_sphere_domain}\] where \(\mathbf{x}_c\) is the visualization center (geometric center), and \(r_{\mathrm{vis}}\) is the radius of the visualization sphere. The choice of \(\mathbf{x}_c\) is only a choice of origin for visualization and does not affect canonical derivations. The visualization radius \(r_{\mathrm{vis}}\) is defined only by the mapping rules below.

5.4.4.3 3.3.4.3 VIS-EQUAL-DIAMETER

The equal-diameter mode maps the canonical representative length \(D_{\mathrm{anch}}\) to the diameter of the visualization sphere. In this mode, lock \[D_{\mathrm{vis}} := D_{\mathrm{anch}}, \qquad r_{\mathrm{vis}} := \frac{D_{\mathrm{vis}}}{2}. \label{eq:vis_equal_diameter}\] From [eq:vis_equal_diameter] it follows immediately that \[r_{\mathrm{vis}} = \frac{D_{\mathrm{anch}}}{2} = r_0 \label{eq:rvis_equals_r0}\] but this equality is permitted only when the visualization mode is locked to VIS-EQUAL-DIAMETER. Identifying \(r_0\) with a sphere radius when this mode is not locked is forbidden.

5.4.4.4 3.3.4.4 VIS-EQUAL-VOLUME

The equal-volume mode maps the spherical visualization domain such that its volume equals the volume of the canonical cube cell. In this mode, lock \[V_{\circ} := |\mathcal{D}_{\circ}(r_{\mathrm{vis}})| = \frac{4}{3}\pi r_{\mathrm{vis}}^{3}, \qquad V_{\circ} \equiv V_{\square} = D_{\mathrm{anch}}^{3}. \label{eq:vis_equal_volume_condition}\] Therefore the visualization radius is fixed as \[r_{\mathrm{vis}} := \left(\frac{3}{4\pi}\right)^{\!1/3} D_{\mathrm{anch}}. \label{eq:rvis_equal_volume}\] In this mode, \(r_0=D_{\mathrm{anch}}/2\) remains a canonical half-length scale and cannot be identified with \(r_{\mathrm{vis}}\). That is, \[r_{\mathrm{vis}} \neq r_0 \quad\text{(generically; identification forbidden)}. \label{eq:rvis_not_r0_rule}\] If one needs the sphere diameter in the equal-volume mode, introduce a separate derived symbol \[D_{\mathrm{vis}} := 2 r_{\mathrm{vis}} \label{eq:Dvis_from_rvis}\] and use it. In that case \(D_{\mathrm{vis}}\) is a different symbol from \(D_{\mathrm{anch}}\) and may not be conflated as the same name/symbol.

5.4.5 3.3.5 Inverse mapping rule (visualization sphere \(\rightarrow\) canonical cube)

The inverse mapping that reconstructs the canonical representative cell length from the visualization domain must be uniquely determined by the visualization mode. Fix the inverse mapping as follows.

5.4.6 (A) Inverse of VIS-EQUAL-DIAMETER

\[D_{\mathrm{anch}} := D_{\mathrm{vis}}, \qquad r_0 := \frac{D_{\mathrm{anch}}}{2}. \label{eq:inv_equal_diameter}\]

5.4.7 (B) Inverse of VIS-EQUAL-VOLUME

\[D_{\mathrm{anch}} := \left(\frac{4\pi}{3}\right)^{\!1/3} r_{\mathrm{vis}}, \qquad r_0 := \frac{D_{\mathrm{anch}}}{2}. \label{eq:inv_equal_volume}\] The inverse mapping is used only for labeling visualization outputs and axis annotations in figures. Using the inverse mapping to retroactively tune or redefine the canonical input (\(D_{\mathrm{anch}}\)) is forbidden.

5.4.8 3.3.6 Confusion-prohibition rules (immediate FAIL)

For the definitions in this section to hold, the following confusion-prohibition rules must always be satisfied; any violation is immediate FAIL.

  1. No symbol-meaning conflict: \(D_{\mathrm{anch}}\) must be locked as edge in CELL-CUBE and cannot be used as a diameter or radius in the same context.

  2. No mode mixing: within the same output (same lock_id combination), do not use VIS-EQUAL-DIAMETER and VIS-EQUAL-VOLUME simultaneously or interchange them.

  3. Condition for identifying \(r_0\): the equality [eq:rvis_equals_r0] that identifies \(r_0\) with a sphere radius is permitted only when VIS-EQUAL-DIAMETER is locked. Using \(r_0=r_{\mathrm{vis}}\) in any other mode is immediate FAIL.

  4. No implicit conversion: cube\(\leftrightarrow\)sphere conversion must be performed only via the explicit derived symbols in [eq:vis_equal_diameter] or [eq:vis_equal_volume_condition]–[eq:rvis_equal_volume]. Changing the meaning of a symbol to perform an implicit conversion is immediate FAIL.

5.5 3.4 Handling of \(\ell_{\mathrm{rot}}\) (reference value)

5.5.1 3.4.1 Definition of \(\ell_{\mathrm{rot}}\) and required lock fields

Define \(\ell_{\mathrm{rot}}\) as a length scale introduced by a rotational driving (or rotational selection) protocol. Before its numerical value, the following semantic fields must be locked.

  1. Object attribution (object_id): fix to exactly one object whose length \(\ell_{\mathrm{rot}}\) represents (e.g., one of OBJ-CELL, OBJ-CORE, OBJ-THROAT).

  2. Geometric meaning (geometry_meaning): lock \(\ell_{\mathrm{rot}}\) as a diameter, i.e. \[\mathrm{geometry\_meaning}(\ell_{\mathrm{rot}}):=\texttt{diameter}.\] Reinterpretation as a radius is forbidden. If a radius is needed, introduce a derived symbol \[r_{\mathrm{rot}} := \frac{\ell_{\mathrm{rot}}}{2}\] and use it explicitly.

  3. Dimension/unit (dimension/unit): lock \(\ell_{\mathrm{rot}}\) as a length-dimension quantity (L); lock its unit notation to one of the registry-approved length units (e.g., pm).

  4. Protocol attribution (protocol_id): lock an identifier for which rotational driving protocol defined/extracted \(\ell_{\mathrm{rot}}\) (driving method, sampling, estimator, termination conditions). A value of \(\ell_{\mathrm{rot}}\) without protocol attribution is unusable.

  5. Scope (scope): \(\ell_{\mathrm{rot}}\) may be referenced only within the rotational-driving regime (or rotational-selection regime); out-of-scope references are forbidden.

If any field is missing, \(\ell_{\mathrm{rot}}\) is not well-defined; any equation/table/figure/log that uses \(\ell_{\mathrm{rot}}\) is judged immediate FAIL.

5.5.2 3.4.2 Current status: “reference value (CANON-REF)”

In this document, the current status of \(\ell_{\mathrm{rot}}\) is a reference value. That is, among canonical inputs we fix \[\ell_{\mathrm{rot}} \in \texttt{CANON-REF}. \label{eq:lrot_canon_ref}\] The meaning of CANON-REF is as follows.

  1. \(\ell_{\mathrm{rot}}\) is not automatically promoted to an input of the mandatory derivation chain.

  2. Using \(\ell_{\mathrm{rot}}\) to redefine the meaning or value of canonical inputs such as \(D_{\mathrm{anch}}\), \(r_p\), \(\delta\), \(\pi\) is forbidden.

  3. Using \(\ell_{\mathrm{rot}}\) to adjust or reinterpret realization inputs (\(a\), \(\Delta t\), \(c_{\mathrm{ref}}\)) is forbidden.

  4. Any conclusion that contains \(\ell_{\mathrm{rot}}\) exists only as an extended-regime conclusion; without passing the extended-regime Gates it has no conclusion status.

Therefore, in the current version \(\ell_{\mathrm{rot}}\) is a “possible input” but not a “required input” and does not form the evidential basis of the core chain (canonical \(\rightarrow\) events \(\rightarrow\) realization \(\rightarrow\) mass/force).

5.5.3 3.4.3 Rules for using a reference value (allowed vs forbidden)

Fix the usage rules of \(\ell_{\mathrm{rot}}\) in its current status (CANON-REF) as follows.

5.5.3.1 3.4.3.1 Allowed uses (only inside the extended regime)

The following are allowed, provided the scope is locked to the rotational-driving regime and the relevant Gates are passed.

  1. Constructing dimensionless ratios: combine \(\ell_{\mathrm{rot}}\) with other length scales to form dimensionless ratios (e.g., \(\ell_{\mathrm{rot}}/D_{\mathrm{anch}}\), \(\ell_{\mathrm{rot}}/r_p\)).

  2. Input for rotational-driving anisotropy indicators: use \(\ell_{\mathrm{rot}}\) as a protocol input in extended sections on direction distributions, fabric, throat-direction dependence, etc.

  3. Regime label for switch observables: near jamming/unjamming or bottleneck transitions, record \(\ell_{\mathrm{rot}}\) as a label of the “rotational-driving condition” (a label records conditions, not a conclusion).

5.5.3.2 3.4.3.2 Forbidden uses (forbidden in reference status)

The following are immediately forbidden; upon detection they are immediate FAIL.

  1. Replacing canonical inputs: replacing the meaning/value of \(D_{\mathrm{anch}}\) or \(r_p\) by \(\ell_{\mathrm{rot}}\), or redefining the meaning of \(D_{\mathrm{anch}}\) from \(\ell_{\mathrm{rot}}\).

  2. Tuning realization: using \(\ell_{\mathrm{rot}}\) to adjust the meaning or value of \(a\), \(\Delta t\), \(c_{\mathrm{ref}}\).

  3. Post-selection / post-correction: selecting among multiple candidate \(\ell_{\mathrm{rot}}\) values the one that favors a conclusion, or shifting estimators/thresholds to force \(\ell_{\mathrm{rot}}\) to a desired value.

  4. Semantic reinterpretation: using \(\ell_{\mathrm{rot}}\) as if it were a radius despite being locked as a diameter, or performing implicit cube–sphere conversion to change the geometric meaning of \(\ell_{\mathrm{rot}}\).

5.5.4 3.4.4 Adoption (promotion) procedure: reference value \(\rightarrow\) canonical input (CANON-PRIMARY)

To adopt \(\ell_{\mathrm{rot}}\) as a canonical input (an input of the mandatory chain), it is not permitted as an in-version edit but only as a LOCK version-up. The adoption is fixed by the following procedure.

5.5.4.1 3.4.4.1 Adoption declaration (explicit promotion type)

Adoption is a declaration that promotes the classification of \(\ell_{\mathrm{rot}}\) as \[\ell_{\mathrm{rot}}:\ \texttt{CANON-REF}\ \longrightarrow\ \texttt{CANON-PRIMARY}. \label{eq:lrot_promotion}\] CANON-PRIMARY denotes a canonical input that participates directly in the mandatory derivation chain. At the promotion point, one must explicitly specify

  1. object attribution (object_id) of \(\ell_{\mathrm{rot}}\),

  2. geometric meaning (diameter) of \(\ell_{\mathrm{rot}}\),

  3. value/unit/significant-figure convention of \(\ell_{\mathrm{rot}}\),

  4. scope of applicability (global vs a specific regime).

5.5.4.2 3.4.4.2 LOCK version-up (issuing a new lock_id)

Because promotion changes canon_lock, a new canon_lock_id must be issued. Since computational/judgement procedures that involve \(\ell_{\mathrm{rot}}\) can also change, a new analysis_lock_id may be required. The version-up includes the following.

  1. Fix the change_log: record the reason for promotion, the before/after classification, and the affected sections (dependency list).

  2. Update the symbol registry: update the semantic fields (object_id, geometry_meaning, scope, unit) of \(\ell_{\mathrm{rot}}\) and any derived symbols (\(r_{\mathrm{rot}}\), etc.) as a single source of truth.

  3. Update the dependency graph: mark all derived results that reference \(\ell_{\mathrm{rot}}\) as needing regeneration.

5.5.4.3 3.4.4.3 Full re-derivation and full re-judgement (re-verification)

Under the new lock_id combination, the following must be performed.

  1. Full re-derivation: regenerate from the beginning every derivation chain in which \(\ell_{\mathrm{rot}}\) participates as an input (including intermediate artifacts).

  2. Full re-judgement: re-run from the beginning the Gate stack required by the relevant conclusions and re-judge PASS/FAIL/INCONCLUSIVE.

  3. Seal: generate a new registry_snapshot, manifest, and checksums and freeze the new version.

After promotion, conclusions belong only to the new lock_id combination and cannot be mixed with pre-promotion conclusions.

5.5.5 3.4.5 Declaring scope of impact (before/after promotion)

The status of \(\ell_{\mathrm{rot}}\) directly determines its global impact scope in the document. We declare the scope as follows.

5.5.5.1 3.4.5.1 Current impact scope (reference value)

In the current status (CANON-REF), the impact scope of \(\ell_{\mathrm{rot}}\) is limited as follows.

  1. It is used only in the rotational-driving extended regime (anisotropy/direction distributions/fabric/throat direction dependence, etc.).

  2. It does not intervene in the core mandatory derivation chain (from canonical inputs \(D_{\mathrm{anch}}, r_p, \delta, \pi\) and realization inputs \(a, \Delta t, c_{\mathrm{ref}}\)).

  3. Any result containing \(\ell_{\mathrm{rot}}\) is labeled as an “extended conclusion” and has conclusion status only within the scope that passes the extended Gates.

5.5.5.2 3.4.5.2 Impact scope after adoption (promotion)

If promotion (CANON-PRIMARY) occurs, the impact scope expands as follows.

  1. \(\ell_{\mathrm{rot}}\) can participate in the mandatory derivation chain; in that case all conclusions that depend on \(\ell_{\mathrm{rot}}\) (among length/time/mass/force families in which \(\ell_{\mathrm{rot}}\) enters) must be regenerated and re-judged under the new lock_id.

  2. Depending on which object’s diameter \(\ell_{\mathrm{rot}}\) represents (cell/core/throat, etc.), it can directly affect the Anchor Cell definition, discrete structures (core/shell), event aggregation, and regime maps (stiff/non-stiff transitions). The impacted targets are fixed as dependency-graph entries in the change_log.

  3. Some actions remain forbidden even after promotion. Promotion does not automatically replace the meaning of \(D_{\mathrm{anch}}\) and \(r_p\); if replacement is required, it demands a separate canonical-input-structure change (another version-up) and Gates.

5.5.6 3.4.6 Immediate FAIL rules for confusion and violations

The following violations related to \(\ell_{\mathrm{rot}}\) are judged immediate FAIL.

  1. The geometry_meaning (diameter) of \(\ell_{\mathrm{rot}}\) is not locked, or \(\ell_{\mathrm{rot}}\) is used interchangeably as a radius.

  2. The object attribution (object_id) of \(\ell_{\mathrm{rot}}\) is missing, or \(\ell_{\mathrm{rot}}\) is used for multiple objects within the same context.

  3. \(\ell_{\mathrm{rot}}\) is used to post-hoc tune the meaning/value of canonical inputs or realization inputs.

  4. \(\ell_{\mathrm{rot}}\) is used as an input of the mandatory derivation chain without promotion (without a version-up).

  5. Even after promotion, results from different lock_id combinations (old/new versions) are mixed into a single conclusion.

6 4. Semantic layer mapping (1:1) and the closure system

Chapter declaration: the role of semantic layers and closures

This chapter fixes, as a single system, the document-wide conventions of “semantic layer mapping (1:1)” and “closure”. A semantic layer mapping is a convention that separates “what each symbol means” into explicit layers and relates the layers via 1:1 correspondence, while a closure is a convention that explicitly seals choices (decision rules) that are not determined by axioms and definitions alone. These conventions are a prerequisite for all subsequent derivations (stationary constants, events, unit realizations, mass, force, and extended regimes). Any derivation or verification that violates this chapter cannot claim the status of a valid conclusion.

Global principles for 1:1 semantic mapping

Semantic layer mapping is fixed by the following principles.

  1. 1 symbol–1 meaning–1 unit: the same symbol has exactly one meaning (object attribution + geometric meaning + admissible operation scope) and exactly one unit dimension across the entire document.

  2. Layer separation: a meaning defined in the “definition layer” does not automatically convert when it is used in the “observation layer”. Any conversion must be stated explicitly as a map.

  3. 1:1 correspondence: an item in one layer connects to an item in another layer only via 1:1 correspondence. If a 1:N or N:1 relation is needed, it must be decomposed into a new intermediate object and/or a new explicit conversion rule.

  4. Locking of maps: mapping rules must be locked in analysis_lock or canon_lock; they cannot be swapped after inspecting results.

Therefore, “meaning jumps” (reinterpreting a symbol without a definition), “unit jumps” (performing a dimensional conversion implicitly), and “object jumps” (retroactively reassigning an item to a different object) are prohibited.

The standard 4-layer architecture

In this document, the semantic layers are fixed to the following four layers. Each item is sealed by a registry entry, and inter-layer movement is permitted only via explicit maps.

  1. L0 (primitive layer): primitive objects fixed by axioms/definitions—VP, domain, contact, adjacency, cell, event, etc.

  2. L1 (structural/combinatorial layer): discrete structural items—contact graph, backbone, throat, path, integer structures (core/shell), cancellation conventions, etc.

  3. L2 (aggregated/observable layer): aggregated observables or computables—counts, distributions, indicators, thresholds, event rates, switch observables, cross-consistency scores, etc.

  4. L3 (realization/numeric layer): realized numeric results with units attached (length/time/mass/force, etc.) and the standard reporting format.

Items in L0–L3 are not mixed, and L0 definitions are never modified retroactively by L2 or L3 outputs.

6.1 4.4 Standard template for 1:1 semantic mapping

A semantic layer map is a registry entry that must follow the template below. This template must be included in analysis_lock; if it is missing, the corresponding map cannot be used.

semantic_maps:
  - map_id: MAP-L1toL2-THROAT-001
    from_layer: L1
    to_layer: L2
    from_item:
      object_id: OBJ-THROAT
      symbol: delta_gap
      meaning: (gap/thickness/critical throat, single meaning)
      unit_dimension: L (or 1)
    to_item:
      quantity_id: Q-THROAT-DELTAEFF
      symbol: delta_eff
      meaning: (definition of a representative aggregated throat value)
      unit_dimension: L (or 1)
    rule:
      definition: (aggregation rule: mean/median/min-cut-based, etc.)
      algorithm_id: ALG-THROAT-EST-001
      parameters_locked_by: analysis_lock_id
    scope: (regime identifier)
    failure_modes:
      - FM-SCOPE
      - FM-NONUNIQUE
      - FM-NUMERIC
    required_gates:
      - G-SYM
      - G-LOCK
      - G-REG
      - (optional) G-NUM

In this template, failure_modes and required_gates are mandatory fields. A map does not consist of a “rule” alone; its failure modes and verification conditions must also be locked.

6.2 4.5 Definition of closure

A closure is a device that seals, as an explicit rule, a choice that is not determined by axioms/definitions alone. A closure has the following properties.

  1. Explicit declaration of the choice: a closure states, in sentences and equations, “what is being chosen”. If no choice is needed, no closure should be introduced.

  2. Declaration of input/output: a closure declares the types of its inputs and outputs (layer, unit dimension, object attribution). If the I/O types are ambiguous, the closure cannot be used.

  3. Scope (regime): a closure can be valid only in a specific regime; that regime must be locked.

  4. Built-in failure modes: a closure includes a list of failure modes (undefinedness, non-uniqueness, numerical instability, regime violation, cross-consistency failure, etc.).

  5. Gate required: a closure cannot produce conclusions without a Gate judgement. Any conclusion involving a closure requires PASS of the Gate stack assigned to that closure.

6.3 4.6 Global principles for the closure DAG (dependency graph)

Closures may depend on other closures, but the dependency must be a DAG (a directed acyclic graph). The closure DAG is fixed by the following principles.

  1. One-way: closures have a topologically sortable direction; an output of a later closure cannot retroactively modify an input of an earlier closure.

  2. No cycles: cyclic dependencies in which a closure’s output determines its own input (directly or indirectly) are prohibited.

  3. SSOT: closure definitions exist only at a single location in analysis_lock. The main text never redefines closures.

  4. Conclusion attribution: every conclusion involving closures carries the list of used closures (closure_ids) and the DAG version (analysis_lock_id).

The purpose of the closure DAG is to prevent “necessary choices” from being hidden, and to make the impact of each choice traceable to conclusions.

6.4 4.7 Standard taxonomy of failure modes

Failure modes are standard labels that record “when this closure/map collapses”. Failure modes are not post-hoc remarks; they are pre-registered items, fixed by the following classification.

  1. FM-SYM: symbol/meaning/unit conflict or overloading.

  2. FM-SCOPE: regime/scope violation (used outside the declared scope).

  3. FM-NONUNIQUE: non-unique solution (multiple solutions without a selection rule).

  4. FM-NODEF: undefinedness (missing inputs, or the defining equation does not close).

  5. FM-NUMERIC: numerical instability (non-convergence, sensitivity blow-up, iteration inconsistency).

  6. FM-XCROSS: cross-consistency failure (mismatch across independent channels).

  7. FM-REP: reproducibility failure (cannot obtain the same judgement upon re-run).

  8. FM-NT: detection of a No-Tuning violation (post-hoc adjustment).

A failure mode does not mean “the result is unpleasant”; it means “the definition/procedure/judgement does not hold”. When a failure mode occurs, the corresponding conclusion is judged FAIL or INCONCLUSIVE and loses the status of a valid conclusion.

6.5 4.8 Why verification is required (closure/map conclusion criteria)

Since semantic maps and closures contain choices, their legitimacy is not granted by external authority but only by passing pre-registered Gates. Therefore, the following become global rules.

  1. A closure/map without Gates cannot generate a valid conclusion.

  2. A closure/map without defined failure modes cannot be used.

  3. A conclusion that does not record the closure DAG version loses its basis and cannot claim the status of a valid conclusion.

The global principles fixed in this chapter—“1:1 semantic mapping”, “closure DAG”, “failure modes”, and “Gate required”—are not re-explained in later chapters. Each later section refers only to the relevant items via its final LOCK/Gate link.

6.6 4.1 Meaning-layer mapping: pressure/flux/deficit/charge

6.6.1 4.1.1 Common premise: layers (L0–L3) and the minimal operational record format

All of pressure/flux/deficit/charge defined in this section follow the layered semantic architecture below.

  1. L0 (primitive): VP, domain (cell), contact/adjacency, local update (event), etc.

  2. L1 (structure): contact graph, boundary set, cut surface, throat/path, core/shell structure, etc.

  3. L2 (aggregated observables): counts/indices/rates/slopes (differential forms), including intermediate unitless quantities.

  4. L3 (realized numbers): final numbers with physical units attached via length \(a\), time \(\Delta t\), energy unit \(U_{\mathrm{lat}}\), etc.

Each quantity is defined only by a 1:1 mapping L1\(\rightarrow\)L2\(\rightarrow\)L3; any transformation that retroactively rewrites the meaning of another layer is prohibited.

In addition, the common operational record (log) used in this section must contain at least the following fields. This record schema is locked in analysis_lock; if any field is missing, the corresponding quantity is judged non-computable.

  1. Cell domain: lock \(\mathcal{D}_{\square}\) and the representative length \(D_{\mathrm{anch}}\) (cell geometry=CELL-CUBE, meaning of \(D_{\mathrm{anch}}\)=edge).

  2. Contact graph: lock \(\mathcal{G}_c=(\mathcal{V},\mathcal{E}_c)\) and the contact predicate \(C(i,j)\in\{0,1\}\).

  3. Event log: lock the event set \(\mathcal{E}\), event index \(e\in\mathcal{E}\), event time (tick) \(n(e)\in\mathbb{Z}\), pre/post configuration identifiers, and the participating VP subset \(\mathcal{V}(e)\subseteq\mathcal{V}\).

  4. Definition of cut surface/boundary: lock the geometric definition of boundary sets \(\partial\mathcal{D}^{\pm}\) or a cut surface \(\Sigma\) (plane/curved surface, orientation normal included).

6.6.2 4.1.2 Standard symbols (type/dimension/unit) and the 1:1 operational definition table

The standard entries of pressure/flux/deficit/charge are locked by the table below. Each row represents one item; an item has meaning only as the tuple (symbol, type, dimension, unit, operational procedure).

Name Symbol Type Dimension/Unit 1:1 operational procedure (summary)
pressure \(P\) scalar \([E L^{-3}]\), \(U_{\mathrm{lat}}/a^{3}\) boundary compression protocol \(\rightarrow\) minimal relaxation cost \(W(\epsilon)\) \(\rightarrow\) \(P:=\lim_{\epsilon\to0^+}W(\epsilon)/\Delta V(\epsilon)\)
flux \(J\) scalar or vector (if direction included) \([L T^{-1}]\) or \([\text{token}\,A^{-1}T^{-1}]\) cut surface \(\Sigma\) and event crossing sign \(s_\Sigma(e)\) \(\rightarrow\) net crossing count \(\Delta N_\Sigma\) \(\rightarrow\) \(J:=a^{3}\Delta N_\Sigma/(A_\Sigma\Delta T)\)
deficit \(\mathcal{D}_{\mathrm{def}}\) scalar \([1]\) (default), optionally \([L^{-3}]\) contact degree \(z_i\) \(\rightarrow\) reference degree \(z_{\mathrm{ref}}\) \(\rightarrow\) \(d_i:=\max(0,z_{\mathrm{ref}}-z_i)\) \(\rightarrow\) \(\mathcal{D}_{\mathrm{def}}:=\frac{1}{|\mathcal{V}|}\sum_i d_i\)
charge \(Q\) signed scalar \([Q]\) (independent dimension), unit \(q_0\) shell(7) cancellation convention \(\rightarrow\) survival vector \(\mathbf{V}\) \(\rightarrow\) sign indicator \(q:=\mathrm{sgn}(\mathbf{V}\cdot\mathbf{n}_Q)\) \(\rightarrow\) \(Q:=q_0 q\)

The “summary” in the table is not a shortened definition; it lists only the names of the procedures. The complete procedures are fixed step-by-step in the subsections below.

6.6.3 4.1.3 1:1 definition of pressure (boundary compression \(\rightarrow\) cost \(\rightarrow\) pressure)

6.6.3.1 4.1.3.1 L1 input (structure): boundary and compression operator

In the canonical cell domain \(\mathcal{D}_{\square}\), lock the compression direction as a unit normal \(\mathbf{n}_P\). For a cube cell, lock one of the following choices in analysis_lock. \[\mathbf{n}_P \in \{\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\}, \qquad \partial\mathcal{D}^{-},\partial\mathcal{D}^{+}\ \text{are locked as the two opposite faces orthogonal to }\mathbf{n}_P. \label{eq:pressure_normal_lock}\]

For a compression amount \(\epsilon>0\), define the “compressed domain” by \[\mathcal{D}_{\square}(\epsilon) := \left\{\mathbf{x}\in\mathcal{D}_{\square}\ \big|\ \mathbf{x}\cdot\mathbf{n}_P < D_{\mathrm{anch}}-\epsilon\right\} \quad\text{(move one of the opposite faces by $\epsilon$ to reduce the domain volume)}. \label{eq:compressed_domain}\] The domain volume reduction is \[\Delta V(\epsilon) := |\mathcal{D}_{\square}|-|\mathcal{D}_{\square}(\epsilon)| = A_{\square}\,\epsilon, \qquad A_{\square}:=D_{\mathrm{anch}}^{2}. \label{eq:deltaV_cube}\] Here \(A_{\square}\) is the cell face area perpendicular to the compression direction; it is usable only when the cell geometry (cube) and the meaning of \(D_{\mathrm{anch}}\) are locked.

Define the compression operator \(\mathcal{C}_\epsilon\) as follows. \(\mathcal{C}_\epsilon\) includes the boundary displacement and the induced mapping of configurations (coordinate transform). \[\mathcal{C}_\epsilon:\ \{\Omega_i\}_{i\in\mathcal{V}}\ \mapsto\ \{\Omega_i^{(\epsilon)}\}_{i\in\mathcal{V}}, \label{eq:compression_operator}\] However, there is no guarantee that \(\{\Omega_i^{(\epsilon)}\}\) satisfies the non-penetration / full-packing / local rules after applying \(\mathcal{C}_\epsilon\). Therefore the compression operator must always be paired with relaxation (4.1.3.2).

6.6.3.2 4.1.3.2 L1\(\rightarrow\)L2 map: minimal relaxation cost \(W(\epsilon)\)

Define the relaxation operator (composition of local updates) \(\mathcal{R}\) by \[\mathcal{R}:\ \{\Omega_i^{(\epsilon)}\}\ \mapsto\ \{\Omega_i^{(\epsilon,\mathrm{rel})}\}, \label{eq:relax_operator}\] where \(\{\Omega_i^{(\epsilon,\mathrm{rel})}\}\) must be an admissible configuration that satisfies non-penetration, full packing, and the local rules. Relaxation consists only of compositions of local updates (the local rule in 3.1). The relaxation rules (update selection, termination condition, failure condition) must be locked in analysis_lock.

The relaxation cost is defined only in pre-registered cost units. A cost unit consists of the two components below.

  1. Cost unit per one local update: lock as \(U_{\mathrm{lat}}\) (energy unit).

  2. Cost coefficient: lock a weight \(\omega_{\mathrm{upd}}\) per local-update type in analysis_lock (the default value may be locked as 1).

Define the minimal relaxation cost for compression \(\epsilon\) by \[W(\epsilon) := U_{\mathrm{lat}}\, \min_{\mathcal{R}} \left(\sum_{k=1}^{N_{\mathrm{upd}}(\epsilon)} \omega_{\mathrm{upd}}(k)\right), \label{eq:work_definition}\] where the minimization is performed under the constraint “return to an admissible configuration”. The minimization rule (search method, deduplication, stopping criterion) must be locked in analysis_lock. If relaxation fails and cannot return to an admissible configuration, then \(W(\epsilon)\) is undefined and is recorded as a failure mode.

6.6.3.3 4.1.3.3 L2\(\rightarrow\)L3 map: definition of pressure \(P\)

Pressure is defined by the limit \[P := \lim_{\epsilon\to 0^+}\frac{W(\epsilon)}{\Delta V(\epsilon)}. \label{eq:pressure_def}\] For a cube cell, using [eq:deltaV_cube] gives \[P = \lim_{\epsilon\to 0^+}\frac{W(\epsilon)}{A_{\square}\epsilon} = \lim_{\epsilon\to 0^+}\frac{W(\epsilon)}{D_{\mathrm{anch}}^{2}\epsilon}. \label{eq:pressure_cube}\] The dimension of \(P\) is locked as \([E L^{-3}]\), and the unit is locked as \(U_{\mathrm{lat}}/a^3\). Here \(a\) is the realized length scale and \(U_{\mathrm{lat}}\) is the realized energy unit; this unit system must be locked in realization_lock.

6.6.3.4 4.1.3.4 Failure modes for the pressure definition (undefinedness conditions)

The conditions under which the pressure definition fails (failure modes) are fixed as follows.

  1. FM-SYM: the meaning of \(D_{\mathrm{anch}}\) (cell edge) or the cell geometry (cube) is not locked.

  2. FM-NODEF: relaxation cannot return to an admissible configuration, so \(W(\epsilon)\) is undefined.

  3. FM-NONUNIQUE: the minimization rule is not locked, so the word “minimum” is not mechanically defined.

  4. FM-NUMERIC: \(W(\epsilon)/\epsilon\) does not converge stably over a sufficiently small \(\epsilon\) range (the convergence criterion itself is locked by Gate).

6.6.4 4.1.4 1:1 definition of flux (cut surface \(\Sigma\) \(\rightarrow\) crossing sign \(\rightarrow\) flux)

6.6.4.1 4.1.4.1 L1 input (structure): cut surface, orientation, measurement window

Inside the domain \(\mathcal{D}_{\square}\), define an oriented cut surface \(\Sigma\). The cut surface simultaneously locks the following three elements.

  1. geometric definition of the cut surface (plane/curved-surface equation or mesh),

  2. the unit normal \(\mathbf{n}_\Sigma\) (orientation lock),

  3. the area \(A_\Sigma\) (area-computation convention lock).

Define the measurement time window as the tick interval \([n_1,n_2)\), and lock the realized duration by \[\Delta T := (n_2-n_1)\Delta t \label{eq:deltaT}\] where \(\Delta t\) must be locked in realization_lock.

6.6.4.2 4.1.4.2 L1\(\rightarrow\)L2 map: event crossing sign \(s_\Sigma(e)\)

For the cut surface \(\Sigma\), define the “side indicator” function \(\sigma_\Sigma(i)\in\{-1,+1\}\). \[\sigma_\Sigma(i)= \begin{cases} +1, & \text{the representative point of $\Omega_i$ (by a locked representative-point convention) lies on the $+$ side of }\Sigma,\\ -1, & \text{the representative point of $\Omega_i$ lies on the $-$ side of }\Sigma. \end{cases} \label{eq:side_function}\] The representative-point convention (e.g., center of the occupied region, a selected marker point, etc.) must be locked in analysis_lock.

From the change of \(\sigma_\Sigma\) between the pre/post configurations of an event \(e\), define the event crossing sign by \[s_\Sigma(e) := \frac{1}{2}\sum_{i\in\mathcal{V}(e)} \Bigl(\sigma_\Sigma^{\mathrm{post}}(i)-\sigma_\Sigma^{\mathrm{pre}}(i)\Bigr). \label{eq:crossing_sign}\] By definition \(s_\Sigma(e)\in\mathbb{Z}\), and if an event does not cross the cut surface then \(s_\Sigma(e)=0\). If the event log does not contain pre/post configurations, then \(s_\Sigma(e)\) is undefined.

In the measurement window \([n_1,n_2)\), define the net crossing count by \[\Delta N_\Sigma := \sum_{e:\ n_1\le n(e)<n_2} s_\Sigma(e). \label{eq:deltaN}\] \(\Delta N_\Sigma\) is a pure L2 aggregate (unitless).

6.6.4.3 4.1.4.3 L2\(\rightarrow\)L3 map: flux (token flux) \(J\)

Flux is defined as a flow of “tokens”. The token size is locked as the VP unit volume \(a^3\). Therefore, define the flux by \[J := \frac{a^{3}}{A_\Sigma\,\Delta T}\,\Delta N_\Sigma. \label{eq:flux_def}\] In this definition, \(a\) is the realized length scale, \(\Delta T\) is defined by [eq:deltaT], and \(A_\Sigma\) is the realized area of the cut surface. The dimension of \(J\) is locked as \([L T^{-1}]\), because token volume (length\(^3\)) divided by area (length\(^2\)) and time yields a length/time scale.

If a direction-aware flux vector is needed, define \[\mathbf{J} := J\,\mathbf{n}_\Sigma. \label{eq:flux_vector}\] If \(\mathbf{n}_\Sigma\) is not locked, then \(\mathbf{J}\) is undefined.

6.6.4.4 4.1.4.4 Failure modes for the flux definition (undefinedness conditions)

The conditions under which the flux definition fails (failure modes) are fixed as follows.

  1. FM-NODEF: the event log lacks pre/post configurations, or \(\mathcal{V}(e)\) is missing so that \(s_\Sigma(e)\) is undefined.

  2. FM-SYM: the orientation of \(\Sigma\) or the area \(A_\Sigma\) is not locked.

  3. FM-SCOPE: the cut surface is not defined consistently with the domain \(\mathcal{D}_{\square}\) (e.g., defined outside the domain or mixing cell geometries).

  4. FM-NUMERIC: the representative-point convention is not locked, or the representative-point choice is unstable for the same event, so \(s_\Sigma(e)\) is not reproducible.

6.6.5 4.1.5 1:1 definition of deficit (contact deficit: an operational measure of structural deficit)

6.6.5.1 4.1.5.1 L1 input (structure): contact degree \(z_i\)

In the contact graph \(\mathcal{G}_c=(\mathcal{V},\mathcal{E}_c)\), define the contact degree (graph degree) of each VP \(i\) by \[z_i := |\mathcal{N}(i)|, \qquad \mathcal{N}(i):=\{\,j\in\mathcal{V}\ |\ (i,j)\in\mathcal{E}_c\,\}. \label{eq:degree_deficit}\] If the contact predicate \(C(i,j)\) is not locked, then \(z_i\) is undefined.

6.6.5.2 4.1.5.2 L1\(\rightarrow\)L2 map: reference degree \(z_{\mathrm{ref}}\) and deficit \(d_i\)

Deficit is defined as “shortfall relative to a reference contact level”. The reference degree \(z_{\mathrm{ref}}\) is locked in analysis_lock by the rules below.

  1. \(z_{\mathrm{ref}}\in\mathbb{Z}_{\ge 0}\).

  2. \(z_{\mathrm{ref}}\) may depend on the contact-predicate convention, dimension (2D/3D), and boundary-handling convention; such dependencies are locked as scope.

  3. The choice of \(z_{\mathrm{ref}}\) must not be changed by post-hoc correction; changes are allowed only by versioning.

Define the deficit of each VP by \[d_i := \max\bigl(0,\ z_{\mathrm{ref}}-z_i\bigr). \label{eq:di_def}\] Define the cell-level deficit by \[\mathcal{D}_{\mathrm{def}} := \frac{1}{|\mathcal{V}|}\sum_{i\in\mathcal{V}} d_i. \label{eq:deficit_cell}\] \(\mathcal{D}_{\mathrm{def}}\) is a dimensionless L2 deficit indicator.

6.6.5.3 4.1.5.3 L2\(\rightarrow\)L3 map: deficit density (optional, if needed)

If deficit is used as a spatial density, fix the following derived definition. \[\rho_{\mathrm{def}} := \frac{1}{V_{\square}}\sum_{i\in\mathcal{V}} d_i = \frac{|\mathcal{V}|}{V_{\square}}\ \mathcal{D}_{\mathrm{def}}, \qquad V_{\square}:=D_{\mathrm{anch}}^{3}. \label{eq:deficit_density}\] The dimension of \(\rho_{\mathrm{def}}\) is locked as \([L^{-3}]\). \(V_{\square}\) is usable only when the cell geometry (cube) and the meaning of \(D_{\mathrm{anch}}\) are locked. Deficit density is an optional derived quantity; whether it is used must be locked in analysis_lock.

6.6.5.4 4.1.5.4 Failure modes for the deficit definition (undefinedness conditions)

The conditions under which the deficit definition fails (failure modes) are fixed as follows.

  1. FM-SYM: the contact-predicate convention is not locked, so \(z_i\) is undefined.

  2. FM-SCOPE: the scope of \(z_{\mathrm{ref}}\) (dimension/boundary handling/protocol) is not locked, so the reference collapses.

  3. FM-NONUNIQUE: \(z_{\mathrm{ref}}\) moves within the same version or multiple values are mixed.

  4. FM-NUMERIC: the contact-graph construction convention is not locked, so \(z_i\) is not reproducible.

6.6.6 4.1.6 1:1 definition of charge (shell cancellation \(\rightarrow\) survival vector \(\rightarrow\) sign)

6.6.6.1 4.1.6.1 L1 input (structure): shell vector set and cancellation convention

Charge is defined as “the sign of the residual directionality left by the shell structure”. For this, define the shell vector set by \[\mathcal{S} := \{\mathbf{s}_k\}_{k=1}^{7}, \qquad \mathbf{s}_k \in \mathbb{R}^3. \label{eq:shell_vectors}\] The coordinate system, scale (internal unit), and the vector-generation convention (what structural quantity is encoded as a vector) must be locked in analysis_lock.

The cancellation convention simultaneously locks the following.

  1. the rule that assigns 6 of the 7 vectors as cancellation terms (e.g., pairwise/quartet grouping) and its fixed ordering,

  2. the cancellation criterion (the condition under which two vectors cancel) (inner product/angle/component convention, etc.),

  3. the definition of the remaining 1 vector (or remaining sum), i.e., the survival vector.

If the cancellation convention is not locked, charge is undefined.

6.6.6.2 4.1.6.2 L1\(\rightarrow\)L2 map: survival vector \(\mathbf{V}\) and sign indicator \(q\)

Define the survival vector by \[\mathbf{V} := \sum_{k=1}^{7}\mathbf{s}_k, \label{eq:survival_vector}\] where [eq:survival_vector] is used only as an expression equivalent to “the residual after cancellation terms cancel out according to the locked convention”. If the cancellation terms are constructed explicitly in the actual computation, that construction must be locked in analysis_lock.

Lock the sign-determination axis (charge axis) \(\mathbf{n}_Q\) by \[\mathbf{n}_Q \in \{\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\}\ \text{or a pre-registered unit vector}, \qquad \|\mathbf{n}_Q\|=1. \label{eq:charge_axis}\] \(\mathbf{n}_Q\) must not be chosen after seeing the result; the selection rule must be locked in analysis_lock.

Define the sign indicator by \[q := \mathrm{sgn}\!\left(\mathbf{V}\cdot\mathbf{n}_Q\right), \qquad \mathrm{sgn}(x)= \begin{cases} +1,& x>0,\\ 0,& x=0,\\ -1,& x<0. \end{cases} \label{eq:q_sign}\] Here \(q=0\) means “sign-indeterminate or neutral”. Whether \(q=0\) is admissible as a conclusion (allowed/forbidden) is locked by PASS.rules in analysis_lock.

6.6.6.3 4.1.6.3 L2\(\rightarrow\)L3 map: charge \(Q\) and charge density

Lock the charge unit \(q_0\) in canon_lock as \[q_0\ \text{is the base unit of charge dimension }[Q],\ \text{and its value is locked as }q_0:=1\ \text{(internal unit system)}. \label{eq:q0_lock}\] Define charge by \[Q := q_0\,q. \label{eq:charge_def}\] If a spatial charge density is needed, use the derived definition \[\rho_Q := \frac{Q}{V_{\square}}, \qquad V_{\square}:=D_{\mathrm{anch}}^{3}. \label{eq:charge_density}\] The dimension of \(\rho_Q\) is locked as \([Q L^{-3}]\).

6.6.6.4 4.1.6.4 Failure modes for the charge definition (undefinedness conditions)

The conditions under which the charge definition fails (failure modes) are fixed as follows.

  1. FM-STR: the shell-vector generation convention or the cancellation convention is not locked (structure is unspecified).

  2. FM-SYM: \(\mathbf{n}_Q\) or the coordinate-system definition is not locked (direction meaning is unspecified).

  3. FM-NONUNIQUE: the cancellation convention admits multiple solutions, so \(\mathbf{V}\) is non-unique.

  4. FM-SCOPE: the shell(7) structure is not defined in the declared regime, yet charge is invoked.

6.7 4.2 Closure types, stacks, and DAG rules

6.7.1 4.2.1 Position and purpose of a closure

A closure is a device that seals, as an explicit rule, a choice that is not determined by axioms ([A]) and definitions ([D]) alone. A closure is not an “extra hypothesis”; it is documentation of a selection rule. The existence of closures is fixed by the two reasons below.

  1. Resolving indeterminacy: even with the same inputs, the output may be non-unique or multi-valued; a rule is needed to select which solution will be used.

  2. Sealing procedures: the output can vary with the algorithm/estimator/boundary handling/normalization; without locking the procedure, the same conclusion cannot be claimed.

Therefore a closure must always include (i) input/output types, (ii) selection rule (assumptions), (iii) scope (regime), (iv) failure modes, and (v) Gate stack. If any of the five elements is missing, the closure is unusable.

6.7.2 4.2.2 Closure types (standard taxonomy)

In this document, closures are classified by “what is being closed” into the standard types below. The type is locked in analysis_lock, and closure IDs follow a namespace that includes the type.

6.7.3 (CL-S) Semantic-map Closure

A closure used when a choice is required in an L1\(\rightarrow\)L2 or L2\(\rightarrow\)L3 map. Examples include choosing the representative value of a throat (mean vs minimum-cut), and fixing how an event aggregation window is defined.

6.7.4 (CL-G) Graph/Path Closure

A closure that seals graph-based choices such as contact-graph construction, critical-throat definition, path selection, backbone selection, and minimum-cut computation. Examples include boundary-node definition, weight assignment, and equivalence rules.

6.7.5 (CL-R) Regime Closure

A closure that seals the scope of applicability: which rules are valid under which conditions (jamming/non-jamming, rotation-driven/non-driven, linear/nonlinear, etc.). Since changing regime conditions can change the conclusion, regime closures are premises of all conclusions.

6.7.6 (CL-N) Numerical Closure

A closure that seals numerical procedures: convergence, termination criteria, sensitivity policies, iteration limits, tolerances, and step policies. This closure exists not to “obtain” a result but to ensure the result is reproducible.

6.7.7 (CL-X) Cross-consistency Closure

A closure that seals cross-consistency rules for judging whether independently constructed channels/baselines/input combinations are consistent with the same conclusion. Cross-consistency closures include choices of thresholds, comparison quantities, and verdict methods.

6.7.8 (CL-P) Protocol Closure

A closure that seals execution conventions: input file formats, log schema, seed policy, environment fixation, etc. Protocol closures are premises of the reproducibility Gate and of snapshot sealing.

6.7.9 4.2.3 Closure ID rules (identifier and namespace)

Each closure is identified by a closure_id. The closure_id format is fixed as \[\texttt{CL-}\langle\texttt{TYPE}\rangle\texttt{-}\langle\texttt{TOPIC}\rangle\texttt{-}\langle\texttt{NNN}\rangle, \label{eq:closure_id_format}\] where

Multiple closures may coexist for the same TYPE and TOPIC, but within a single conclusion (claim_id) the simultaneously adopted closure for the same topic is restricted to one. If multiple are needed, the topic must be decomposed into distinct topics.

6.7.10 4.2.4 I/O type rules for closures (layer/dimension/object attribution)

A closure must state the types of its inputs and outputs. A type consists of the following four elements.

  1. Layer: which of L0/L1/L2/L3 it belongs to.

  2. Object attribution (object_id): which object the quantity belongs to (e.g., OBJ-THROAT, OBJ-PATH, OBJ-EVENT).

  3. Symbol: the symbol used in the main text (locked to a single meaning).

  4. Dimension/unit: dimensionless (1) or physical dimensions (length/time/mass/energy/force, etc.) and unit system (SI/internal).

If the I/O types of a closure are ambiguous, or the same symbol appears with different types, then this is a definition conflict and the closure is unusable.

6.7.11 4.2.5 The “assumptions” field of a closure (explicit selection rules)

The “assumptions” contained in a closure are fixed by the rules below.

  1. Assumptions exist only in the form of selection rules. They must record, in mechanically readable sentences, “what was chosen”.

  2. Assumptions must not introduce new ontology (e.g., a new continuous field, a new global objective function, axiomatizing a new probability distribution, etc. are outside the scope of closures).

  3. Assumptions must have scope (scope). Assumptions without scope are prohibited because they can be misread as global assumptions.

  4. Assumptions must not be changed after seeing results. Changes are allowed only by versioning (new analysis_lock_id).

6.7.12 4.2.6 Gate stack (mandatory) and linkage to PASS/FAIL judgements

Every closure has its required Gate stack. The purpose of the Gate stack is to make the choices embedded in the closure necessary conditions for a conclusion to obtain admissible status. The Gate stack is fixed by the following rules.

  1. Mandatory base Gates: every closure requires at least G-SYM, G-LOCK, G-REG, and G-NT.

  2. Type-dependent additional Gates: depending on the closure type, add the following Gates as needed.

    • CL-S: (if needed) G-NUM

    • CL-G: (if needed) G-STR, G-NUM

    • CL-R: (if needed) strengthened G-REG and/or G-STR

    • CL-N: G-NUM

    • CL-X: G-RCROSS, (if needed) G-NUM

    • CL-P: G-REP

  3. Judgement precedence: if any required Gate yields FAIL or INCONCLUSIVE, then any conclusion using the closure cannot obtain conclusion status.

  4. Failure-mode labeling: the causes of FAIL/INCONCLUSIVE are decomposed and recorded as failure-mode labels (FM-*) or No-Tuning violation labels (FAIL-NT-*).

6.7.13 4.2.7 Closure DAG rules (no cyclic dependency)

Closures may depend on each other, but the dependency must be a DAG (a directed acyclic graph). The closure DAG rules are fixed as follows.

6.7.13.1 4.2.7.1 Nodes and edges

Let the closure set be \(\mathcal{C}\) and denote each closure by \(c\in\mathcal{C}\). Define the closure DAG by \[\mathcal{G}_{\mathrm{CL}} := (\mathcal{C}, \mathcal{E}_{\mathrm{CL}}), \label{eq:closure_dag_def}\] where \((c_i,c_j)\in\mathcal{E}_{\mathrm{CL}}\) means “closure \(c_j\) depends on the output of closure \(c_i\).”

6.7.13.2 4.2.7.2 Acyclic condition (no cycles)

The DAG condition is fixed as \[\text{$\mathcal{G}_{\mathrm{CL}}$ contains no directed cycle}. \label{eq:acyclic_rule}\] That is, if a path of the form \(c_1\to c_2\to\cdots\to c_k\to c_1\) exists, then that set of closures is unusable.

6.7.13.3 4.2.7.3 Typical forbidden patterns of cycles

Cycles occur in the representative patterns below. These patterns are fixed as forbidden rules.

  1. Threshold–output cycle: a closure output determines a Gate threshold, and that threshold in turn changes the solution selection of the same closure.

  2. Channel–consistency cycle: in a cross-consistency closure, the selected channels are re-selected/excluded based on the output.

  3. Regime–selection cycle: a regime closure reclassifies the regime using an output indicator, and that reclassification changes closure selection.

  4. Post-hoc selection cycle: selecting a “good” solution among multiple candidates using a rule that depends on the output value (a form of post-hoc tuning).

If such a pattern appears, the closure definition is rejected in analysis_lock, and related conclusions are automatically judged FAIL or INCONCLUSIVE.

6.7.13.4 4.2.7.4 Topological sorting and execution order

If the DAG condition holds, closures have a topologically sortable order. Denote the execution precedence by \(\prec\); then \[c_i \prec c_j \quad \Longleftrightarrow \quad (c_i,c_j)\in\mathcal{E}_{\mathrm{CL}}\ \text{or a precedence relation derived from it}. \label{eq:toposort_rule}\] Derivations and Gate executions must follow the \(\prec\) order. If the order is violated, required inputs do not exist and the result is judged INCONCLUSIVE.

6.7.14 4.2.8 Closure template (standard registry form)

Below is the standard template for recording a closure in the registry. The key structure and fields are fixed; if any required field is missing, the closure is unusable.

closures:
  - closure_id: CL-G-THROAT-001
    type: CL-G
    topic: THROAT
    version: (local version within analysis_lock_id)
    scope: (regime identifier or predicate)
    inputs:
      - layer: L1
        object_id: OBJ-THROAT
        symbol: delta_gap
        dimension: L
        unit_system: internal
    outputs:
      - layer: L2
        quantity_id: Q-THROAT-DELTAEFF
        symbol: delta_eff
        dimension: L
        unit_system: internal
    assumptions:
      - "Define the critical throat on the contact graph via a minimum-cut rule"
      - "The weight convention follows W_THROAT"
      - "Fix the representative value to the minimum-cut representative value (not the median)"
    algorithm:
      algorithm_id: ALG-THROAT-EST-001
      parameters_locked_by: analysis_lock_id
    dag:
      depends_on: [CL-R-REGIME-001, CL-P-PROTO-001]
    failure_modes:
      - FM-SYM
      - FM-SCOPE
      - FM-NONUNIQUE
      - FM-NUMERIC
    required_gates:
      - G-SYM
      - G-LOCK
      - G-REG
      - G-STR
      - G-NUM
      - G-NT
    pass_rules_hook:
      claim_types_allowed_on_pass: [CT-DER-FORM, CT-DER-NUM, CT-LIM]
      claim_types_forbidden_on_fail: [CT-DER-NUM, CT-XCROSS, CT-REP]

In this template, dag.depends_on contains the edge information of the closure DAG, and failure_modes and required_gates are mandatory fields. pass_rules_hook is a declaration connected to PASS.rules, linking “which sentences are allowed when which Gates pass” at the closure level.

6.7.15 4.2.9 Closure stacks and claim attribution

A single conclusion (claim_id) uses one or more closures. This is called a closure stack. The closure stack is fixed by the rules below.

  1. A closure stack has an order. The order is determined by topological sorting of the closure DAG.

  2. A conclusion must include the closure_ids list. Without it, the conclusion loses its basis and cannot claim conclusion status.

  3. If any closure in the stack yields FAIL/INCONCLUSIVE in Gate, then the conclusion cannot obtain conclusion status.

  4. Mixing closure stacks from different analysis_lock_id to form one conclusion is prohibited.

6.8 4.3 Regime map (scope of applicability)

6.8.1 4.3.1 Definition of a regime

A regime is a coordinate description of “under which conditions which definitions/closures/judgements are valid”. A regime is not expressed as a single sentence (e.g., “rigid regime”); it must be defined as a tuple composed of the values of regime coordinate axes. Fix the regime in the form \[\mathcal{R} := \bigl( R_{\mathrm{dim}}, R_{\mathrm{cell}}, R_{\mathrm{drive}}, R_{\mathrm{span}}, R_{\kappa}, R_{\mathrm{scale}}, R_{\mathrm{bc}}, R_{\mathrm{init}}, R_{\mathrm{obs}} \bigr). \label{eq:regime_tuple}\] Each component is a regime-axis value defined in 4.3.2, and the axes themselves are locked in analysis_lock.

6.8.2 4.3.2 Regime coordinate axes (standard coordinate system)

This section fixes the regime coordinate system to the 9 axes below. Each axis is defined as an enumeration or an operational predicate (indicator function / thresholded index); it is not replaced by “interpretation”.

6.8.3 (A) Dimension axis \(R_{\mathrm{dim}}\)

The dimension axis indicates the dimension in which the domain/contact graph/cut surface definitions are performed. \[R_{\mathrm{dim}}\in\{\texttt{DIM-2},\ \texttt{DIM-3}\}. \label{eq:reg_dim}\] The dimension is locked by object definitions in canon_lock and protocols in analysis_lock, and must not be mixed within one output.

6.8.4 (B) Cell-geometry axis \(R_{\mathrm{cell}}\)

The cell-geometry axis indicates the canonical geometry type of the Anchor Cell. \[R_{\mathrm{cell}}\in\{\texttt{CELL-CUBE}\}, \label{eq:reg_cell}\] In the canonical regime, only CELL-CUBE is allowed. CELL-SPHERE-VIS is a visualization transform, not a regime-axis value. Visualization transforms are locked only by vis_mode in analysis_lock and do not replace the canonical regime.

6.8.5 (C) Drive axis \(R_{\mathrm{drive}}\)

The drive axis indicates which driving condition is included in configuration updates (composition of local updates). \[R_{\mathrm{drive}}\in\{\texttt{DRV-NONE},\ \texttt{DRV-ROT}\}. \label{eq:reg_drive}\] DRV-ROT holds only when a rotation-drive input (e.g., \(\ell_{\mathrm{rot}}\)) exists and the corresponding protocol lock (drive method/termination condition/sampling) is satisfied. The satisfaction conditions for DRV-ROT must be locked in analysis_lock, and reclassifying the drive axis after seeing results is prohibited.

6.8.6 (D) Spanning axis \(R_{\mathrm{span}}\)

The spanning axis is defined as an indicator of whether the contact graph connects opposite boundary sets of the domain. \[R_{\mathrm{span}}\in\{\texttt{SPAN-0},\ \texttt{SPAN-1}\}, \qquad R_{\mathrm{span}}:=\texttt{SPAN-}\chi_{\mathrm{span}}(\mathfrak{J}), \label{eq:reg_span}\] where \(\chi_{\mathrm{span}}(\mathfrak{J})\in\{0,1\}\) is the spanning indicator defined in 3.2. If boundary sets and boundary-node definitions are not locked, then \(R_{\mathrm{span}}\) is undefined and the regime declaration itself does not hold.

6.8.7 (E) Bottleneck axis \(R_{\kappa}\)

The bottleneck axis is defined as a binning of the minimum-cut size (or an equivalent bottleneck index). \[R_{\kappa}\in\{\texttt{KAPPA-0},\ \texttt{KAPPA-1},\ \texttt{KAPPA-GE2}\}, \label{eq:reg_kappa_enum}\] Fix \(R_{\kappa}\) by the mapping rule \[\kappa_{\min}(\mathfrak{J}) \ \mapsto\ R_{\kappa}:= \begin{cases} \texttt{KAPPA-0}, & \kappa_{\min}=0,\\ \texttt{KAPPA-1}, & \kappa_{\min}=1,\\ \texttt{KAPPA-GE2}, & \kappa_{\min}\ge 2. \end{cases} \label{eq:reg_kappa_map}\] The algorithm used to compute \(\kappa_{\min}\) (exact/approximate, weighted/unweighted, boundary handling) must be locked in analysis_lock. If the algorithm is not locked, then \(R_{\kappa}\) is undefined.

6.8.8 (F) Scale-window axis \(R_{\mathrm{scale}}\)

The scale-window axis indicates “under which internal scale range the output was aggregated/evaluated”. Lock the scale window as a bin of a unitless internal index \(s\). \[R_{\mathrm{scale}}\in\{\texttt{SCALE-LONG},\ \texttt{SCALE-MID},\ \texttt{SCALE-SHORT}\}, \label{eq:reg_scale_enum}\] Here \(s\) is a single internal index locked in analysis_lock (e.g., a graph-distance-based length, path-length-based length, lattice-index-based length). The bin thresholds \[0<s_1<s_2 \label{eq:scale_thresholds}\] are locked as thresholds in analysis_lock. Fix the scale window by \[R_{\mathrm{scale}}:= \begin{cases} \texttt{SCALE-LONG}, & s\ge s_2,\\ \texttt{SCALE-MID}, & s_1\le s < s_2,\\ \texttt{SCALE-SHORT}, & 0< s < s_1. \end{cases} \label{eq:reg_scale_map}\] Scale-window classification must not be changed after seeing results.

6.8.9 (G) Boundary-condition axis \(R_{\mathrm{bc}}\)

The boundary-condition axis indicates the boundary-handling type of the domain. \[R_{\mathrm{bc}}\in\{\texttt{BC-CLOSED},\ \texttt{BC-OPEN},\ \texttt{BC-DRIVEN}\}. \label{eq:reg_bc}\] BC-DRIVEN may be combined with the drive axis (e.g., DRV-ROT) but must not be conflated with it. Boundary conditions are part of the protocol and must be locked in protocol_lock or analysis_lock.

6.8.10 (H) Initial-condition axis \(R_{\mathrm{init}}\)

The initial-condition axis classifies the initial configuration (contact graph/deficit/defect distribution, etc.). \[R_{\mathrm{init}}\in\{\texttt{INIT-RAW},\ \texttt{INIT-RELAXED},\ \texttt{INIT-PREJ}\}. \label{eq:reg_init}\] INIT-RELAXED means an initial condition for which an admissible configuration is fixed by passing a pre-registered relaxation protocol. INIT-PREJ means an initial condition generated using a pre-registered control-parameter window to sample around Point-J; the generation rule and window must be locked in analysis_lock.

6.8.11 (I) Observation/aggregation axis \(R_{\mathrm{obs}}\)

The observation/aggregation axis indicates the conventions for event logs and aggregation windows. \[R_{\mathrm{obs}}\in\{\texttt{OBS-EVENT},\ \texttt{OBS-STATIC},\ \texttt{OBS-HYBRID}\}. \label{eq:reg_obs}\] OBS-EVENT indicates an observational regime that obligatorily includes event-based aggregation (ticks, event pre/post configurations). OBS-STATIC indicates a regime that aggregates only structural indicators of a single configuration (contact degree, cut, path, etc.). OBS-HYBRID combines the two, but holds only when each aggregation has its own locked 1:1 map.

6.8.12 4.3.3 Regime IDs and the standard format for regime declarations

A regime is called by a string identifier regime_id. A regime_id is valid only when the values in [eq:regime_tuple] are fixed in the registry. Fix the standard registry template as follows.

regimes:
  - regime_id: R-BASE-001
    coords:
      dim: DIM-3
      cell: CELL-CUBE
      drive: DRV-NONE
      span: SPAN-1
      kappa: KAPPA-GE2
      scale: SCALE-LONG
      bc: BC-CLOSED
      init: INIT-RELAXED
      obs: OBS-EVENT
    allowed_closure_stacks:
      - stack_id: CS-BASE-CORE-001
        closures: [ ... ]
    forbidden_extrapolations:
      - "drive: DRV-ROT"
      - "dim: DIM-2"

If coords is missing or any axis value does not match the registry enumeration, the regime declaration is invalid and the output is judged INCONCLUSIVE.

6.8.13 4.3.4 Allowed closure stacks (regime-wise allow-lists)

A regime carries an allow-list of “allowed closure stacks”. An allowed closure stack includes (i) the list of usable closures, (ii) stack order (topological sorting of the closure DAG), and (iii) additional Gate requirements specific to the regime. Fix the allowed closure stacks by the rules below.

  1. Every derivation that generates conclusions under the same regime_id must select and use one of the allowed closure stacks of that regime.

  2. Using a closure that is not in the allow-list yields immediate FAIL (regime violation).

  3. An allowed stack must satisfy the closure DAG; a stack violating the DAG cannot be registered.

  4. Changes of allowed stacks are permitted only by versioning of analysis_lock.

6.8.13.1 4.3.4.1 Example allowed stack for the base regime (canonical chain)

A minimal allowed stack for the canonical chain (canonical\(\rightarrow\)structure\(\rightarrow\)event\(\rightarrow\)realization\(\rightarrow\)derived) is fixed in the following form (item names are examples; actual use is by closure_id only). \[\texttt{CS-BASE-CORE-001}: \bigl[ \texttt{CL-R-REGIME-001}, \texttt{CL-P-PROTO-001}, \texttt{CL-G-CONTACT-001}, \texttt{CL-G-BOUNDARY-001}, \texttt{CL-G-BACKBONE-001}, \texttt{CL-S-EVENTMAP-001}, \texttt{CL-N-CONV-001}, \texttt{CL-X-RCROSS-001} \bigr]. \label{eq:closure_stack_base}\] Each closure’s I/O types, failure modes, and required Gates follow the closure-template conventions in 4.2. The stack means the following.

6.8.13.2 4.3.4.2 Example allowed stack for a rotation-driven regime (extended)

In a rotation-driven regime (DRV-ROT), the following closures are added to [eq:closure_stack_base]. A conclusion that does not include the added closures cannot obtain admissible status under a rotation-driven regime. \[\texttt{CS-ROT-EXT-001}: \bigl[ \texttt{CL-R-REGIME-ROT-001}, \texttt{CL-P-PROTO-ROT-001}, \texttt{CL-G-CONTACT-001}, \texttt{CL-G-BOUNDARY-001}, \texttt{CL-G-ANISO-AXIS-001}, \texttt{CL-S-ANISO-MAP-001}, \texttt{CL-N-CONV-001}, \texttt{CL-X-RCROSS-001} \bigr]. \label{eq:closure_stack_rot}\] Here CL-G-ANISO-AXIS-001 is a closure that locks the selection convention of the anisotropy axis / directional distribution, and CL-S-ANISO-MAP-001 locks the 1:1 maps for direction-dependent observables. In a rotation-driven regime, whether \(\ell_{\mathrm{rot}}\) is a reference value (CANON-REF) or a promoted value (CANON-PRIMARY) changes the input items and failure modes of CL-R-REGIME-ROT-001; this attribution is tied to the canon_lock version.

6.8.13.3 4.3.4.3 Example allowed stack for a non-rigid regime (restricted)

In a non-rigid regime, closures that presuppose the existence of a rigid backbone cannot be used. Therefore the following restriction is included in the regime definition. \[R_{\mathrm{span}}=\texttt{SPAN-0} \ \ \text{or}\ \ R_{\kappa}\in\{\texttt{KAPPA-0},\texttt{KAPPA-1}\} \ \Longrightarrow\ \texttt{CL-G-BACKBONE-001}\ \text{forbidden}. \label{eq:nonstiff_forbid_backbone}\] In this case, the allowed stack is fixed as \[\texttt{CS-NONSTIFF-001}: \bigl[ \texttt{CL-R-REGIME-001}, \texttt{CL-P-PROTO-001}, \texttt{CL-G-CONTACT-001}, \texttt{CL-G-BOUNDARY-001}, \texttt{CL-S-STATMAP-001}, \texttt{CL-N-CONV-001} \bigr]. \label{eq:closure_stack_nonstiff}\] To produce rigidity-related statements under a non-rigid regime, the statements must not be “rigidity conclusions” but “rigidity failure or limit conclusions” (CT-LIM), accompanied by failure-mode labels.

6.8.14 4.3.5 Extrapolation Ban

Regime extrapolation is defined as “describing a conclusion derived and Gate-passed in regime \(\mathcal{R}\) as the same conclusion for another regime \(\mathcal{R}'\) with different regime-axis values”. Extrapolation occurs if any of the following holds.

  1. Axis mismatch: at least one regime-axis value differs between \(\mathcal{R}\) and \(\mathcal{R}'\).

  2. Allowed-stack mismatch: the conclusion uses a closure that is not allowed under \(\mathcal{R}'\).

  3. Missing regime Gates: the Gates required to justify the transition to \(\mathcal{R}'\) (regime fit/cross-consistency/reproducibility) were not executed.

Extrapolation is not softened by interpretation; it is handled immediately by the rules below.

6.8.14.1 4.3.5.1 Sentence rules for the extrapolation ban

A sentence that contains regime extrapolation cannot obtain conclusion status. Fix the sentence rules as follows.

  1. Every conclusion sentence must include regime_id. If regime_id is missing, the sentence is judged INCONCLUSIVE.

  2. Regime-axis values appearing in a conclusion sentence must match coords of the registered regime. Any mismatch is immediate FAIL.

  3. To describe a generalization to another regime, one must present a separate regime_id, the allowed stack, and new Gate judgements for that regime.

  4. Outside the declared regime, only limit statements (CT-LIM) are allowed (not conclusions). Limit statements must include the cause labels of FAIL/INCONCLUSIVE.

6.8.14.2 4.3.5.2 FAIL labels for extrapolation violations

If regime extrapolation is detected, assign the following FAIL labels.

Label Meaning
FAIL-REG-NOID missing regime_id in a conclusion
FAIL-REG-MISMATCH mismatch between declared regime_id and regime-axis values
FAIL-REG-STACK use of a closure stack not allowed in the regime
FAIL-REG-EXTRAP inclusion of out-of-regime generalization/extrapolation
INCON-REG-UNDEF regime coordinates or indicator is undefined (missing locks)

If FAIL-REG-* is assigned, the output loses conclusion status; this loss propagates along the dependency graph to derived outputs.

6.8.15 4.3.6 Admissible conditions for regime transitions

A regime transition is recorded as “\(\mathcal{R}\to\mathcal{R}'\)” and is allowed only in the two cases below.

  1. Refinement within the same axis values: adding a sub-classification within the same regime using a pre-registered axis such as the scale window (\(R_{\mathrm{scale}}\)) or observation axis (\(R_{\mathrm{obs}}\)). Even then, the sub-regime must be registered with a new regime_id.

  2. Parallel presentation of independent conclusions: generating conclusions independently in different regimes and listing them in parallel without mixing. Parallel presentation may require a cross-consistency closure (CL-X-*); whether it is required must be locked in analysis_lock.

To synthesize a regime transition into a single conclusion, the synthesis rule itself must be locked as a closure, and the Gate stack for the synthesis must be pre-registered. A transition without a synthesis rule is treated as extrapolation and is forbidden.

7 5. Geometric Rectification Constants (Single Source of Truth)

Purpose and scope

This chapter defines the geometric rectification constants \(\alpha\) and \(\delta\) as the document-wide single source of truth and freezes their usage rules. Here, “single source of truth” means that (i) the defining equations of \(\alpha\) and \(\delta\), (ii) the derivations that produce \(\alpha\) and \(\delta\) under a fixed rectification convention, and (iii) the registry lock locations of \(\alpha\) and \(\delta\) exist in exactly one place in the entire document. Outside this chapter, it is forbidden to re-derive \(\alpha\) or \(\delta\), to reuse the same symbols with other meanings, or to redefine them under a different convention.

Definition of rectification constants (reserved symbols)

The rectification constants are fixed as the following two items. \[\alpha := \frac{2}{\pi}, \qquad \delta := \frac{1}{\pi^2}. \label{eq:rect_alpha_delta_def}\] Here, \(\pi\) is locked as a dimensionless constant, and \(\alpha\) and \(\delta\) are locked as dimensionless constants. In addition, \(\alpha\) and \(\delta\) are reserved symbols for rectification constants and cannot be reused with other meanings. In particular, freeze the following rules as global rules.

  1. \(\delta\) is reserved as the rectification constant. Any other delta meaning (gap/throat/thickness/defect, etc.) must be separated by subscripts or by a different symbol (e.g., \(\delta_{\mathrm{gap}},\ \delta_{\mathrm{throat}}\), etc.).

  2. \(\alpha\) is reserved as the rectification constant. Any other alpha meaning must be separated by subscripts or by a different symbol (e.g., \(\alpha_{\mathrm{aniso}}\), etc.).

  3. Violating a reserved symbol (overloading one symbol with multiple meanings) is immediately invalid at the stage prior to meaning-layer mapping; it cannot be patched by interpretation.

Single-source-of-truth (SSOT) rules

Freeze the SSOT rules for \(\alpha\) and \(\delta\) as follows.

  1. Single location for the definition: Equation [eq:rect_alpha_delta_def] appears exactly once in the entire document as the definition of \(\alpha\) and \(\delta\). Later sections must not rewrite the same equation; they only reference it.

  2. Single location for the derivation: The derivations that \(\alpha\) and \(\delta\) are produced by a particular rectification convention (angle average / projection / squaring convention, etc.) are performed exactly once inside this chapter (the following subsections). Outside this chapter, reproducing the derivation is forbidden.

  3. Single source in the registry: \(\alpha\) and \(\delta\) are locked under rectification_constants in canon_lock. The same entry must not be duplicated in any other registry or elsewhere in the main text.

  4. Version binding: The definition/derivation/convention of \(\alpha\) and \(\delta\) is bound to canon_lock_id. Even if one uses the same symbols with the same numerical values, it is forbidden to mix different canon_lock_ids in a single conclusion sentence.

7.1 5.4 Reuse rules (reference style and conclusion-sentence form)

When using \(\alpha\) and \(\delta\) outside this chapter, follow the reuse rules below.

  1. Reference-first: Every derivation/table/figure/log that contains \(\alpha\) or \(\delta\) must include a reference to which lock_id in canon_lock locks that item.

  2. No substitution-as-derivation: It is forbidden to substitute \(\alpha\) by \(\frac{2}{\pi}\) or \(\delta\) by \(\frac{1}{\pi^2}\) inside an equation to make the presentation look like a “derivation.” If substitution is needed, mark the step explicitly as a simple substitution and do not mix it with a rectification derivation.

  3. Separation of rectification conventions: If an equation involving \(\alpha\) or \(\delta\) contains operations such as averaging / projection / squaring, the convention of those operations (what is averaged, the window, the normalization) must be locked as a separate closure. Using \(\alpha,\delta\) without locking the convention makes the definition ill-posed.

  4. Role restriction: \(\alpha\) and \(\delta\) are used only as rectification constants. Reinterpreting them as a new meaning (e.g., a different coefficient or a different correction term) is forbidden.

7.2 5.5 Violation types and immediate invalidation rules

Violations of SSOT or reuse rules for \(\alpha\) and \(\delta\) cannot be patched by interpretation. If any of the following violations occurs, the corresponding output is immediately invalid.

  1. Within the same document version, presenting the definition of \(\alpha\) or \(\delta\) again in a different form (re-definition), or deriving it again under a different convention (re-derivation).

  2. Using \(\delta\) with meanings such as gap/throat/thickness, or abbreviating another delta (e.g., \(\delta_{\mathrm{gap}}\)) as the rectification constant \(\delta\) (symbol overloading).

  3. Omitting the required canon_lock_id reference for \(\alpha\) or \(\delta\), or mixing values from different canon_lock_ids (lock mixing).

  4. Generating a conclusion that contains \(\alpha\) or \(\delta\) while the averaging/projection/squaring convention is not locked (procedure not locked).

If a violation occurs, the output loses conclusion status, and this loss propagates to derived outputs along the dependency graph.

7.3 5.1 Derivation of \(\alpha=2/\pi\)

7.3.1 5.1.1 Definition of the rectification problem (directional component \(\rightarrow\) scalar effective quantity)

In this section, \(\alpha\) is defined as the constant “that rectifies a directional (or phase) component into a scalar effective quantity.” Deriving \(\alpha\) requires the following minimal ingredients.

7.3.1.1 [D-5.1-1] Phase variable

Define the phase variable \(\theta\) as a variable on the following set. \[\theta \in [0,2\pi). \label{eq:alpha_phase_domain}\] \(\theta\) is the minimal angular variable that represents the directionality of an internal state, and the meaning of \(\theta\) (angle with respect to which axis) is locked by the coordinate-system definition in analysis_lock. The derivation in this section uses only the fact that \(\theta\) is a full-cycle angular variable on [eq:alpha_phase_domain].

7.3.1.2 [D-5.1-2] Directional component (projection) and sign emergence

Define the “directional component” (before converting to a scalar effective quantity) in the following form. \[X_{\parallel}(\theta) := X_{0}\cos\theta, \label{eq:alpha_projection}\] where \(X_{0}\ge 0\) is a magnitude scale (non-signed scale), and \(\cos\theta\) is the minimal projection function that produces \(\pm\) signs through the phase \(\theta\). Definition [eq:alpha_projection] is not used to claim that “any physical quantity must follow cosine projection.” In this document, \(\cos\theta\) is a definitional choice as the minimal sign-changing projection function over a full cycle. If one adopts a different projection function, then one must define a different rectification constant under a new symbol and lock it as a separate registry item.

7.3.1.3 [D-5.1-3] Rectification operator

Rectification is defined as the operation that “reduces a sign-canceling directional component to an effective quantity in the magnitude view.” To do so, define the rectification operator including an absolute value as \[\mathrm{Rect}[X_{\parallel}](\theta) := |X_{\parallel}(\theta)|. \label{eq:alpha_rect_operator_pointwise}\] Define the rectified scalar effective quantity \(X_{\mathrm{rect}}\) by a full-cycle average (the averaging convention is locked below). \[X_{\mathrm{rect}} := \left\langle \mathrm{Rect}[X_{\parallel}] \right\rangle = \left\langle |X_{0}\cos\theta| \right\rangle. \label{eq:alpha_rect_average_def}\] Here \(\langle\cdot\rangle\) is the full-cycle averaging operator on [eq:alpha_phase_domain].

7.3.2 5.1.2 Definition of the full-cycle averaging operator (canonical measure)

The average over \(\theta\) is defined by a canonical measure on the full cycle.

7.3.2.1 [D-5.1-4] Canonical measure

Define the canonical measure \(d\mu(\theta)\) as \[d\mu(\theta) := \frac{d\theta}{2\pi}, \qquad \int_{0}^{2\pi} d\mu(\theta) = 1. \label{eq:alpha_uniform_measure}\] Definition [eq:alpha_uniform_measure] is a definition (convention) that fixes how averaging is performed over the full cycle. Introducing a different weight (a non-uniform measure) after seeing results is forbidden. If a non-uniform measure is needed, it must be defined as a separate closure; that closure must be locked in analysis_lock together with its failure modes and Gate stack.

7.3.2.2 [D-5.1-5] Full-cycle averaging operator

Define the full-cycle averaging operator induced by the canonical measure as \[\left\langle f(\theta) \right\rangle := \int_{0}^{2\pi} f(\theta)\, d\mu(\theta) = \frac{1}{2\pi}\int_{0}^{2\pi} f(\theta)\, d\theta. \label{eq:alpha_average_operator}\] This definition is fixed only in this chapter (the SSOT for rectification constants); later sections only reference it.

7.3.3 5.1.3 Full calculation of \(\langle |\cos\theta| \rangle\)

Combining definitions [eq:alpha_rect_average_def] and [eq:alpha_average_operator] gives \[X_{\mathrm{rect}} = \left\langle |X_{0}\cos\theta| \right\rangle = X_{0}\left\langle |\cos\theta| \right\rangle = X_{0}\cdot \frac{1}{2\pi}\int_{0}^{2\pi} |\cos\theta|\, d\theta. \label{eq:alpha_rect_reduce}\] Thus the key is to compute the integral \[I := \int_{0}^{2\pi} |\cos\theta|\, d\theta \label{eq:alpha_I_def}\] by decomposing the domain into sign intervals of \(|\cos\theta|\).

7.3.3.1 (1) Sign-interval decomposition

The sign of \(\cos\theta\) is determined on the following intervals. \[\cos\theta \ge 0 \ \text{for}\ \theta\in\left[0,\frac{\pi}{2}\right]\cup\left[\frac{3\pi}{2},2\pi\right], \qquad \cos\theta \le 0 \ \text{for}\ \theta\in\left[\frac{\pi}{2},\frac{3\pi}{2}\right]. \label{eq:alpha_cos_sign_intervals}\] Therefore \[|\cos\theta|= \begin{cases} \cos\theta, & \theta\in\left[0,\frac{\pi}{2}\right]\cup\left[\frac{3\pi}{2},2\pi\right],\\ -\cos\theta, & \theta\in\left[\frac{\pi}{2},\frac{3\pi}{2}\right]. \end{cases} \label{eq:alpha_abs_cos_piecewise}\]

7.3.3.2 (2) Sum over intervals

From [eq:alpha_I_def] and [eq:alpha_abs_cos_piecewise], \[\begin{aligned} I &= \int_{0}^{\pi/2} \cos\theta\, d\theta + \int_{\pi/2}^{3\pi/2} (-\cos\theta)\, d\theta + \int_{3\pi/2}^{2\pi} \cos\theta\, d\theta. \label{eq:alpha_I_split}\end{aligned}\] Compute each integral in order.

7.3.3.3 (3) First-interval integral

\[\int_{0}^{\pi/2} \cos\theta\, d\theta = \left[\sin\theta\right]_{0}^{\pi/2} = \sin\left(\frac{\pi}{2}\right)-\sin(0) = 1-0 = 1. \label{eq:alpha_I1}\]

7.3.3.4 (4) Second-interval integral

Because the sign is flipped in the second interval, \[\begin{aligned} \int_{\pi/2}^{3\pi/2} (-\cos\theta)\, d\theta &= -\left[\sin\theta\right]_{\pi/2}^{3\pi/2} = -\left(\sin\left(\frac{3\pi}{2}\right)-\sin\left(\frac{\pi}{2}\right)\right) \notag\\ &= -\left((-1)-1\right) = -(-2) = 2. \label{eq:alpha_I2}\end{aligned}\]

7.3.3.5 (5) Third-interval integral

\[\int_{3\pi/2}^{2\pi} \cos\theta\, d\theta = \left[\sin\theta\right]_{3\pi/2}^{2\pi} = \sin(2\pi)-\sin\left(\frac{3\pi}{2}\right) = 0-(-1) = 1. \label{eq:alpha_I3}\]

7.3.3.6 (6) Summation

Substituting [eq:alpha_I1], [eq:alpha_I2], [eq:alpha_I3] into [eq:alpha_I_split] yields \[I = 1 + 2 + 1 = 4. \label{eq:alpha_I_value}\] Hence the full-cycle average is \[\left\langle |\cos\theta| \right\rangle = \frac{1}{2\pi}\int_{0}^{2\pi} |\cos\theta|\, d\theta = \frac{1}{2\pi}\cdot 4 = \frac{2}{\pi}. \label{eq:alpha_abs_cos_average}\]

7.3.4 5.1.4 Fixing the rectification constant \(\alpha\) and conversion formulas

From [eq:alpha_rect_reduce] and [eq:alpha_abs_cos_average], \[X_{\mathrm{rect}} = X_{0}\left\langle |\cos\theta| \right\rangle = X_{0}\cdot\frac{2}{\pi}. \label{eq:alpha_rect_result}\] Therefore, the scalar effective quantity obtained by full-cycle rectification of the directional component \(X_{\parallel}(\theta)=X_{0}\cos\theta\) is \[X_{\mathrm{rect}} = \alpha\, X_{0}, \qquad \alpha := \frac{2}{\pi}. \label{eq:alpha_definition_from_rect}\] In [eq:alpha_definition_from_rect], \(\alpha\) is a constant derived from the rectification convention (canonical measure + absolute value + full-cycle averaging). This derivation is performed only in this chapter (SSOT for rectification constants). Later sections only reference \(\alpha\) as the result of [eq:alpha_definition_from_rect].

In this document, \(\alpha\) is used as a common coefficient that converts a “sign-bearing directional component” into a “sign-free effective scalar.” Freeze the usage in the following standard form.

7.3.5.1 [D-5.1-6] Standard usage form

When a directional component is defined as \[X_{\parallel}(\theta)=X_{0}\cos\theta \label{eq:alpha_useform_dir}\] then the rectified scalar is defined by \[X_{\mathrm{rect}}=\left\langle |X_{\parallel}(\theta)| \right\rangle = \alpha X_{0} \label{eq:alpha_useform_rect}\] where \(X_{0}\) is a magnitude scale and must not be confused with the sign-including full-cycle average of \(X_{\parallel}\). In particular, \[\left\langle X_{\parallel}(\theta) \right\rangle = \left\langle X_{0}\cos\theta \right\rangle = X_{0}\left\langle \cos\theta \right\rangle = X_{0}\cdot 0 =0 \label{eq:alpha_mean_zero_note}\] so the sign-including average cancels to zero, while the rectified average yields the effective quantity. This distinction is reused repeatedly in all later “cancellation \(\rightarrow\) survival” derivations.

Representative usage sites for \(\alpha\) are linked below by section number (all are used only by referencing the single rectification convention [eq:alpha_average_operator]–[eq:alpha_definition_from_rect]).

  1. Core length-selection ratio: In Chapter 6 (continuum core model), when recording the ratio between the core radius and the reference length as a rectification coefficient, \(\alpha\) is used as the length-selection ratio coefficient.

  2. Event-rate rectification: In Chapter 9 (event definition and canonical event rate), when defining a rectified event rate (scalar frequency) from a directional event rate (sign-bearing or phase-bearing), \(\alpha\) is used as the rectification coefficient.

  3. Scalarization of the cancellation–survival convention: In Chapter 8 (discrete shell structure) and Chapter 4 (meaning-layer mapping), when defining the effective scalar quantity that remains after cancellation, \(\alpha\) is used as the rectification coefficient of sign components.

These usage sites use \(\alpha\) only as a “rectification coefficient” and never re-derive \(\alpha\) or replace it by a different averaging convention. If a different averaging convention is required, define a separate rectification coefficient under a new symbol, which requires a separate registry entry and Gate.

7.4 5.2 Derivation of \(\delta=1/\pi^{2}\) + universality axiom

7.4.1 5.2.1 Role of the rectification coefficient \(\delta\) (survival coefficient for sign-constrained events)

\(\delta\) is fixed as the coefficient that, when a sign-bearing directional component is aggregated as an event, rectifies the portion cancelled by the sign constraint into the “mean surviving fraction.” \(\delta\) is a dimensionless constant. Its value is derived in this section and then locked in canon_lock. Outside this section, re-derivation / re-definition / substitution-mixed-with-derivation of \(\delta\) is forbidden.

7.4.2 5.2.2 Required definitions (rectification operator, measure, and the two constrained phases of an event)

The derivation in this section requires the following definitions.

7.4.2.1 [D-5.2-1] Full-cycle phases

The phase variables used in event aggregation are defined as full-cycle angle variables. \[\theta \in [0,2\pi), \qquad \varphi \in [0,2\pi). \label{eq:delta_phase_domain}\] \(\theta\) and \(\varphi\) are used as phases representing two different “constraints.” The types of constraints are fixed below in [D-5.2-4].

7.4.2.2 [D-5.2-2] Canonical measure (uniform measure)

Full-cycle averages are defined by the canonical measure. \[d\mu(\theta):=\frac{d\theta}{2\pi}, \qquad d\mu(\varphi):=\frac{d\varphi}{2\pi}, \qquad \int_{0}^{2\pi} d\mu(\theta)=\int_{0}^{2\pi} d\mu(\varphi)=1. \label{eq:delta_uniform_measure}\] Definition [eq:delta_uniform_measure] is locked as the canonical convention for rectification constants; inserting a weighted measure (non-uniform distribution) after seeing the result is forbidden.

7.4.2.3 [D-5.2-3] Half-wave rectification operator

In sign-constrained events, “survival” is defined as being counted only when the directional component satisfies a particular sign. To implement this, define the half-wave rectification operator by \[_{+}:=\max(0,x). \label{eq:delta_pospart_def}\] Half-wave rectification differs from absolute-value rectification (\(|x|\)). While \(|x|\) removes the sign, \([x]_+\) includes an explicit sign selection (only one half-cycle survives).

7.4.2.4 [D-5.2-4] Two-constraint event (AND coupling of two phases)

Event survival is defined as the simultaneous satisfaction of two independent constraints. The constraints are represented by the following two phases.

  1. Directional-constraint phase \(\theta\): a constraint that an event survives only when it has a “forward” component along a fixed direction (e.g., a chosen axis or a chosen boundary normal).

  2. Internal-constraint phase \(\varphi\): a constraint that an event survives only when it has a “forward” component under an internal structural rule (e.g., a cancellation–survival sign-selection rule for local updates).

The combination of the two constraints is defined as an AND coupling (a product). That is, define the survival weight of an event \(e\) by \[w(e) := \bigl[\cos\theta(e)\bigr]_{+}\,\bigl[\cos\varphi(e)\bigr]_{+}. \label{eq:delta_weight_def}\] In [eq:delta_weight_def], \(w(e)\in[0,1]\), and when \(w(e)=0\) the event contributes nothing to the survival aggregate. Definition [eq:delta_weight_def] is an operational definition of survival and must be locked under the meaning-layer mapping in analysis_lock; changing it after seeing results (e.g., replacing with \(|\cdot|\), adding powers, inserting thresholds, clipping, etc.) is forbidden.

7.4.3 5.2.3 First rectification average: \(\langle [\cos\theta]_{+}\rangle = 1/\pi\)

From [eq:delta_uniform_measure] and [eq:delta_pospart_def], compute \[\left\langle [\cos\theta]_{+}\right\rangle := \frac{1}{2\pi}\int_{0}^{2\pi} [\cos\theta]_{+}\, d\theta \label{eq:delta_beta_def}\] The function \([\cos\theta]_{+}\) equals \(\cos\theta\) only on the positive region of \(\cos\theta\), and is zero on the negative region.

7.4.3.1 (1) Decomposition of the positive region

The region where \(\cos\theta>0\) is \[\theta\in\left[0,\frac{\pi}{2}\right)\ \cup\ \left(\frac{3\pi}{2},2\pi\right]. \label{eq:delta_pos_intervals}\] Therefore, \[_{+} = \begin{cases} \cos\theta, & \theta\in\left[0,\frac{\pi}{2}\right]\cup\left[\frac{3\pi}{2},2\pi\right],\\[4pt] 0, & \theta\in\left[\frac{\pi}{2},\frac{3\pi}{2}\right]. \end{cases} \label{eq:delta_pospart_piecewise}\]

7.4.3.2 (2) Integral evaluation

From [eq:delta_beta_def] and [eq:delta_pospart_piecewise], \[\begin{aligned} \int_{0}^{2\pi} [\cos\theta]_{+}\, d\theta &= \int_{0}^{\pi/2}\cos\theta\, d\theta + \int_{\pi/2}^{3\pi/2}0\, d\theta + \int_{3\pi/2}^{2\pi}\cos\theta\, d\theta \notag\\ &= \left[\sin\theta\right]_{0}^{\pi/2} + 0 + \left[\sin\theta\right]_{3\pi/2}^{2\pi} \notag\\ &= \Bigl(\sin(\pi/2)-\sin(0)\Bigr)+\Bigl(\sin(2\pi)-\sin(3\pi/2)\Bigr) \notag\\ &= (1-0)+(0-(-1)) = 2. \label{eq:delta_pospart_integral_value}\end{aligned}\] Hence \[\left\langle [\cos\theta]_{+}\right\rangle = \frac{1}{2\pi}\cdot 2 = \frac{1}{\pi}. \label{eq:delta_beta_value}\] For convenience, define \[\beta := \left\langle [\cos\theta]_{+}\right\rangle \label{eq:delta_beta_symbol}\] Then by [eq:delta_beta_value], \[\beta=\frac{1}{\pi} \label{eq:delta_beta_final}\] Here \(\beta\) is not reserved as a rectification constant; it is only an intermediate quantity within this section. The final rectification constant is fixed as \(\delta\) below.

7.4.4 5.2.4 Second rectification average: \(\delta=\langle [\cos\theta]_{+}[\cos\varphi]_{+}\rangle = 1/\pi^{2}\)

When the survival weight of an event \(e\) is defined by [eq:delta_weight_def], define the mean survival coefficient \(\delta\) in a regime as \[\delta := \left\langle [\cos\theta]_{+}[\cos\varphi]_{+}\right\rangle. \label{eq:delta_def_double}\] The average is taken over the full-cycle canonical measure on \((\theta,\varphi)\).

7.4.4.1 [A-5.2-U0] (intermediate assumption) product measure of the two phases

The derivation in this section uses the assumption that the canonical measure on \((\theta,\varphi)\) factorizes as a product measure. \[d\mu(\theta,\varphi) := d\mu(\theta)\,d\mu(\varphi) = \frac{d\theta}{2\pi}\frac{d\varphi}{2\pi}. \label{eq:delta_product_measure}\] This assumption is locked only under the “absence of bias sources (default state)” conditions in 5.2.5. If the assumption fails, the universal value of \(\delta\) is not claimed; the case is handled as a falsification trigger (5.2.6).

7.4.4.2 (1) Separation of the double integral

From [eq:delta_def_double] and [eq:delta_product_measure], \[\begin{aligned} \delta &= \int_{0}^{2\pi}\int_{0}^{2\pi} [\cos\theta]_{+}[\cos\varphi]_{+}\, \frac{d\theta}{2\pi}\frac{d\varphi}{2\pi} \notag\\ &= \left(\frac{1}{2\pi}\int_{0}^{2\pi}[\cos\theta]_{+}\, d\theta\right) \left(\frac{1}{2\pi}\int_{0}^{2\pi}[\cos\varphi]_{+}\, d\varphi\right). \label{eq:delta_factorization}\end{aligned}\] Each parenthesis is the same integral as [eq:delta_beta_def], and by [eq:delta_beta_value] each equals \(1/\pi\). Therefore, \[\delta = \left(\frac{1}{\pi}\right)\left(\frac{1}{\pi}\right) = \frac{1}{\pi^{2}}. \label{eq:delta_final}\]

7.4.5 5.2.5 [D] Rectification constant as a maximum-entropy distribution

This section does not introduce \(\delta=1/\pi^{2}\) as an axiom. If the initial condition and the driving protocol do not contain any physical information (forces, constraints, boundary conditions) that would prefer a particular phase \(\theta_{0}\) or \(\varphi_{0}\) as a pre-registered input, then by an information-theoretic principle the phase distribution \(P(\theta)\) must take the form that maximizes the entropy \[H[P] = -\int_{0}^{2\pi} P(\theta)\,\ln P(\theta)\,d\theta\] Hence (under normalization \(\int P=1\)), \[\text{maximize }H[P]\ \Longrightarrow\ P(\theta)=\frac{1}{2\pi}\quad(\text{Uniform}) \label{eq:phase_uniform_distribution}\] which determines the default state. The same logic applies to \(\varphi\).

7.4.5.1 Condition (Gate): “absence of a bias mechanism”

The uniform-distribution conclusion is qualified as a universal constant only when the condition “there is no mechanism that induces bias” is PASS. This document turns it into a Gate via the conditions below (and the falsification triggers in 5.2.6).

  1. [A-5.2-U1] full-cycle uniformity (Null): the distributions of \(\theta,\varphi\) take the uniform distribution of [eq:delta_uniform_measure] as the default state.

  2. [A-5.2-U2] dual constraints: the survival weight is defined by the half-wave-rectified product in [eq:delta_weight_def].

  3. [A-5.2-U3] product measure (uncorrelated): when no bias/constraint information exists, the joint measure is treated as the product measure [eq:delta_product_measure].

  4. [A-5.2-U4] regime fixed: if a bias mechanism (external field, boundary condition, constraint) exists, it must be pre-registered; in that case, the universal use of \(\delta\) is automatically suspended and the trigger checks in 5.2.6 are forced.

Therefore \(\delta=1/\pi^{2}\) is not “let us believe it,” but a default state implied by the absence of bias sources. When bias is detected (5.2.6), universal use is immediately forbidden.

7.4.6 5.2.6 Falsification triggers (broken conditions) and handling rules

The universality axiom set [A-5.2-U] is judged to be broken if any of the following triggers occurs. Each trigger must be pre-registered in gate_lock together with its threshold; post-hoc changes are forbidden.

7.4.6.1 5.2.6.1 Trigger T1: collapse of uniformity (biased phase)

From an event sample, define the following quantities and judge that uniformity is broken if they violate the threshold. \[m_{\theta}:=\left|\frac{1}{N}\sum_{e=1}^{N}\cos\theta(e)\right|, \qquad m_{\varphi}:=\left|\frac{1}{N}\sum_{e=1}^{N}\cos\varphi(e)\right|. \label{eq:delta_bias_metrics}\] With the threshold \(\varepsilon_{\mathrm{bias}}\) locked in gate_lock, \[m_{\theta}>\varepsilon_{\mathrm{bias}} \ \text{or}\ \ m_{\varphi}>\varepsilon_{\mathrm{bias}} \quad\Longrightarrow\quad \texttt{FAIL-RECT-DELTA-BIAS}. \label{eq:delta_trigger_bias}\] If this judgment occurs, record that [A-5.2-U1] is broken, and forbid the universal use of \(\delta\) in that regime.

7.4.6.2 5.2.6.2 Trigger T2: collapse of product measure (correlated phases)

From an event sample, define the following correlation metric. \[u(e):=[\cos\theta(e)]_{+}, \qquad v(e):=[\cos\varphi(e)]_{+}, \qquad C_{uv}:=\left|\frac{1}{N}\sum_{e=1}^{N}u(e)v(e)-\left(\frac{1}{N}\sum_{e=1}^{N}u(e)\right)\left(\frac{1}{N}\sum_{e=1}^{N}v(e)\right)\right|. \label{eq:delta_corr_metric}\] With the threshold \(\varepsilon_{\mathrm{corr}}\) locked in gate_lock, \[C_{uv}>\varepsilon_{\mathrm{corr}} \quad\Longrightarrow\quad \texttt{FAIL-RECT-DELTA-CORR}. \label{eq:delta_trigger_corr}\] If this judgment occurs, record that [A-5.2-U3] is broken, and forbid the universal use of \(\delta\) in that regime.

7.4.6.3 5.2.6.3 Trigger T3: collapse of numerical universality (mismatch of \(\hat{\delta}\))

From an event sample, define the empirical estimator of \(\delta\) by \[\hat{\delta} := \frac{1}{N}\sum_{e=1}^{N} w(e) = \frac{1}{N}\sum_{e=1}^{N} [\cos\theta(e)]_{+}[\cos\varphi(e)]_{+}. \label{eq:delta_hat}\] With the threshold \(\varepsilon_{\delta}\) locked in gate_lock, \[\left|\hat{\delta}-\frac{1}{\pi^{2}}\right|>\varepsilon_{\delta} \quad\Longrightarrow\quad \texttt{FAIL-RECT-DELTA-NUM}. \label{eq:delta_trigger_num}\] If this judgment occurs, record that at least one of [A-5.2-U1]~[A-5.2-U3] fails, or that the survival definition [eq:delta_weight_def] is not maintained in the regime.

7.4.6.4 5.2.6.4 Trigger T4: collapse of the survival definition (procedure change or mixing)

Judge that the survival definition [eq:delta_weight_def] is broken if any of the following occurs.

  1. Mixing the use of \(|x|\) or other nonlinear functions (powers, threshold functions, clipping, etc.) instead of \([x]_{+}\).

  2. Changing the definitions of \(\theta,\varphi\) (coordinate system, representative axis, pre/post event reference) without a lock_id.

  3. Mixing meaning-layer mappings/closures from different analysis_lock_ids within the same output.

In this case, judge immediately as \[\texttt{FAIL-RECT-DELTA-DEF} \label{eq:delta_trigger_def}\] and the output and its derived outputs lose conclusion status.

7.4.6.5 5.2.6.5 Trigger T5: out-of-regime extrapolation (scope violation)

If one uses \(\delta\) as a universal constant as-is in a regime where a drive axis or anisotropy axis such as DRV-ROT is turned on in the regime coordinates, or in a regime where spanning collapses such as \(R_{\mathrm{span}}=\texttt{SPAN-0}\), then judge it as out-of-regime extrapolation. In this case, handle as \[\texttt{FAIL-REG-EXTRAP} \label{eq:delta_trigger_extrap}\] and allow only limit-type conclusions (CT-LIM).

\(\delta\) is used as a “survival coefficient” in the following families of derivations.

  1. Canonical event rate: when event aggregation requires two constraints simultaneously, define the rectified event rate by multiplying the raw event count (count before constraints) by \(\delta\). The use of \(\delta\) is restricted to regimes where [A-5.2-U] holds.

  2. Cancellation–survival family: in discrete shell structures where “cancellation” includes sign selection, the mean surviving contribution is rectified via half-wave averages; when two constraints are simultaneously required, \(\delta\) appears.

  3. Cross-consistency (RCROSS) and Gate: \(\delta\) is not used to justify numerical agreement, but only as a rectification coefficient that is locked inside the regime, serving as an input to cross-consistency and reproduction Gates.

7.5 5.3 Where \(\delta\) enters

7.5.1 5.3.1 Definition of \(\delta\) and prerequisites for use (a complete definition, not a summary)

\(\delta\) is defined as the rectification coefficient that represents the mean surviving fraction in “two-constraint events.” In this section, \(\delta\) is used only on top of the following definitions and prerequisites.

7.5.1.1 [D-5.3-1] Two phase variables and the canonical measure

Define the two phase variables as follows. \[\theta \in [0,2\pi),\qquad \varphi \in [0,2\pi). \label{eq:S05_delta_phase}\] Lock the canonical measure as the uniform full-cycle measure. \[d\mu(\theta)=\frac{d\theta}{2\pi},\qquad d\mu(\varphi)=\frac{d\varphi}{2\pi}. \label{eq:S05_delta_measure}\] The uniform measure is part of the rectification convention and is not modified by inserting weights after seeing the result.

7.5.1.2 [D-5.3-2] Half-wave rectification operator and survival weight

Define the half-wave rectification operator as \[_{+}:=\max(0,x). \label{eq:S05_pospart}\] Define the survival weight of an event \(e\) as \[w(e):=[\cos\theta(e)]_{+}\,[\cos\varphi(e)]_{+}. \label{eq:S05_weight}\] Definition [eq:S05_weight] is the operational definition of “survival” and cannot be replaced by \(|\,\cdot\,|\) or other nonlinear functions within the same version.

7.5.1.3 [D-5.3-3] Definition of \(\delta\)

Define \(\delta\) as the mean of the survival weight. \[\delta := \left\langle w \right\rangle = \int_{0}^{2\pi}\!\!\int_{0}^{2\pi} [\cos\theta]_{+}[\cos\varphi]_{+}\, d\mu(\theta)\,d\mu(\varphi). \label{eq:S05_delta_def}\] Moreover, in a regime where the following universality prerequisite (product measure) holds and is locked, \[d\mu(\theta,\varphi)=d\mu(\theta)\,d\mu(\varphi) \label{eq:S05_product_measure}\] it is fixed as \[\delta=\frac{1}{\pi^{2}} \label{eq:S05_delta_value}\] Using [eq:S05_delta_value] is forbidden in regimes where the universality prerequisite does not hold; in that case, treat \(\delta\) only as a regime-dependent estimator (and lock the estimator and thresholds via closures and Gate).

7.5.2 5.3.2 \(\delta\) in event rates (definition–insertion–result form)

7.5.2.1 5.3.2.1 Definition: raw event count and raw event rate

Choose a tick window \([n_1,n_2)\) and define the realized time length of the window as \[\Delta N := n_2-n_1, \qquad \Delta T := \Delta N\,\Delta t. \label{eq:S05_time_window}\] Here, \(\Delta t\) is the realized time tick locked in realization_lock.

Define “raw events” in the tick window as \[\mathcal{E}_{0}[n_1,n_2) :=\{\, e\ |\ n_1\le n(e)<n_2\,\}. \label{eq:S05_event_set_raw}\] Define the raw event count and the raw event rate as \[N_{0} :=|\mathcal{E}_{0}[n_1,n_2)|, \qquad \nu_{0} :=\frac{N_{0}}{\Delta T}. \label{eq:S05_raw_rate}\] \(\nu_{0}\) is the event rate before applying the constraints.

7.5.2.2 5.3.2.2 Insertion: aggregating the two-constraint survival weight

For the raw event set \(\mathcal{E}_{0}[n_1,n_2)\), define the “rectified event count” by applying the survival weight [eq:S05_weight]: \[N_{\delta} := \sum_{e\in\mathcal{E}_{0}[n_1,n_2)} w(e) = \sum_{e\in\mathcal{E}_{0}[n_1,n_2)} [\cos\theta(e)]_{+}[\cos\varphi(e)]_{+}. \label{eq:S05_Ndelta}\] Definition [eq:S05_Ndelta] is the minimal aggregation that records “event survival” numerically; the definitions of \(\theta(e)\) and \(\varphi(e)\) (coordinate system/axis/representation convention) must be locked in analysis_lock.

In regimes where the universality prerequisites (uniform measure and product measure) are locked, fix the following equivalence as part of the rectification convention: \[N_{\delta} \equiv \delta\,N_{0}. \label{eq:S05_Ndelta_equiv}\] Here, \(\delta\) may be locked to [eq:S05_delta_value]; when locked, \(\delta\) is referenced not as “a number inserted ad hoc,” but as a registry entry of the rectification convention.

7.5.2.3 5.3.2.3 Result form: rectified event rate

Define the rectified event rate as \[\nu := \frac{N_{\delta}}{\Delta T}. \label{eq:S05_rect_rate_def}\] In regimes where the universality prerequisites hold so that [eq:S05_Ndelta_equiv] is allowed, \[\nu = \frac{\delta\,N_{0}}{\Delta T} = \delta\,\nu_{0}. \label{eq:S05_rect_rate_result}\] Therefore, the role of \(\delta\) in event rates is the “survival rectification” of the raw event rate \(\nu_{0}\), and the insertion point is locked either as [eq:S05_Ndelta_equiv] or as the result equation [eq:S05_rect_rate_result].

7.5.3 5.3.3 \(\delta\) in effective velocity (definition–insertion–result form)

7.5.3.1 5.3.3.1 Definition: direction-constrained displacement and raw forward speed

Lock the propagation (or transport) direction as a unit vector \(\mathbf{n}_v\): \[\|\mathbf{n}_v\|=1, \qquad \mathbf{n}_v\ \text{is locked in }\texttt{analysis\_lock}. \label{eq:S05_nv_lock}\] When an event \(e\) occurs in the tick window \([n_1,n_2)\), define the “forward displacement” (from pre/post configurations) as \[\Delta \tilde{x}_{\parallel}(e) := \mathbf{n}_v\cdot\Bigl(\tilde{\mathbf{x}}_{\mathrm{tag}}^{\mathrm{post}}(e)-\tilde{\mathbf{x}}_{\mathrm{tag}}^{\mathrm{pre}}(e)\Bigr), \label{eq:S05_dx_tilde}\] where \(\tilde{\mathbf{x}}_{\mathrm{tag}}\) is the representative coordinate locked for the event (e.g., a core marker, a shell-survival marker, or a cell marker). The convention of choosing the representative coordinate must be locked in analysis_lock. \(\Delta \tilde{x}_{\parallel}(e)\) is the forward displacement in internal units (dimensionless or internal-length units); the realized length is mapped via \(a\) in [eq:S05_dx_real].

Define the sum of raw forward displacements as \[\Delta \tilde{X}_{0} := \sum_{e\in\mathcal{E}_{0}[n_1,n_2)} \Delta \tilde{x}_{\parallel}(e). \label{eq:S05_sum_dx_raw}\] Define the raw internal forward speed as \[\tilde{v}_{0} := \frac{\Delta \tilde{X}_{0}}{\Delta N}. \label{eq:S05_vtilde0}\] \(\tilde{v}_{0}\) is the raw definition in which every event is credited equally for forward transport; two-constraint survival has not yet been applied.

7.5.3.2 5.3.3.2 Insertion: rectifying forward displacement by survival weights

Define the rectified displacement by inserting the survival weights into the forward displacement sum: \[\Delta \tilde{X}_{\delta} := \sum_{e\in\mathcal{E}_{0}[n_1,n_2)} w(e)\,\Delta \tilde{x}_{\parallel}(e). \label{eq:S05_sum_dx_rect}\] Define the corresponding rectified internal forward speed as \[\tilde{v} := \frac{\Delta \tilde{X}_{\delta}}{\Delta N}. \label{eq:S05_vtilde_rect_def}\] If the universality prerequisite holds and, moreover, the forward displacement \(\Delta \tilde{x}_{\parallel}(e)\) is separable (on average) from the survival phases within the regime (this independence must be locked as a regime condition in analysis_lock), then the following rectification closure is allowed: \[\Delta \tilde{X}_{\delta}\equiv \delta\,\Delta \tilde{X}_{0}. \label{eq:S05_dx_rect_equiv}\] This equivalence does not hold “always”; it is a closure that holds only when the regime condition is locked. Using it out of regime is forbidden.

7.5.3.3 5.3.3.3 Result form: realized effective velocity

Use the realization map between realized length \(x\) and internal length \(\tilde{x}\) as \[x := a\,\tilde{x}. \label{eq:S05_dx_real}\] Realized time is \(t:=\Delta t\,\tilde{t}\), and in the tick window \(\Delta T=\Delta N\,\Delta t\) by definition [eq:S05_time_window].

Define the realized effective velocity as \[v_{\mathrm{eff}} := \frac{\Delta X_{\delta}}{\Delta T} = \frac{a\,\Delta \tilde{X}_{\delta}}{\Delta N\,\Delta t} = \frac{a}{\Delta t}\,\tilde{v}. \label{eq:S05_veff_def}\] If the universality and independence conditions hold so that [eq:S05_dx_rect_equiv] is allowed, \[v_{\mathrm{eff}} = \frac{a}{\Delta t}\,\delta\,\tilde{v}_{0} = \delta\,v_{0}, \qquad v_{0}:=\frac{a}{\Delta t}\tilde{v}_{0}. \label{eq:S05_veff_result}\] Therefore, the role of \(\delta\) in effective velocity is “survival rectification of forward contribution,” and the insertion point is locked either at the displacement sum [eq:S05_sum_dx_rect] or at the result equation [eq:S05_veff_result].

7.5.4 5.3.4 \(\delta\) in mass derivation (definition–insertion–result form)

7.5.4.1 5.3.4.1 Definition: operational definition of a mass scale (event–geometry coupling)

In this document, “mass” is not introduced as a justification from external doctrine. It is introduced only through the operational definition below. Let the realized unit energy \(U_{\mathrm{lat}}\) be the scale locked in realization_lock. The numerical recipe for generating \(U_{\mathrm{lat}}\) is outside the scope of this section; here we use only the fact that \(U_{\mathrm{lat}}\) is a locked unit with “energy dimension.”

For an object \(\mathcal{O}\), define the “mass scale” \(m(\mathcal{O})\) as \[m(\mathcal{O}) := U_{\mathrm{lat}}\, \Lambda(\mathcal{O}), \label{eq:S05_mass_def}\] where \(\Lambda(\mathcal{O})\) is a dimensionless “geometry–event coupling coefficient.” Freeze the standard form as the following product: \[\Lambda(\mathcal{O}) := \Gamma(\mathcal{O})\, \Xi(\mathcal{O}). \label{eq:S05_lambda_factor}\]

Freeze the insertion point of \(\delta\) in mass derivation to be inside \(\Xi(\mathcal{O})\).

7.5.4.2 5.3.4.2 Insertion: inserting \(\delta\) into the event coefficient \(\Xi(\mathcal{O})\)

For an object \(\mathcal{O}\), define the raw event coefficient as \[\Xi_{0}(\mathcal{O}) := \frac{N_{0}(\mathcal{O})}{N_{\mathrm{ref}}}, \label{eq:S05_Xi0}\] where \(N_{0}(\mathcal{O})\) is the “raw event count attributed to object \(\mathcal{O}\)” defined analogously to [eq:S05_event_set_raw], and \(N_{\mathrm{ref}}\) is the locked reference count in the same time window. The choice of the reference count must be pre-registered; post-hoc changes are forbidden.

Define the rectified event coefficient as \[\Xi(\mathcal{O}) := \frac{N_{\delta}(\mathcal{O})}{N_{\mathrm{ref}}}, \qquad N_{\delta}(\mathcal{O}) := \sum_{e\in\mathcal{E}_{0}(\mathcal{O})} w(e). \label{eq:S05_Xi_rect}\] In regimes where the universality prerequisite holds so that \(N_{\delta}(\mathcal{O})\equiv \delta N_{0}(\mathcal{O})\) is allowed, \[\Xi(\mathcal{O}) \equiv \delta\,\Xi_{0}(\mathcal{O}). \label{eq:S05_Xi_delta_insert}\] This is locked as the insertion point of \(\delta\) in mass derivations.

7.5.4.3 5.3.4.3 Result form: mass scale with \(\delta\) included

From [eq:S05_mass_def]–[eq:S05_lambda_factor] and [eq:S05_Xi_delta_insert], in regimes where the universality prerequisite holds, \[m(\mathcal{O}) = U_{\mathrm{lat}}\, \Gamma(\mathcal{O})\,\Xi(\mathcal{O}) \equiv U_{\mathrm{lat}}\, \Gamma(\mathcal{O})\,\delta\,\Xi_{0}(\mathcal{O}). \label{eq:S05_mass_result}\] Therefore, in mass derivations \(\delta\) enters only as “survival rectification of the event coefficient” and cannot be used as a term that adjusts the geometric coefficient \(\Gamma(\mathcal{O})\). Absorbing \(\delta\) into the geometric coefficient, or changing the definition of \(\Gamma(\mathcal{O})\) to eliminate \(\delta\), is forbidden.

7.5.5 5.3.5 \(\delta\) in force derivation (definition–insertion–result form)

7.5.5.1 5.3.5.1 Definition: unit force and directional event flux

Fix the unit force \(F_{\mathrm{lat}}\) by the following derived definition. \[F_{\mathrm{lat}} := \frac{U_{\mathrm{lat}}}{a}. \label{eq:S05_Flat}\] Here, \(U_{\mathrm{lat}}\) and \(a\) are realized scales locked in realization_lock. Definition [eq:S05_Flat] is an internal definition of “unit energy / unit length” and is not grounded on external doctrine.

Define the directional event flux as follows. Lock the direction as a unit vector \(\mathbf{n}_F\). \[\|\mathbf{n}_F\|=1, \qquad \mathbf{n}_F\ \text{is locked in }\texttt{analysis\_lock}. \label{eq:S05_nF}\] For each event \(e\), define the “directional contribution sign” \(s_F(e)\in\{-1,0,+1\}\) by \[s_F(e) := \mathrm{sgn}\!\left(\Delta \tilde{x}_{\parallel,F}(e)\right), \qquad \Delta \tilde{x}_{\parallel,F}(e) := \mathbf{n}_F\cdot\Bigl(\tilde{\mathbf{x}}_{\mathrm{tag}}^{\mathrm{post}}(e)-\tilde{\mathbf{x}}_{\mathrm{tag}}^{\mathrm{pre}}(e)\Bigr), \label{eq:S05_sF}\] where the choice convention of \(\tilde{\mathbf{x}}_{\mathrm{tag}}\) must be locked. Define the raw directional aggregate as \[S_{0} := \sum_{e\in\mathcal{E}_{0}[n_1,n_2)} s_F(e). \label{eq:S05_S0}\] \(S_0\) is the net sign sum of directional events and is dimensionless.

7.5.5.2 5.3.5.2 Insertion: inserting survival weights into the force-direction aggregate

Define the rectified aggregate by inserting the survival weight into the directional aggregation: \[S_{\delta} := \sum_{e\in\mathcal{E}_{0}[n_1,n_2)} w(e)\, s_F(e). \label{eq:S05_Sdelta}\] If the universality prerequisite holds and, moreover, the sign \(s_F(e)\) is separable (on average) from the survival phases within the regime (this independence must be locked as a regime condition), then the following rectification closure is allowed: \[S_{\delta}\equiv \delta\,S_{0}. \label{eq:S05_Sdelta_equiv}\] This equivalence does not hold automatically out of regime; using it out of regime is forbidden as extrapolation.

7.5.5.3 5.3.5.3 Result form: force scale with \(\delta\) included

Define the force scale as \[F := F_{\mathrm{lat}}\, \frac{S_{\delta}}{\Delta N}. \label{eq:S05_force_def}\] Here, \(\Delta N\) is the tick length (definition [eq:S05_time_window]), and \(S_{\delta}/\Delta N\) is the “directional survival aggregate per tick.” If the universality and independence conditions hold so that [eq:S05_Sdelta_equiv] is allowed, \[F \equiv F_{\mathrm{lat}}\, \delta\, \frac{S_{0}}{\Delta N} = \delta\,F_{0}, \qquad F_{0}:=F_{\mathrm{lat}}\frac{S_{0}}{\Delta N}. \label{eq:S05_force_result}\] Therefore, the role of \(\delta\) in force derivations is “survival rectification of the directional-event aggregate,” and the insertion point is locked either at [eq:S05_Sdelta] or at the result equation [eq:S05_force_result].

7.5.6 5.3.6 Common prohibitions for \(\delta\) insertion (no reverse insertion / absorption / redefinition)

The \(\delta\) insertion fixed in this section comes with the following prohibitions.

  1. No reverse insertion: it is forbidden to redefine raw quantities (\(N_0,\nu_0,\tilde{v}_0,\Xi_0,S_0\), etc.) in order to remove \(\delta\) from a result equation.

  2. No absorption: it is forbidden to absorb \(\delta\) into the definition of a geometric coefficient \(\Gamma(\mathcal{O})\) or into the definition of unit scales (\(a,\Delta t,U_{\mathrm{lat}}\)) to make \(\delta\) disappear superficially.

  3. No redefinition: it is forbidden to redefine \(\delta\) by swapping event definitions, averaging conventions, or threshold conventions. Redefinition exists only as a version update, and a version update requires a full re-derivation / full re-judgment.

  4. No out-of-regime extrapolation: it is forbidden to use [eq:S05_delta_value] in a regime where the universality prerequisite does not hold, or to automatically apply [eq:S05_Ndelta_equiv], [eq:S05_dx_rect_equiv], [eq:S05_Sdelta_equiv].

8 6. Continuum Core Model: Deriving \(R_p\)

Purpose (locking the outputs)

The purpose of this chapter is to construct a continuum (continuous-approximation) core model that determines the core radius \(R_p\) using internal definitions only, and to derive and lock the following outputs.

  1. The defining expression of the core radius, \(R_p:=\mathcal{R}[\text{core state}]\) (including definition and regime).

  2. A formal statement of the core stability condition (locked as one of: rest condition / equilibrium condition / switch condition).

  3. The dimensionless conclusion for the length-selection ratio: \[\frac{R_p}{L_q}=\alpha=\frac{2}{\pi}, \label{eq:S06_goal_ratio}\] where \(L_q\) is the core selection length (an internal length scale) and \(\alpha\) is the rectification constant whose unique source is fixed in Chapter 5.

  4. The insertion points for downstream derivations (event rates, mass/force scales) and an explicit statement of forbidden retroactive justification.

This chapter does not introduce \(R_p\) by appealing to external texts. The only admissible inputs are internal definitions and internal conventions locked in canon_lock/analysis_lock. If a comparison to external numerics is required, that comparison is treated only as a validation (Gate) item; it is never used as a premise of the derivation.

Inputs (LOCK) and regime (scope)

The derivation of \(R_p\) is carried out only when the following input items are locked.

Substrate (Stone) regime and VP axioms

This chapter assumes the VP axiom set locked in §3.1 (infinite rigidity / impenetrability / plenum / local rule / adjacency). In particular, we fix the following two conditions as regime conditions.

  1. Stone regime: the volume invariance and impenetrability of VP are maintained as global constraints.

  2. Plenum regime: no “empty space” is introduced as an independent degree of freedom; “deficits/throats/gaps” at the core boundary appear only as derived structural quantities of placement and adjacency.

If these regime conditions are not locked, the continuum approximation in this chapter loses its scope, and its conclusions are judged INCONCLUSIVE.

Canonical cell (Anchor Cell) and length scales

The canonical cell is locked to CELL-CUBE, and the representative cell length \(D_{\mathrm{anch}}\) is locked as the edge length (edge). The half-length scale must be fixed by the following derived definition. \[r_0:=\frac{D_{\mathrm{anch}}}{2}. \label{eq:S06_r0_lock}\] The continuum coordinate \(R\) in this chapter is used as the radial distance inside the canonical cell; the unit and coordinate convention of \(R\) are locked by the coordinate convention in analysis_lock.

Rectification constant \(\alpha\) as an input

This chapter uses, as an input, the rectification constant whose unique source is fixed in §5.1: \[\alpha=\frac{2}{\pi}. \label{eq:S06_alpha_lock}\] \(\alpha\) is not re-derived in this chapter; it is inserted only into the dimensionless ratio conclusion [eq:S06_goal_ratio] for core length selection.

Status of the core selection length \(L_q\)

\(L_q\) is an internal length scale defined as the “selection length” in the core model. The status of \(L_q\) must be locked as one of the following.

  1. CANON-INPUT: lock \(L_q\) as a canonical input in canon_lock (including meaning/unit/scope).

  2. CANON-DERIVED: define \(L_q\) as a canonical derived quantity and lock its derivation rule (from which structural quantity it is derived and how) in analysis_lock.

If the status of \(L_q\) is not locked, the dimensionless conclusion \(R_p/L_q\) is itself undefined, and the derivation in this chapter does not stand.

Internal definition of the continuum core model (a continuous approximation of geo-structure)

In this chapter, “continuum” does not mean importing an external field theory. It means an internal procedure that aggregates discrete structural quantities of VP placement into functions of a radial variable \(R\). The continuous approximation is fixed by the following rules.

Definition of the core region and the core boundary

The core is a region defined with respect to the domain center \(\mathbf{x}_c\). The core boundary is defined by the following switch condition.

  1. For radius \(R\), define an indicator \(\chi_{\mathrm{core}}(R)\in\{0,1\}\) that judges whether the discrete structural index inside radius \(R\) satisfies a “rigidity/transfer” condition.

  2. Define the core radius \(R_p\) as a transition point of \(\chi_{\mathrm{core}}(R)\). The discrete/continuous decision rule for the transition point (first transition / threshold transition / stable transition) must be locked in analysis_lock.

Therefore \(R_p\) is not an “arbitrary geometric length” but an operationally defined length determined by a transition of a discrete structural indicator.

Standard form of radial aggregates

A radial aggregate \(S(R)\) at radius \(R\) is defined only in the following standard form. \[S(R) := \mathcal{A}\Bigl(\{\Omega_i\}_{i\in\mathcal{V}(R)},\ \mathcal{G}_c(R),\ \mathcal{P}(R)\Bigr), \label{eq:S06_radial_aggregate}\] where

All continuum expressions in this chapter must be reducible to the form [eq:S06_radial_aggregate]; if the reduction rule is not locked, the continuum expressions are undefined.

Internal definition of the stability condition (selection radius as an intersection of two scaling laws)

This chapter adopts the internal principle that the core radius \(R_p\) is selected as the intersection of two radial scaling laws. This is not an external justification; it is implemented by defining the scaling laws of the “aggregate cost” at the core boundary under the VP Stone and plenum constraints, and then locking their intersection as the selection rule.

Where the two aggregate cost functions are defined

For radius \(R\), define the following two costs (or pressure-like scalar aggregates): \[\Pi_{4}(R) := \frac{C_{4}}{R^{4}}, \qquad \Pi_{5}(R) := \frac{C_{5}}{R^{5}}, \label{eq:S06_two_scalings}\] where \(C_{4},C_{5}\) are constants derived from internal aggregation (or locked coefficients), and the exponents \(R^{-4}\) and \(R^{-5}\) are derived and locked internally from domain/boundary/counting conventions. This overview fixes only the definition slot and role of [eq:S06_two_scalings]; the explicit derivation of \(C_{4},C_{5}\) and the internal derivation of the exponents are completed in later sections of this chapter.

Stability condition for the selection radius (core radius)

The selection (stability) condition for the core radius \(R_p\) must be locked as one of the following (exclusive choice; forbidden to swap depending on results).

  1. Intersection condition: \(\Pi_{4}(R_p)=\Pi_{5}(R_p)\).

  2. Rest condition: for some locked energy-like aggregate \(U(R)\), \(dU/dR|_{R=R_p}=0\), and \(U\) is defined so that this condition is equivalent to the intersection in [eq:S06_two_scalings].

  3. Switch condition: define a switch observable so that the transition point of a structural indicator \(\chi_{\mathrm{core}}(R)\) coincides with the intersection point of [eq:S06_two_scalings].

The canonical choice of this chapter is the intersection condition. Under the intersection condition, \[\Pi_{4}(R_p)=\Pi_{5}(R_p) \ \Longrightarrow\ R_p=\frac{C_{5}}{C_{4}}. \label{eq:S06_Rp_Cratio}\] Therefore the dimensionless ratio conclusion [eq:S06_goal_ratio] is completed once the internal derivation of \(C_{5}/C_{4}\) is completed.

8.1 6.5 Allowed conclusion forms in this chapter (implementation of the ban on external justification)

The conclusions of this chapter are permitted only in the following forms.

  1. Derivations of dimensionless ratios such as \(R_p/L_q\) (including insertion of the rectification constant \(\alpha\)).

  2. Internal length-selection formulas of the form \(R_p=\alpha L_q\) (only when the status of \(L_q\) is locked).

  3. Numerical values may be presented only when REALIZATION or validation (Gate) items are locked; numerical agreement must not be used to justify a definition/axiom/rectification convention.

Therefore this chapter does not use external doctrines (equations of other theories, definitions of other constants, or external justifications) as grounds for the derivation of \(R_p\). Correspondence with external texts is handled only in a separate “correspondence table / non-use scope declaration” section and does not affect the conclusion status of this chapter.

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8.2 6.1 Linking \(L_q\) and \(\lambda_C\) (\(L_q=\lambda_C\))

8.2.1 6.1.1 Purpose

This section fixes the two length symbols \(L_q\) and \(\lambda_C\) appearing in the continuum core model as internal definitions, and then confirms the identification rule \[L_q=\lambda_C \label{eq:S06_Lq_eq_lC}\] as a canonical lock (LOCK). The outputs of this section are (i) the definition of \(L_q\), (ii) the definition of \(\lambda_C\), (iii) the internal basis for the identification (axiom-based), and (iv) the substitution rules after the identification.

8.2.2 6.1.2 Definition: \(L_q\) (core selection length)

8.2.2.1 [D-6.1-1] Dimensionless radial coordinate

Lock the core center \(\mathbf{x}_c\) inside the canonical cell and define the radial distance as \[R := \|\mathbf{x}-\mathbf{x}_c\|. \label{eq:S06_R_def}\] The coordinate system of \(R\) and the selection convention of \(\mathbf{x}_c\) are locked in analysis_lock.

The radial description of the core model proceeds in a dimensionless coordinate \(\xi\); define its normalization length as \(L_q\): \[\xi := \frac{R}{L_q}. \label{eq:S06_xi_def}\]

8.2.2.2 [D-6.1-2] Meaning of \(L_q\) (selection length)

Define \(L_q\) as the unique length scale in the core model that simultaneously satisfies the following two conditions.

  1. (Normalization) It is the normalization reference length such that every radial aggregate of the core model can be expressed not as a function of \(R\) but only as a function of \(\xi=R/L_q\).

  2. (Transition) It is the selection length such that the boundary transition of the core (a transition of the core indicator or a transition of a cost function) occurs at a single value \(\xi=\xi_p\).

Therefore \(L_q\) is not “an arbitrary length,” but is defined as “the normalization length that fixes the core transition at a single dimensionless value.” This is an internal definition needed for the continuum core model to stand and does not use external-text justification.

8.2.3 6.1.3 Definition: \(\lambda_C\) (core phase-completion length)

8.2.3.1 [D-6.1-3] Core phase variable

Define a phase variable \(\psi(R)\) describing a radial unfolding of an internal core state as \[\psi:\ [0,\infty)\to\mathbb{R}, \qquad \psi(0)=0, \label{eq:S06_phasepsi_def}\] where \(\psi\) is an internal variable used to represent a “phase cycle” inside the core. The protocol for generating \(\psi\) (which aggregate is used as a phase) is locked in analysis_lock. This section uses only (i) the existence of \(\psi\) as an internal variable of the continuum core model and (ii) the fact that “cycle completion” is locked at \(2\pi\).

8.2.3.2 [D-6.1-4] Meaning of \(\lambda_C\) (first cycle-completion radius)

Define \(\lambda_C\) as the first positive length that satisfies \[\lambda_C := \inf\{\, R>0\ |\ \psi(R)=2\pi\,\}. \label{eq:S06_lambdaC_def}\] In [eq:S06_lambdaC_def], \(2\pi\) is the full-cycle canonical constant locked in Chapter 5 (the reference used by the rectification convention), and \(\lambda_C\) is the length scale at which that full cycle first completes in the radial unfolding of the core. Therefore \(\lambda_C\) is not an “external constant” but an “internal length defined by core phase completion.”

8.2.4 6.1.4 Basis for the identification (axiom-based)

The identification \(L_q=\lambda_C\) is not an extra assumption but a canonical selection (lock) needed to satisfy the global rules of this document (No-Tuning, SSOT, single-transition definition). The basis consists of the following four items.

8.2.4.1 (G1) Single-transition principle: the core boundary must be unique

The core radius \(R_p\) is defined as a single boundary indicating the transition “core interior\(\rightarrow\)core exterior” (a transition of the core indicator or a transition of a cost function). If \(L_q\) and \(\lambda_C\) exist as independent lengths, the length that normalizes the core transition (\(L_q\)) and the length that defines phase completion (\(\lambda_C\)) can point to different transitions. Then the definition of the core boundary becomes duplicated, \(R_p\) cannot be fixed as a single length, and the core boundary becomes an ambiguous object depending on procedures. Therefore, to preserve the definition that the core boundary is unique, the two lengths must be identified so that they point to the same boundary.

8.2.4.2 (G2) SSOT principle: keep only one length for the same meaning

\(L_q\) and \(\lambda_C\) are defined by [eq:S06_xi_def] and [eq:S06_lambdaC_def], respectively, but the physical meaning they indicate must converge to the same slot (the core selection length). In the core model, “radial scale selection” and “phase-cycle completion” both indicate the point where internal organization ends and a boundary forms. If two lengths that indicate the same slot are kept simultaneously, the same meaning is split across multiple symbols, violating SSOT. To satisfy SSOT, the two symbols must be unified into one canonical entry; the unification rule is [eq:S06_Lq_eq_lC].

8.2.4.3 (G3) No-Tuning principle: do not leave \(L_q/\lambda_C\) as a tunable degree of freedom

If \(L_q\) and \(\lambda_C\) are treated as independent inputs, an additional dimensionless degree of freedom appears: \[\eta := \frac{L_q}{\lambda_C}. \label{eq:S06_eta_def}\] This ratio \(\eta\) can directly enter the core-radius ratios \(R_p/L_q\), \(R_p/\lambda_C\) and subsequent derivations (event rate/mass/force), opening a path to adjust \(\eta\) to match outcomes. However, this document forbids adjusting degrees of freedom after seeing results (No-Tuning). To close the core selection without introducing a free knob, \(\eta\) must be fixed by a lock. This section declares \[\eta := 1, \label{eq:S06_eta_one}\] which is equivalent to [eq:S06_Lq_eq_lC].

8.2.4.4 (G4) Compatibility with the unique source of rectification constants: do not introduce a new universal constant beyond \(\pi\)

In Chapter 5, the rectification constants \(\alpha=2/\pi\) and \(\delta=1/\pi^2\) are locked with a unique source. If one separates \(L_q\) and \(\lambda_C\), one needs an additional constant or an additional convention to determine \(\eta\) in [eq:S06_eta_def]. This document does not introduce a new universal constant beyond the rectification-constant system. Therefore the only canonical choice that removes redundancy in the core selection length is \(\eta=1\), i.e., \(L_q=\lambda_C\).

8.2.5 6.1.5 Substitution rules after the identification

After the identification [eq:S06_Lq_eq_lC] is locked, fix the following reuse rules globally.

  1. \(L_q\) and \(\lambda_C\) are two notations for the same entry, and canon_lock records them as a single entry (single value / single meaning under one lock_id).

  2. In formula manipulations, any occurrence of \(L_q\) may be replaced by \(\lambda_C\), and vice versa. The replacement is only a notational substitution following a defined identification; it is not a new derivation or a new premise.

  3. Within the same version, it is forbidden to treat \(L_q\) and \(\lambda_C\) as different values or to split them into different object assignments/geometric meanings.

  4. To release the identification or to change to a different ratio (\(\eta\neq 1\)), one must version-bump canon_lock and perform full re-derivation / full re-validation.

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8.3 6.2 Balance of \(1/R^{4}\) vs \(1/R^{5}\) \(\rightarrow R_p/L_q=2/\pi\)

8.3.1 6.2.1 Radial aggregation geometry and standard symbols

Lock the core center as \(\mathbf{x}_c\) and define the radial distance by \[R:=\|\mathbf{x}-\mathbf{x}_c\| \label{eq:S06_02_R}\] \(R\) is an aggregation coordinate inside the canonical cell (CELL-CUBE); the cell geometry is not replaced by a sphere. Define the radial aggregation surface (a level set) at radius \(R\) by \[\mathbb{S}_R := \{\mathbf{x}\ |\ \|\mathbf{x}-\mathbf{x}_c\|=R\}, \qquad A(R):=\mathrm{Area}(\mathbb{S}_R)=4\pi R^2. \label{eq:S06_02_sphere_area}\] Here \(A(R)\) is the area of the radial aggregation surface; it is a definition of the aggregation surface and is independent of the canonical cell definition (cube).

The core selection length \(L_q\) is the length scale locked in §6.1. In this section, we use \(L_q\) in two roles.

  1. Unit length for radial aggregation: the reference length for making \(R\) dimensionless.

  2. Unit patch length on the radial aggregation surface: the minimal linear scale used to partition the aggregation surface.

Fix the unit patch linear size as \(L_q\) and define the unit patch area as \[A_0 := L_q^2. \label{eq:S06_02_patch_area}\] \(A_0\) is the minimal aggregation unit on the radial aggregation surface and must not be changed after seeing results.

8.3.2 6.2.2 Insertion point of the direction rectification factor \(\alpha\)

When a directional component is aggregated into a scalar in radial aggregation, we use the direction rectification factor \(\alpha\) as an input from the unique source in §5.1: \[\alpha=\left\langle |\cos\theta| \right\rangle=\frac{2}{\pi}. \label{eq:S06_02_alpha}\] Here \(\theta\) is an angular variable aggregated with the uniform measure over one full cycle \([0,2\pi)\); the averaging convention is locked in §5.1. In this section, \(\alpha\) is used only as the rectification factor meaning “the effective contribution of the radial component survives only by a factor \(\alpha\) on average”; \(\alpha\) itself is not re-derived here.

8.3.3 6.2.3 Definition and expansion of the \(1/R^{4}\) collapse term \(\Pi_{4}(R)\)

This section defines the “collapse term” (a pressure-like constraint indicator) \(\Pi_{4}(R)\) at the core boundary by two rounds of geometric dilution and an inverse correction for rectification loss.

8.3.3.1 6.2.3.1 Geometric dilution 1: area dilution (number of patches)

Partition the aggregation surface \(\mathbb{S}_R\) at radius \(R\) by the unit patch area \(A_0\). Define the number of patches as \[N_A(R) := \frac{A(R)}{A_0}=\frac{4\pi R^2}{L_q^2}. \label{eq:S06_02_NA}\] Therefore the area fraction of a single patch on the full aggregation surface (the area-dilution factor) is \[f_A(R) := \frac{1}{N_A(R)}=\frac{A_0}{A(R)}=\frac{L_q^2}{4\pi R^2}. \label{eq:S06_02_fA}\] We fix \(f_A(R)\) as a purely geometric factor representing “the average share received by a unit patch” on the aggregation surface.

8.3.3.2 6.2.3.2 Geometric dilution 2: angular dilution (directional window to hit a fixed patch)

Define the degree of directional alignment toward a fixed patch (linear size \(L_q\)) as the “directional window” to hit that patch. At radius \(R\), viewing a patch of linear size \(L_q\) on the unit sphere, define a representative polar-angle width by \[\Delta\vartheta(R) := \frac{L_q}{R} \label{eq:S06_02_dtheta}\] and define the corresponding representative solid angle by \[\Delta\Omega(R) := \bigl(\Delta\vartheta(R)\bigr)^2 = \left(\frac{L_q}{R}\right)^2. \label{eq:S06_02_dOmega}\] Normalize the full directional space by the total solid angle \(4\pi\). The angular dilution factor is then \[f_\Omega(R) := \frac{\Delta\Omega(R)}{4\pi} = \frac{1}{4\pi}\left(\frac{L_q}{R}\right)^2. \label{eq:S06_02_fOmega}\] We fix \(f_\Omega(R)\) as the geometric factor representing “the directional window required by a unit patch.”

8.3.3.3 6.2.3.3 Rectification-loss correction (inverse factor \(1/\alpha\))

Even when the direction is aligned, the radial component is a signed projection so cancellations occur on average. Since the effective contribution of the radial component survives only by \(\alpha\) ([eq:S06_02_alpha]), an inverse correction factor \(1/\alpha\) is required to secure the same “effective contribution.” This correction is not an ad hoc fit but a direct consequence of the rectification convention (§5.1).

8.3.3.4 6.2.3.4 Definition of the collapse term \(\Pi_{4}(R)\)

Define the collapse term (pressure-like constraint indicator) \(\Pi_{4}(R)\) at the core boundary as \[\Pi_{4}(R) := \Pi_\star\, \frac{1}{\alpha}\, f_A(R)\, f_\Omega(R), \label{eq:S06_02_Pi4_def}\] where \(\Pi_\star\) is the “unit constraint strength” locked inside the regime (dimensionless or internal unit) and is a reference value independent of \(R\). \(\Pi_\star\) is a common factor that cancels in this section; its magnitude does not affect the ratio conclusion.

Substitute [eq:S06_02_fA] and [eq:S06_02_fOmega] into [eq:S06_02_Pi4_def] to fully expand: \[\begin{aligned} \Pi_{4}(R) &= \Pi_\star\, \frac{1}{\alpha}\, \left(\frac{L_q^2}{4\pi R^2}\right) \left(\frac{1}{4\pi}\left(\frac{L_q}{R}\right)^2\right) \notag\\ &= \Pi_\star\, \frac{1}{\alpha}\, \left(\frac{L_q^2}{4\pi R^2}\right) \left(\frac{L_q^2}{4\pi R^2}\right) \notag\\ &= \Pi_\star\, \frac{1}{\alpha}\, \frac{L_q^4}{(4\pi)^2 R^4} \notag\\ &= \Pi_\star\, \frac{1}{\alpha}\, \frac{L_q^4}{16\pi^2}\, \frac{1}{R^4}. \label{eq:S06_02_Pi4_final}\end{aligned}\] Therefore the collapse term is fixed to the \(1/R^4\) scaling, and its coefficient includes the inverse factor \(1/\alpha\).

8.3.4 6.2.4 Definition and expansion of the \(1/R^{5}\) rigidity term \(\Pi_{5}(R)\)

This section defines the “rigidity term” (a pressure-like constraint indicator) \(\Pi_{5}(R)\) in the substrate (Stone) regime by adding a radial-chain lock factor to the same two rounds of geometric dilution.

8.3.4.1 6.2.4.1 Radial-chain lock factor \(\eta_R(R)\)

Define the number of unit lengths \(L_q\) stacked along the radial direction up to radius \(R\) (the number of radial layers) by \[N_R(R) := \frac{R}{L_q}. \label{eq:S06_02_NR}\] Under the local rule (§3.1), radial transfer is represented as a chain of unit layers, and the average contribution of the chain is fixed by a rectification factor inversely proportional to the layer count. Define this as the “radial-chain lock factor” \[\eta_R(R) := \frac{1}{N_R(R)}=\frac{L_q}{R}. \label{eq:S06_02_etaR}\] \(\eta_R(R)\) is a geometric factor determined by the definition [eq:S06_02_NR] and must not be adjusted after seeing results.

8.3.4.2 6.2.4.2 Definition of the rigidity term \(\Pi_{5}(R)\)

Define the rigidity term \(\Pi_{5}(R)\) as \[\Pi_{5}(R) := \Pi_\star\, f_A(R)\, f_\Omega(R)\,\eta_R(R). \label{eq:S06_02_Pi5_def}\] Unlike the collapse term, the rigidity term does not include \(1/\alpha\). The attenuation of the rigidity term is determined not by “rectification loss of a directional projection” but only by “radial-chain locking.”

Substitute [eq:S06_02_fA], [eq:S06_02_fOmega], and [eq:S06_02_etaR] into [eq:S06_02_Pi5_def] to fully expand: \[\begin{aligned} \Pi_{5}(R) &= \Pi_\star\, \left(\frac{L_q^2}{4\pi R^2}\right) \left(\frac{1}{4\pi}\left(\frac{L_q}{R}\right)^2\right) \left(\frac{L_q}{R}\right) \notag\\ &= \Pi_\star\, \left(\frac{L_q^2}{4\pi R^2}\right) \left(\frac{L_q^2}{4\pi R^2}\right) \left(\frac{L_q}{R}\right) \notag\\ &= \Pi_\star\, \frac{L_q^5}{(4\pi)^2 R^5} \notag\\ &= \Pi_\star\, \frac{L_q^5}{16\pi^2}\, \frac{1}{R^5}. \label{eq:S06_02_Pi5_final}\end{aligned}\] Therefore the rigidity term is fixed to the \(1/R^5\) scaling. It shares the same geometric coefficient \((4\pi)^{-2}\) with the collapse term and has one additional order of attenuation by the factor \(L_q/R\).

8.3.5 6.2.5 Balance condition and the complete derivation of \(R_p/L_q=2/\pi\)

Define the core radius \(R_p\) as the transition point where the “collapse term” and the “rigidity term” balance. Lock the balance condition as \[\Pi_{4}(R_p)=\Pi_{5}(R_p). \label{eq:S06_02_balance_condition}\] Substitute [eq:S06_02_Pi4_final] and [eq:S06_02_Pi5_final] into [eq:S06_02_balance_condition]: \[\begin{aligned} \Pi_\star\, \frac{1}{\alpha}\, \frac{L_q^4}{16\pi^2}\, \frac{1}{R_p^4} &= \Pi_\star\, \frac{L_q^5}{16\pi^2}\, \frac{1}{R_p^5}. \label{eq:S06_02_balance_sub}\end{aligned}\] Cancel the common factors \(\Pi_\star\) and \(16\pi^2\) on both sides: \[\frac{1}{\alpha}\,L_q^4\,\frac{1}{R_p^4} = L_q^5\,\frac{1}{R_p^5}. \label{eq:S06_02_balance_cancel1}\] Divide both sides by \(L_q^4/R_p^4\): \[\frac{1}{\alpha} = \frac{L_q}{R_p}. \label{eq:S06_02_balance_cancel2}\] Therefore the dimensionless ratio of the core radius is \[\frac{R_p}{L_q}=\alpha. \label{eq:S06_02_Rp_over_Lq_alpha}\] Since \(\alpha\) is fixed in §5.1 as \(\alpha=2/\pi\), \[\frac{R_p}{L_q}=\frac{2}{\pi}. \label{eq:S06_02_Rp_over_Lq_final}\] Moreover, since \(L_q=\lambda_C\) is locked in §6.1, the following substitution is allowed within the same locked version: \[R_p=\frac{2}{\pi}\,L_q=\frac{2}{\pi}\,\lambda_C. \label{eq:S06_02_Rp_lambdaC}\] Equations [eq:S06_02_Rp_over_Lq_final] and [eq:S06_02_Rp_lambdaC] are fixed as the conclusions of this section.

8.3.6 6.2.6 Validity conditions (regime prerequisites) and handling violations

For the conclusion [eq:S06_02_Rp_over_Lq_final] to hold, the following items must be locked.

  1. The aggregation convention that adopts \(L_q\) as the unit patch length must be locked (\(A_0=L_q^2\)).

  2. The direction rectification factor \(\alpha\) must be locked by the canonical convention in §5.1 (\(\alpha=2/\pi\)).

  3. The radial-chain lock factor \(\eta_R(L_q/R)\) must be locked by the regime convention.

If any of the above items is not locked, or if they are mixed within the same output, the definitions [eq:S06_02_Pi4_def]–[eq:S06_02_Pi5_def] collapse and the corresponding conclusion loses conclusion status.

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8.4 6.3 Numeric substitution for \(R_p\) and a summary of invariants

8.4.1 6.3.1 Inputs (LOCK) and reference equations (starting point)

In this section, lock the following inputs and reference equations.

  1. Rectification constant: \[\alpha=\frac{2}{\pi}. \label{eq:S06_03_alpha}\]

  2. Length identification: \[L_q=\lambda_C. \label{eq:S06_03_Lq_eq_lC}\]

  3. Ratio conclusion for the core radius (derived in §6.2): \[\frac{R_p}{L_q}=\alpha. \label{eq:S06_03_Rp_over_Lq}\]

The three equations above are not re-derived here. This section combines them to (i) compute a numerical value of \(R_p\), (ii) define and compute the geometric cross section \(\sigma_{\mathrm{geom}}\), and (iii) summarize the invariant \(4/\pi\).

8.4.2 6.3.2 Full expansion of the \(R_p\) formula

From [eq:S06_03_Rp_over_Lq], \[\frac{R_p}{L_q}=\alpha \quad\Longrightarrow\quad R_p=\alpha\,L_q. \label{eq:S06_03_Rp_alpha_Lq}\] Substitute [eq:S06_03_Lq_eq_lC] into [eq:S06_03_Rp_alpha_Lq]: \[R_p=\alpha\,\lambda_C. \label{eq:S06_03_Rp_alpha_lC}\] Substitute [eq:S06_03_alpha] into [eq:S06_03_Rp_alpha_lC]: \[R_p=\frac{2}{\pi}\,\lambda_C. \label{eq:S06_03_Rp_final_form}\] Therefore, the numerical substitution for \(R_p\) in this section is fixed as the procedure “evaluate [eq:S06_03_Rp_final_form] using the locked value of \(\lambda_C\).”

8.4.3 6.3.3 Substituting the locked value of \(\lambda_C\) \(\rightarrow R_p=0.8412\,\mathrm{fm}\)

\(\lambda_C\) is the “core phase-completion length” defined in §6.1. In this section, assume that the following value is locked by canon_lock: \[\lambda_C = 1.3213538700998668\ \mathrm{fm}. \label{eq:S06_03_lC_value_lock}\] Substitute [eq:S06_03_lC_value_lock] into [eq:S06_03_Rp_final_form]: \[\begin{aligned} R_p &=\frac{2}{\pi}\,\lambda_C =\frac{2}{\pi}\times 1.3213538700998668\ \mathrm{fm} \notag\\ &= 0.8412\ \mathrm{fm}. \label{eq:S06_03_Rp_value_fm}\end{aligned}\] With the unit-conversion convention \(1\,\mathrm{fm}=10^{-15}\,\mathrm{m}\), \[R_p = 0.8412\times 10^{-15}\ \mathrm{m} = 8.412\times 10^{-16}\ \mathrm{m}. \label{eq:S06_03_Rp_value_m}\]

8.4.4 6.3.4 Back-calculation (consistency): reconstructing \(\lambda_C\) from \(R_p=0.8412\,\mathrm{fm}\)

Equation [eq:S06_03_Rp_final_form] is equivalent to \[\lambda_C=\frac{\pi}{2}\,R_p. \label{eq:S06_03_lC_from_Rp}\] Substituting [eq:S06_03_Rp_value_fm] into [eq:S06_03_lC_from_Rp] gives \[\begin{aligned} \lambda_C &=\frac{\pi}{2}\times 0.8412\ \mathrm{fm} \notag\\ &= 1.3213538700998668\ \mathrm{fm}, \label{eq:S06_03_lC_backcalc}\end{aligned}\] which matches [eq:S06_03_lC_value_lock]. Therefore the locks \(R_p/L_q=\alpha\) and \(L_q=\lambda_C\) are numerically consistent within the same version (this check is used only as a Gate input and not as a justification).

8.4.5 6.3.5 Definition and numerics of the geometric cross section \(\sigma_{\mathrm{geom}}\)

8.4.5.1 [D-6.3-1] Definition of geometric cross section

Once the core radius \(R_p\) is defined, define the geometric cross section (geometric area) as \[\sigma_{\mathrm{geom}} := \pi R_p^2. \label{eq:S06_03_sigma_def}\] Definition [eq:S06_03_sigma_def] is purely geometric and must not be reinterpreted as a different notion (effective cross section, etc.). If an effective cross section is needed, it must be introduced as a separate symbol with a separate definition.

8.4.5.2 Numerical substitution

Substitute [eq:S06_03_Rp_value_fm] into [eq:S06_03_sigma_def]: \[\begin{aligned} \sigma_{\mathrm{geom}} &=\pi\,(0.8412\ \mathrm{fm})^2 \notag\\ &=\pi\times 0.70761744\ \mathrm{fm}^2 \notag\\ &=2.223045751056016\ \mathrm{fm}^2. \label{eq:S06_03_sigma_value_fm2}\end{aligned}\] With the unit-conversion convention \((1\,\mathrm{fm})^2=10^{-30}\,\mathrm{m}^2\), \[\sigma_{\mathrm{geom}} = 2.223045751056016\times 10^{-30}\ \mathrm{m}^2. \label{eq:S06_03_sigma_value_m2}\]

8.4.6 6.3.6 Summary of the \(4/\pi\) invariant (definition–expansion–conclusion)

In this section, an “invariant” means a dimensionless or normalized combination that remains valid inside the regime regardless of unit choices as long as the lock on the ratio \(R_p/L_q\) is maintained.

8.4.6.1 6.3.6.1 Invariant I: \(\sigma_{\mathrm{geom}}/L_q^2=4/\pi\)

From the conclusion [eq:S06_03_Rp_over_Lq] of §6.2, \[R_p=\alpha L_q \label{eq:S06_03_Rp_alphaLq_repeat}\] and since \(\alpha=2/\pi\), \[R_p=\frac{2}{\pi}L_q. \label{eq:S06_03_Rp_2overpi_Lq}\] Substitute [eq:S06_03_Rp_2overpi_Lq] into [eq:S06_03_sigma_def]: \[\begin{aligned} \sigma_{\mathrm{geom}} &=\pi R_p^2 =\pi\left(\frac{2}{\pi}L_q\right)^2 \notag\\ &=\pi\left(\frac{4}{\pi^2}\right)L_q^2 \notag\\ &=\frac{4}{\pi}\,L_q^2. \label{eq:S06_03_sigma_in_Lq}\end{aligned}\] Therefore the following invariant holds. \[\boxed{ \frac{\sigma_{\mathrm{geom}}}{L_q^2} = \frac{4}{\pi} } \qquad (\text{Invariant I}). \label{eq:S06_03_invariant_4overpi}\] Applying the lock \(L_q=\lambda_C\) from §6.1 gives the equivalent form \[\boxed{ \frac{\sigma_{\mathrm{geom}}}{\lambda_C^2} = \frac{4}{\pi} } \qquad (\text{Invariant I, equivalent form}). \label{eq:S06_03_invariant_4overpi_lC}\]

8.4.6.2 6.3.6.2 Numerical check (Invariant I)

From [eq:S06_03_lC_value_lock], since \(L_q=\lambda_C\), we have \[L_q^2 = (1.3213538700998668\ \mathrm{fm})^2 = 1.7459760500278958\ \mathrm{fm}^2. \label{eq:S06_03_Lq2_value}\] Using [eq:S06_03_sigma_value_fm2] and [eq:S06_03_Lq2_value], \[\begin{aligned} \frac{\sigma_{\mathrm{geom}}}{L_q^2} &= \frac{2.223045751056016}{1.7459760500278958} \notag\\ &= 1.2732395447351628, \label{eq:S06_03_ratio_numeric}\end{aligned}\] and since \[\frac{4}{\pi}=1.2732395447351628, \label{eq:S06_03_4overpi_numeric}\] the numerical ratio agrees with [eq:S06_03_invariant_4overpi]. This check is a validation computation; it does not use external texts as a basis for the invariant.

8.4.7 6.3.7 Where the invariant is used (locking insertion points)

Invariant I can be used as a geometric normalization in derivations of the following form, and its use must have its insertion point locked.

  1. In every expression where \(\sigma_{\mathrm{geom}}\) appears, when normalizing \(\sigma_{\mathrm{geom}}\) by \(L_q^2\), use [eq:S06_03_invariant_4overpi] to fix the constant as \(4/\pi\).

  2. For expressions written in terms of \(\lambda_C\), use the equivalent form [eq:S06_03_invariant_4overpi_lC].

This use is permitted only as a geometric consequence of the already-locked ratio \(R_p/L_q=\alpha\) and \(\alpha=2/\pi\), not as an “ad hoc constant correction.”

LOCK/Gate connections for this section (none if empty)

8.5 6.4 Continuum\(\rightarrow\)Discrete (82+7) stability conditions

8.5.1 6.4.1 Purpose

This section declares a list of minimal stability conditions required when the core radius \(R_p\) and related invariants derived in Chapter 6 (e.g., \(R_p/L_q=2/\pi\), \(\sigma_{\mathrm{geom}}/L_q^2=4/\pi\)) descend to the discrete structure “core 82 + shell 7” in Chapter 8. This section does not derive the detailed coordinates or coupling conventions of the discrete structure. It fixes only the necessary conditions imposed by the continuum results on the discrete structure, in the form of (i) condition identifier, (ii) condition content, and (iii) handling on violation.

8.5.2 6.4.2 Premises (locked continuum-side results)

The continuum-side locked results referenced in this section are as follows.

  1. Core radius ratio: \[\frac{R_p}{L_q}=\frac{2}{\pi}. \label{eq:S06_04_Rp_ratio}\]

  2. Length identification: \[L_q=\lambda_C. \label{eq:S06_04_Lq_eq_lC}\]

  3. Geometric cross-section invariant: \[\frac{\sigma_{\mathrm{geom}}}{L_q^2}=\frac{4}{\pi}, \qquad \sigma_{\mathrm{geom}}:=\pi R_p^2. \label{eq:S06_04_invariant_sigma}\]

These results are chapter outputs of the continuum model and belong to canon_lock. If the discrete structure fails to satisfy them, the continuum\(\rightarrow\)discrete link does not hold.

8.5.3 6.4.3 Minimal stability conditions for the discrete structure (82+7)

The discrete structure consists of the combination “core 82” and “shell 7.” The continuum results require the following conditions as mandatory. Each condition is independent; violating any one of them judges the continuum\(\rightarrow\)discrete link as FAIL.

8.5.4 [C-82/7-01] Core-boundary radius consistency (radius lock condition)

Given a discrete core (82) coordinate set \(\{\mathbf{x}_i\}_{i=1}^{82}\) and a center \(\mathbf{x}_c\), the radial aggregate of the key boundary layer (boundary-candidate set \(\mathcal{B}_{82}\subseteq\{1,\ldots,82\}\)) must be consistent with \(R_p\). Declare the consistency in the following form: \[R_{82} := \mathrm{Agg}\Bigl(\{\|\mathbf{x}_i-\mathbf{x}_c\|\}_{i\in\mathcal{B}_{82}}\Bigr) \equiv R_p, \label{eq:S06_04_R82_match}\] where the aggregation operator \(\mathrm{Agg}\) (e.g., median, mean, center of a min–max band) must be locked in analysis_lock. If \(\mathrm{Agg}\) is not locked, \(R_{82}\) is non-unique; the condition is undefined and judged INCONCLUSIVE. The tolerance (allowable error) \(\varepsilon_{R}\) must be registered in advance in gate_lock, and \[|R_{82}-R_p|>\varepsilon_{R} \quad\Longrightarrow\quad \texttt{FAIL-CORE82-RADIUS}. \label{eq:S06_04_R82_fail}\]

8.5.5 [C-82/7-02] Cross-section invariant consistency (geometric cross-section condition)

In the discrete core (82), it must be possible to define a “discrete cross section” through the boundary-candidate set \(\mathcal{B}_{82}\), and its normalization must be consistent with the invariant [eq:S06_04_invariant_sigma]. Define the discrete cross section \(\sigma_{82}\) in the following form: \[\sigma_{82} := \mathrm{ProjArea}\Bigl(\{\mathbf{x}_i\}_{i\in\mathcal{B}_{82}};\ \mathbf{n}_\sigma\Bigr), \label{eq:S06_04_sigma82_def}\] where \(\mathbf{n}_\sigma\) is the projection axis and is locked in analysis_lock. The definition of the projection-area operator \(\mathrm{ProjArea}\) (convex hull / lattice-cell count / pixelization, etc.) must also be locked in analysis_lock.

Define the normalized cross-section ratio as \[I_{\sigma,82}:=\frac{\sigma_{82}}{L_q^2}. \label{eq:S06_04_I_sigma82}\] The requirement from the continuum invariant is declared as \[\left|I_{\sigma,82}-\frac{4}{\pi}\right|>\varepsilon_{\sigma} \quad\Longrightarrow\quad \texttt{FAIL-CORE82-SIGMA}. \label{eq:S06_04_sigma_fail}\] where \(\varepsilon_{\sigma}\) is a threshold locked in gate_lock.

8.5.6 [C-82/7-03] Uniqueness of boundary transition (single core-boundary condition)

The continuum model locks the core boundary as a single transition point. The discrete structure must also have a single transition. Define a “core indicator” \(\chi_{82}(r)\) for radius \(r\) and require that there is only one transition point: \[\chi_{82}(r)\in\{0,1\}, \qquad r\mapsto \chi_{82}(r)\ \text{has exactly one }0\to 1\text{ transition}. \label{eq:S06_04_single_transition}\] The decision rule for the transition (which structural quantity defines \(\chi_{82}\), and the thresholds that judge “one transition”) must be locked in analysis_lock. If two or more transitions occur, or if there is no transition, treat it as a failure of the continuum\(\rightarrow\)discrete link: \[\text{transition count}\neq 1 \quad\Longrightarrow\quad \texttt{FAIL-CORE82-TRANSITION}. \label{eq:S06_04_transition_fail}\]

8.5.7 [C-82/7-04] Locality of shell (7) attachment (preservation of the local rule)

Shell (7) must form attachment/cancellation/survival structures locally near the core boundary. This is the requirement that the discrete construction is compatible with the local-rule axiom in §3.1. Declare the following condition: \[\forall k\in\{1,\ldots,7\},\ \exists i(k)\in\mathcal{B}_{82}\ \text{s.t.}\ \|\mathbf{s}_k-\mathbf{x}_{i(k)}\|\le \rho_{\mathrm{attach}}. \label{eq:S06_04_shell_local_attach}\] Here \(\mathbf{s}_k\) is a shell vector or shell marker point (locked in Chapter 8), and \(\rho_{\mathrm{attach}}\) is the attachment-radius threshold. Lock \(\rho_{\mathrm{attach}}\) by normalizing it with the length scale \(L_q\): \[\rho_{\mathrm{attach}} := \eta_{\mathrm{attach}}\,L_q, \qquad \eta_{\mathrm{attach}}>0\ \text{is locked}. \label{eq:S06_04_attach_radius_lock}\] Handle violation as \[\exists k\ \text{s.t.}\ \min_{i\in\mathcal{B}_{82}}\|\mathbf{s}_k-\mathbf{x}_i\|>\rho_{\mathrm{attach}} \quad\Longrightarrow\quad \texttt{FAIL-SHELL7-LOCAL}. \label{eq:S06_04_shell_local_fail}\]

8.5.8 [C-82/7-05] Non-degeneracy of the cancellation–survival convention (existence of a survival vector)

When the continuum results descend to the discrete structure, the cancellation–survival convention of shell (7) must remain non-degenerate. This means that a survival vector \(\mathbf{V}\) is definable (not non-unique/ambiguous) and must not collapse to the zero vector. \[\mathbf{V}:=\sum_{k=1}^{7}\mathbf{s}_k, \qquad \|\mathbf{V}\|\ge V_{\min}. \label{eq:S06_04_survival_nondeg}\] \(V_{\min}\) is a threshold locked in gate_lock in internal units or normalized by \(L_q\). If \(\|\mathbf{V}\|<V_{\min}\), the definition of “survival sign/charge/emission direction” collapses, so treat it as a link failure: \[\|\mathbf{V}\|<V_{\min} \quad\Longrightarrow\quad \texttt{FAIL-SHELL7-DEGEN}. \label{eq:S06_04_survival_fail}\]

8.5.9 [C-82/7-06] Regime consistency (Stone/plenum/jamming prerequisites)

The continuum core model assumes the Stone/plenum regime. The discrete core (82+7) must also be defined under the same regime. Declare the following conditions.

  1. The discrete placement does not violate impenetrability.

  2. The discrete placement does not conflict with the plenum convention (do not treat void as an independent object).

  3. The contact graph and backbone decision are consistent with the locked regime coordinate axes (§4.3).

If any of the above is violated, treat it as a regime mismatch: \[\text{regime mismatch} \quad\Longrightarrow\quad \texttt{FAIL-REG-MISMATCH}. \label{eq:S06_04_regime_fail}\]

8.5.10 [C-82/7-07] Scale-normalization consistency (single length unit)

All coordinates and length judgments of the discrete structure must be normalized by the same length scale \(L_q\) (or the identified \(\lambda_C\)). The following mixes are forbidden within the same output.

  1. Using \(L_q\) and \(\lambda_C\) as different values.

  2. Re-introducing \(R_p\) via a separate definition to evade the \(R_{82}\) matching.

  3. Mixing the cell geometry (cube) and a visualization sphere when computing cross sections or radii.

If a mix is detected, it is an immediate FAIL as a lock_id mix or a meaning conflict: \[\text{normalization mix} \quad\Longrightarrow\quad \texttt{FAIL-LOCK-MIX}\ \text{or}\ \texttt{FAIL-GEO-CONF}. \label{eq:S06_04_norm_fail}\]

8.5.11 [C-82/7-08] Existence of tetrahedral locking (a 4-point rigidity certificate)

This condition locks the minimal structural certificate needed to state that “core (82) has full jamming rigidity” (i.e., that one can logically claim the \(c^2\) stiffness corresponding to \(\Psi_{\rm yield}\) in the unjamming trigger [eq:unjamming_trigger]).

In 3D, four points are the minimal non-coplanar simplex. If the contact network does not contain this tetrahedral (simplex) backbone, shear rigidity can remain zero or regime-out. Therefore the contact graph of core 82, \(\mathcal{G}_c=(\mathcal{V}_c,\mathcal{E}_c)\), must satisfy \[\exists\ (i_1,i_2,i_3,i_4)\subset \mathcal{B}_{82} \ \text{s.t.}\ (i_a,i_b)\in \mathcal{E}_c\ \forall\ a<b. \label{eq:S06_04_tetra_lock_exist}\] That is, there must exist a 4-point backbone corresponding to the complete graph \(K_4\) within \(\mathcal{B}_{82}\). Denote the corresponding 4-point set by \(\mathcal{T}_4\).

To claim “rigidity in full,” this tetrahedron must also be non-degenerate (not coplanar or near-coplanar). In coordinate form, judge this by a threshold on tetrahedral volume (mixed product): \[V_{\mathrm{tet}}(i_1,i_2,i_3,i_4) :=\frac{1}{6}\left|\det\left[\mathbf{x}_{i_2}-\mathbf{x}_{i_1},\ \mathbf{x}_{i_3}-\mathbf{x}_{i_1},\ \mathbf{x}_{i_4}-\mathbf{x}_{i_1}\right]\right| \ge V_{\min}^{\mathrm{tet}}. \label{eq:S06_04_tetra_volume}\] Here \(V_{\min}^{\mathrm{tet}}\) is a threshold locked in gate_lock after normalization by \(L_q^3\).

If the condition holds, the fact that core (82) has a jamming backbone containing “tetrahedral locking” is secured, and it becomes the minimal basis to claim the jamming regime (\(\xi\approx 1\)) in the dynamic rigidity-mixing narrative (\(\xi\in[0,1]\)) of this document. Conversely, if this condition is violated, sentences such as “core rigidity = \(c^2\)” or “no further contraction” lose conclusion status; only limit claims (CT-LIM) are permitted. \[\neg\exists\ \mathcal{T}_4\ \ \text{or}\ \ V_{\mathrm{tet}}<V_{\min}^{\mathrm{tet}} \quad\Longrightarrow\quad \texttt{FAIL-CORE82-TETRA}. \label{eq:S06_04_tetra_fail}\]

8.5.12 6.4.4 Handling violations (conclusion status and limit statements)

If any of the conditions [C-82/7-01]~[C-82/7-08] is violated, the continuum\(\rightarrow\)discrete link loses conclusion status. In that case, only “limit statements (CT-LIM)” are permitted, and one must record the FAIL label together with the causal condition ID. Patching violations by “interpretation,” or changing condition definitions (aggregation operator, thresholds, boundary-candidate set, etc.) after seeing results, violates No-Tuning and is forbidden. If changes are necessary, the only allowed path is a version bump followed by full re-validation.

LOCK/Gate connections for this section (none if empty)

  \item LOCK: Fix the continuum outputs ($R_p/L_q=2/\pi$, $L_q=\lambda_C$, $\sigma_{\mathrm{geom}}/L_q^2=4/\pi$) as upstream inputs.
  \item LOCK: Lock the definitions of the discrete stability-condition list [C-82/7-01]\textasciitilde[C-82/7-08] (aggregation operators, projection axis, thresholds, normalization conventions) in \texttt{analysis\_lock}/\texttt{gate\_lock}.
  \item Gate: On any condition violation, immediately assign \texttt{FAIL-CORE82-*}/\texttt{FAIL-SHELL7-*}/\texttt{FAIL-REG-*} labels and revoke conclusion status.
  \item Gate: If condition definitions are not locked (aggregation/threshold/boundary-candidate missing), judge \texttt{INCONCLUSIVE}; if conditions are modified post hoc based on outcomes, judge \texttt{FAIL} in G-NT.
  \item Gate: Mixing different \texttt{lock\_id} combinations or confusing cell geometry is immediate \texttt{FAIL} (G-LOCK/G-SYM).

9 7. 3-Sector Integerization (120) and Build Time

Topological necessity of 3 sectors

To physically enclose a center point (Core) in a 2D cross section, at least three vectors are required (two vectors are linear; four vectors are overcomplete). Therefore, the minimum geometric cost to distinguish states such as charge (\(\pm,0\)) inevitably reduces to a 3-sector structure (120).

9.0.0.1 (Proposition) \(3\) is minimal and \(120^\circ\) is forced

(i) Minimality. With only two vectors one cannot form an enclosure around the center point (except for the degenerate opposite-pair case), and even in the opposite-pair case only a “line” remains, so one cannot generate three sectors (states). Hence the minimal topological cost to enclose the center is \(3\).
(ii) \(120^\circ\) is forced. Let the direction axes be unit vectors \(\{\mathbf{n}_1,\mathbf{n}_2,\mathbf{n}_3\}\) and lock the sum-zero closure as \[\mathbf{n}_1+\mathbf{n}_2+\mathbf{n}_3=\mathbf{0},\qquad \|\mathbf{n}_i\|=1 \label{eq:S07_00_nsum0_unit}\] Then \(\mathbf{n}_3=-(\mathbf{n}_1+\mathbf{n}_2)\), hence \[\|\mathbf{n}_3\|^2=\|\mathbf{n}_1+\mathbf{n}_2\|^2=2+2\,\mathbf{n}_1\cdot\mathbf{n}_2.\] For the left-hand side to be \(1\), we must have \(\mathbf{n}_1\cdot\mathbf{n}_2=-\tfrac{1}{2}\), so the angle between the two axes is \(120^\circ\). By permutation symmetry all pairs are \(120^\circ\), and thus the 3-sector (120) structure is not a choice but a geometric consequence of “minimality + sum-zero closure.”

Purpose and outputs of the chapter

This chapter (i) defines the 120 3-sector coordinate axes, (ii) locks an integerization rule that converts continuous (real-valued) directional/phase contributions into three integer sector counts, and (iii) defines build time by coupling the integerized counts to time ticks. The outputs of this chapter are locked to the following four items.

  1. Definition of the 3-sector axes \(\{\mathbf{n}_1,\mathbf{n}_2,\mathbf{n}_3\}\) (including the 120 conditions).

  2. Definition of the integerization output \((k_1,k_2,k_3)\in\mathbb{Z}_{\ge 0}^3\) and the sum-preservation rule \(k_1+k_2+k_3=N\).

  3. Definition of the residual (non-cancelled) vector \(\mathbf{V}\) and its non-degeneracy condition (including the prohibition of collapse to the zero vector).

  4. Definition of the build time \(T_{\mathrm{build}}\) and auxiliary build times (tick-based / event-rate based).

The integerization rule connects directly, in later chapters, to (a) the shell cancellation–survival label (“6 cancel + 1 survive”), (b) the charge-sign label, and (c) the electron (survival) label. Therefore, if the integerization rule is not locked here, the downstream charge/electron labels become undefined.

Definition of the 3-sector (120) axes

2D sector plane and unit axes

3-sector integerization is performed on a specific plane \(\Pi\). The choice of the plane (which 2D subspace of which coordinate system) is locked in analysis_lock. On the plane \(\Pi\), define three unit vectors as \[\mathbf{n}_1,\mathbf{n}_2,\mathbf{n}_3 \in \Pi,\qquad \|\mathbf{n}_1\|=\|\mathbf{n}_2\|=\|\mathbf{n}_3\|=1. \label{eq:S07_n_unit}\]

120 condition (inner-product convention)

Lock the 3-sector axes to have mutual spacing of 120 by the inner-product convention \[\mathbf{n}_i\cdot \mathbf{n}_j= \begin{cases} 1,& i=j,\\ -\dfrac{1}{2},& i\neq j. \end{cases} \label{eq:S07_120deg_inner}\] Also lock their sum to be zero (center cancellation): \[\mathbf{n}_1+\mathbf{n}_2+\mathbf{n}_3=\mathbf{0}. \label{eq:S07_n_sum_zero}\] Equations [eq:S07_120deg_inner]–[eq:S07_n_sum_zero] are the geometric base of 3-sector integerization. After seeing results, the axes may not be rotated or replaced. A permutation (reordering) of axes may be allowed, but the permutation rule (e.g., \(\mathbf{n}_1\mapsto \mathbf{n}_2\)) must be pre-registered in analysis_lock.

Integerization rule: continuous contributions \(\rightarrow\) \((k_1,k_2,k_3)\)

Inputs: direction contribution vector and total count \(N\)

The inputs of integerization are a vector \(\mathbf{u}\) on the plane \(\Pi\) and a total count \(N\): \[\mathbf{u}\in \Pi,\qquad N\in \mathbb{Z}_{\ge 0}. \label{eq:S07_inputs}\] \(\mathbf{u}\) represents a “directional contribution” vector derived from event aggregation, shell structure, or core–shell coupling. The generation procedure of \(\mathbf{u}\) (which landmarks, which before/after placement, which aggregation window) is locked in analysis_lock. \(N\) is the conserved total integer resource (total sector count) at the given step; its provenance (e.g., total number in a structure, total mass in a given event window) is locked in analysis_lock or canon_lock.

Definition of real-valued sector scores \(s_i\) (nonnegativization)

Define projection scores onto the sector axes: \[p_i := \mathbf{u}\cdot \mathbf{n}_i\qquad (i=1,2,3). \label{eq:S07_projection_pi}\] Since \(p_i\) can be positive or negative, shift them to nonnegative scores for conversion into integer counts. Define the shift amount as \[p_{\min}:=\min\{p_1,p_2,p_3\}, \qquad s_i:=p_i-p_{\min}\qquad (i=1,2,3). \label{eq:S07_shift_si}\] Therefore \[s_i\ge 0\quad (i=1,2,3), \qquad \text{and at least one } s_i \text{ is } 0. \label{eq:S07_si_nonneg}\] If \(\mathbf{u}\) is perfectly symmetric with respect to the three axes, then \(p_1=p_2=p_3\) and thus \(s_1=s_2=s_3=0\); in that case normalization is impossible and a separate handling rule is required. Lock the degeneracy indicator \[S:=s_1+s_2+s_3. \label{eq:S07_S_sum}\] If \(S=0\), classify the input as a “perfectly symmetric input,” and lock the integerization output as \[(k_1,k_2,k_3)= \begin{cases} \left(\dfrac{N}{3},\dfrac{N}{3},\dfrac{N}{3}\right),& N\equiv 0\ (\mathrm{mod}\ 3),\\[6pt] \left(\left\lfloor\dfrac{N}{3}\right\rfloor,\left\lfloor\dfrac{N}{3}\right\rfloor,\left\lceil\dfrac{N}{3}\right\rceil\right)\ \text{and its permutations},& N\not\equiv 0\ (\mathrm{mod}\ 3), \end{cases} \label{eq:S07_degenerate_rule}\] where the permutation-selection rule must be pre-registered in analysis_lock and cannot be chosen after seeing outcomes.

Definition of normalized fractions \(f_i\)

For \(S>0\), define sector fractions by \[f_i := \frac{s_i}{S}\qquad (i=1,2,3), \label{eq:S07_fi_def}\] so that \[f_i\ge 0,\qquad f_1+f_2+f_3=1. \label{eq:S07_fi_simplex}\]

Integerization (sum-preserving) rule

The goal of integerization is to select an integer triple \((k_1,k_2,k_3)\) that satisfies simultaneously \[k_i\in\mathbb{Z}_{\ge 0},\qquad k_1+k_2+k_3=N. \label{eq:S07_sum_constraint}\] Define integerization as “the integer allocation closest to the fractions \(f_i\),” and lock the procedure as follows.

9.0.0.2 (1) Floor allocation

\[\tilde{k}_i := \left\lfloor N f_i \right\rfloor,\qquad r_i := N f_i - \tilde{k}_i \in [0,1), \label{eq:S07_floor_and_residual}\] where \(r_i\) is the residual (fractional part).

9.0.0.3 (2) Compute the deficit

\[\Delta := N - (\tilde{k}_1+\tilde{k}_2+\tilde{k}_3). \label{eq:S07_deficit_Delta}\] By construction, \(\Delta\in\{0,1,2\}\).

9.0.0.4 (3) Correct by the largest residuals

Select \(\Delta\) indices in decreasing order of \(r_i\), and add 1 to \(\tilde{k}_i\) for those indices. Formally, define the index set \(\mathcal{I}_\Delta\subset\{1,2,3\}\) by \[\mathcal{I}_\Delta := \operatorname{TopK}\bigl(\{r_1,r_2,r_3\};\Delta\bigr) \label{eq:S07_topk_def}\] and define the final integerization output by \[k_i := \tilde{k}_i + \mathbf{1}_{\{i\in\mathcal{I}_\Delta\}} \qquad (i=1,2,3) \label{eq:S07_final_ki}\] where \(\mathbf{1}_{\{\cdot\}}\) is the indicator function.

9.0.0.5 (4) Tie-handling (tie-break)

Because equal residuals can occur in \(\operatorname{TopK}\), a tie-break rule must be pre-registered and locked. Only one of the following tie-break modes is allowed (one must be selected and locked).

  1. TB-LEX: prioritize the index order (1\(\rightarrow\)2\(\rightarrow\)3).

  2. TB-AXIS: prioritize a pre-registered preferred axis (e.g., \(\mathbf{n}_1\) first).

  3. TB-HASH: hash-based decision computed from an event/structure identifier (same input \(\Rightarrow\) same output).

If the tie-break rule is not locked, the integerization result is non-unique and is judged INCONCLUSIVE.

First-order invariants of the integerization output

The integerization output satisfies the following invariants.

  1. Sum preservation: \(k_1+k_2+k_3=N\).

  2. Nonnegativity: \(k_i\ge 0\).

  3. Fraction approximation: it is constructed so that \(\left|k_i/N - f_i\right|\) is minimized (ties are resolved only by the pre-registered tie-break rule).

These invariants are used downstream as the integer basis of charge/electron labels. If any invariant is violated, the label becomes undefined.

Definition of the residual vector \(\mathbf{V}\)

From the integerization output \((k_1,k_2,k_3)\) define the sector residual (non-cancelled) vector by \[\mathbf{V} := k_1\mathbf{n}_1+k_2\mathbf{n}_2+k_3\mathbf{n}_3. \label{eq:S07_V_def}\] By [eq:S07_n_sum_zero], if \(k_1=k_2=k_3\) then \(\mathbf{V}=\mathbf{0}\). Hence \(\mathbf{V}\) is locked as an integer-based residual vector that encodes the magnitude and direction of deviation from “perfect cancellation.”

Non-degeneracy condition (label definability condition)

Lock that charge/electron labels are definable only when \(\mathbf{V}\) is nonzero (non-degenerate). Define the threshold as \[\|\mathbf{V}\|\ge V_{\min}, \label{eq:S07_Vmin}\] where \(V_{\min}\) is a pre-registered threshold in gate_lock; changing \(V_{\min}\) is allowed only by a version bump. If \(\|\mathbf{V}\|<V_{\min}\), labels are treated as neutral (or undefined), and the handling rule for that case (whether it is CT-LIM or CT-DEF) must be locked by PASS.rules.

The integerization rule is linked to downstream labels in the following forms.

  1. Charge-sign label: for a pre-registered charge axis \(\mathbf{n}_Q\), \[q := \mathrm{sgn}(\mathbf{V}\cdot\mathbf{n}_Q) \label{eq:S07_charge_label_decl}\] defines the sign label. The choice of \(\mathbf{n}_Q\) must be locked in analysis_lock and cannot be chosen after seeing outcomes.

  2. Electron (survival) label: under the shell(7) cancellation–survival convention, the “survival” residual direction is defined from \(\mathbf{V}\), and the electron label is assigned according to the existence and signed direction of that survival residual. The details of the survival convention are locked in the discrete-structure chapter; here we only declare that integerization is the integer basis for the label.

Therefore the integerization rule in this chapter is a “super-convention” for charge/electron labels: labels cannot be introduced separately by bypassing the integerization rule.

9.1 7.5 Definition of Build Time

9.1.1 7.5.1 Tick-based build time

When the realization time tick \(\Delta t\) is locked in realization_lock, define the build time corresponding to total count \(N\) as \[T_{\mathrm{build}} := N\,\Delta t. \label{eq:S07_Tbuild_tick}\] Definition [eq:S07_Tbuild_tick] adopts “1 count per tick.” If a different aggregation convention is required (e.g., multiple counts per tick or sparse counts), it must be defined by a separate closure.

9.1.2 7.5.2 Event-rate-based build time

If a stationary event rate \(\nu\) is defined (event definition and \(\delta\) insertion are locked), and the mean time to accumulate \(N\) counts in the same regime is used as build time, lock \[T_{\mathrm{build}} := \frac{N}{\nu}. \label{eq:S07_Tbuild_rate}\] The use of [eq:S07_Tbuild_rate] is regime-dependent. If \(\nu\) is undefined or judged FAIL/INCONCLUSIVE by Gate, then [eq:S07_Tbuild_rate] is forbidden and only [eq:S07_Tbuild_tick] is permitted.

9.1.3 7.5.3 Auxiliary build times via 3-sector decomposition

When the integerization output \((k_1,k_2,k_3)\) is locked, define the sector-wise build times by \[T_i := k_i\,\Delta t \qquad (i=1,2,3), \label{eq:S07_Ti_def}\] so that \[T_1+T_2+T_3 = (k_1+k_2+k_3)\Delta t = N\Delta t = T_{\mathrm{build}}. \label{eq:S07_Tsum}\] \(T_i\) is the “accumulated time per sector” and can be used later as a temporal representation of sector asymmetry (e.g., how a sector dominance affects the charge/electron label). In all cases \(T_i\) is fixed by [eq:S07_Ti_def] and cannot be redefined after seeing results.

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9.2 7.1 Minimum-variance integerization (\(N=3m+r\)): 89, 82

9.2.1 7.1.1 Definition of the integerization problem

The goal of 3-sector integerization is to distribute a total integer resource \(N\in\mathbb{Z}_{\ge 0}\) into three nonnegative integer sector counts \[(k_1,k_2,k_3)\in\mathbb{Z}_{\ge 0}^3 \label{eq:S07_01_k_domain}\] so that \[k_1+k_2+k_3=N \label{eq:S07_01_sumN}\] while minimizing “bias (asymmetry).”

In this section, define bias by the variance and lock the integerization rule that minimizes this variance as minimum-variance integerization. Let the mean be \[\mu := \frac{N}{3} \label{eq:S07_01_mu}\] and define the (3-sector) variance by \[\mathrm{Var}(k_1,k_2,k_3) := \frac{1}{3}\sum_{i=1}^{3}(k_i-\mu)^2. \label{eq:S07_01_variance}\] Under the fixed-sum constraint [eq:S07_01_sumN], minimizing [eq:S07_01_variance] is equivalent to minimizing the following second moment. \[S_2(k_1,k_2,k_3):=\sum_{i=1}^{3}k_i^2. \label{eq:S07_01_S2}\] Indeed, using [eq:S07_01_sumN], \[\begin{aligned} \mathrm{Var} &=\frac{1}{3}\sum_{i=1}^{3}(k_i^2-2\mu k_i+\mu^2) =\frac{1}{3}\sum_{i=1}^{3}k_i^2-\frac{2\mu}{3}\sum_{i=1}^{3}k_i+\mu^2 \notag\\ &=\frac{1}{3}S_2-\frac{2\mu}{3}N+\mu^2 =\frac{1}{3}S_2-\frac{2}{3}\left(\frac{N}{3}\right)N+\left(\frac{N}{3}\right)^2 =\frac{1}{3}S_2-\frac{N^2}{9}. \label{eq:S07_01_var_S2_relation}\end{aligned}\] Hence, for fixed \(N\), minimizing \(\mathrm{Var}\) is fully equivalent to minimizing \(S_2\).

9.2.2 7.1.2 Minimum-variance integerization theorem (equalization principle)

9.2.2.1 7.1.2.1 Equalization lemma

Let an integer triple \((k_1,k_2,k_3)\) satisfy [eq:S07_01_sumN]. If some pair \((k_a,k_b)\) satisfies \[k_a \ge k_b + 2 \label{eq:S07_01_gap_ge2}\] then define the following “equalization move”: \[k_a' := k_a-1,\qquad k_b' := k_b+1, \qquad k_c' := k_c\ (c\neq a,b). \label{eq:S07_01_balance_move}\] The sum is preserved: \[k_1'+k_2'+k_3' = (k_1+k_2+k_3) = N. \label{eq:S07_01_sum_preserve}\] Moreover, the second moment strictly decreases: \[\begin{aligned} S_2(k_1,k_2,k_3)-S_2(k_1',k_2',k_3') &= (k_a^2+k_b^2)-\bigl((k_a-1)^2+(k_b+1)^2\bigr)\notag\\ &= k_a^2+k_b^2-\bigl(k_a^2-2k_a+1+k_b^2+2k_b+1\bigr)\notag\\ &= 2(k_a-k_b)-2. \label{eq:S07_01_S2_decrease}\end{aligned}\] By assumption [eq:S07_01_gap_ge2], \(k_a-k_b\ge 2\), hence \[2(k_a-k_b)-2 \ge 2 > 0, \label{eq:S07_01_S2_strict}\] so \(S_2\) must decrease. Therefore, if any solution contains a gap of the form [eq:S07_01_gap_ge2], it cannot be a minimum-variance solution.

9.2.2.2 7.1.2.2 Necessary and sufficient condition for a minimum-variance solution

By the lemma, a solution that minimizes \(S_2\) must have no pairwise gap exceeding 1: \[|k_i-k_j|\le 1\qquad (i,j\in\{1,2,3\}). \label{eq:S07_01_gap_le1}\] Conversely, any solution satisfying [eq:S07_01_gap_le1] admits no further equalization move [eq:S07_01_balance_move] that decreases \(S_2\). Hence this condition is both necessary and sufficient (on the integer set, a local minimum is a global minimum).

9.2.3 7.1.3 \(N=3m+r\) decomposition and the closed form of the minimum-variance solution

9.2.3.1 7.1.3.1 Definition of quotient and remainder

Define the quotient and remainder of dividing \(N\) by 3: \[N = 3m + r, \qquad m:=\left\lfloor\frac{N}{3}\right\rfloor, \qquad r:=N-3m\in\{0,1,2\}. \label{eq:S07_01_division}\]

9.2.3.2 7.1.3.2 Form of the minimum-variance solution (unique up to permutation)

The integer triples that satisfy both [eq:S07_01_gap_le1] and [eq:S07_01_sumN] exist only in the following forms (unique up to permutation).

  1. \(r=0\): \[(k_1,k_2,k_3)=(m,m,m). \label{eq:S07_01_case_r0}\]

  2. \(r=1\): \[(k_1,k_2,k_3)\ \text{is a permutation of } (m,m,m+1). \label{eq:S07_01_case_r1}\]

  3. \(r=2\): \[(k_1,k_2,k_3)\ \text{is a permutation of } (m,m+1,m+1). \label{eq:S07_01_case_r2}\]

These three cases fully classify all solutions satisfying [eq:S07_01_gap_le1].

9.2.3.3 7.1.3.3 Closed form of the minimum second moment

Lock the minimum value of the second moment for the minimum-variance solution as \[S_{2,\min}(N)=3m^2+2mr+r, \qquad (N=3m+r,\ r\in\{0,1,2\}). \label{eq:S07_01_S2min}\] Indeed, for \(r=0\) one has \(S_2=3m^2\); for \(r=1\), \(S_2=2m^2+(m+1)^2=3m^2+2m+1\); for \(r=2\), \(S_2=m^2+2(m+1)^2=3m^2+4m+2\).

9.2.4 7.1.4 Sector priority (tie-break) rule

When \(r\neq 0\), one must choose which sector receives \(m+1\) (i.e., which permutation is adopted). Because this choice cannot be changed after seeing outcomes, pre-register and lock the following “priority permutation.” \[\pi_{\mathrm{sec}}=(i_1,i_2,i_3) \quad\text{is a permutation of } \{1,2,3\}\ \text{and is locked in } \texttt{analysis\_lock}. \label{eq:S07_01_priority_perm}\] Given \(\pi_{\mathrm{sec}}\), define the minimum-variance integerization output by \[k_{i_j}:= \begin{cases} m+1,& 1\le j\le r,\\ m,& r<j\le 3. \end{cases} \label{eq:S07_01_priority_assign}\] That is, the first \(r\) sectors in the priority permutation receive \(m+1\), and the remaining sectors receive \(m\). If \(\pi_{\mathrm{sec}}\) is not locked, the result is non-unique for \(r\neq 0\), hence INCONCLUSIVE.

9.2.5 7.1.5 Examples: \(N=89\) and \(N=82\)

9.2.5.1 7.1.5.1 \(N=89\)

By [eq:S07_01_division], \[89=3\cdot 29 + 2, \qquad m=29,\quad r=2. \label{eq:S07_01_89_div}\] Hence the minimum-variance solution, by [eq:S07_01_case_r2], is \[(k_1,k_2,k_3)\ \text{is a permutation of } (29,30,30). \label{eq:S07_01_89_triplet}\] If the priority permutation is locked as \(\pi_{\mathrm{sec}}=(1,2,3)\), then by [eq:S07_01_priority_assign] the output is fixed as \[(k_1,k_2,k_3)=(30,30,29) \label{eq:S07_01_89_example_order}\]

9.2.5.2 7.1.5.2 \(N=82\)

By [eq:S07_01_division], \[82=3\cdot 27 + 1, \qquad m=27,\quad r=1. \label{eq:S07_01_82_div}\] Hence the minimum-variance solution, by [eq:S07_01_case_r1], is \[(k_1,k_2,k_3)\ \text{is a permutation of } (27,27,28). \label{eq:S07_01_82_triplet}\] If the priority permutation is locked as \(\pi_{\mathrm{sec}}=(1,2,3)\), then \[(k_1,k_2,k_3)=(28,27,27) \label{eq:S07_01_82_example_order}\]

9.2.6.1 7.1.6.1 Minimum-bias principle (internal rule)

Because the 3-sector structure is locked as a 120 symmetric structure (see the chapter overview and §7.0), the three sectors have equal status. Therefore integerization must satisfy the following internal rules.

  1. Symmetry preservation: if the input does not distinguish sectors, the output should distinguish sectors as little as possible.

  2. Bias minimization: minimize sector imbalance (variance or gaps) in the output.

  3. Allow only unavoidable residuals: leave only the imbalance that is unavoidable due to \(N\ (\mathrm{mod}\ 3)\), and forbid any larger imbalance.

A partition satisfying [eq:S07_01_gap_le1] is the unique form that satisfies all three rules simultaneously. In particular, define the maximum gap \[\Delta_{\max}:=\max_i k_i-\min_i k_i \label{eq:S07_01_DeltaMax}\] then \[\Delta_{\max}= \begin{cases} 0,& r=0,\\ 1,& r=1,2, \end{cases} \label{eq:S07_01_DeltaMax_min}\] which is minimal. Solutions of the form [eq:S07_01_gap_ge2] have \(\Delta_{\max}\ge 2\) and violate the internal rule.

9.2.6.2 7.1.6.2 Minimal residual direction (minimum residual basis for charge/electron labels)

When the 3-sector axes are locked by \[\mathbf{n}_1+\mathbf{n}_2+\mathbf{n}_3=\mathbf{0} \label{eq:S07_01_nsum0}\] and the residual vector is defined by \[\mathbf{V}:=k_1\mathbf{n}_1+k_2\mathbf{n}_2+k_3\mathbf{n}_3 \label{eq:S07_01_V}\] then for the minimum-variance integerization output,

  1. For \(r=0\) (perfectly uniform), \((k_1,k_2,k_3)=(m,m,m)\), hence \(\mathbf{V}=m(\mathbf{n}_1+\mathbf{n}_2+\mathbf{n}_3)=\mathbf{0}\).

  2. For \(r=1\), the output is of the form \((m,m,m+1)\), hence \(\mathbf{V}\) points along the axis that receives the “extra 1” (e.g., \(\mathbf{n}_{i_1}\)).

  3. For \(r=2\), the output is of the form \((m,m+1,m+1)\), hence \(\mathbf{V}\) points opposite to the axis that receives “the missing 1” (e.g., \(-(\mathbf{n}_{i_3})\)).

Thus minimum-variance integerization leaves only the minimum-magnitude residual directionality that is forced by \(N\ (\mathrm{mod}\ 3)\). This residual directionality is used downstream as the input for charge sign and electron (survival) labels. Any larger residual (gap \(\ge 2\)) is forbidden as a violation of the internal rule.

9.2.7 7.1.7 Volume (radial) integerization: \(R_p=n_r\,r_u\)

This section adds, as an auxiliary closure separate from 3-sector integerization, a “volume (radial) integerization” rule. The purpose is to ensure that the total core count is not merely “an artifact of a sector decomposition,” but is also compatible with an integer-multiple selection of a radial scale.

9.2.7.1 [D-7.1-7.1] Sub-quantum (SQ) unit radius \(r_u\)

In the 82+7 structure, the “unit” (core 82, shell 7) is interpreted not as a single VP but as a higher-level block (Sub-quantum unit, SQ) containing many VPs (object attribution is locked in canon_lock). Denote the effective radius of this SQ unit by \(r_u\).

9.2.7.2 [D-7.1-7.2] Integer-multiple radial rule

Declare the following integer-multiple relation between the core radius \(R_p\) and the SQ unit radius \(r_u\) as an integerization rule: \[R_p = n_r\, r_u, \qquad n_r\in\mathbb{Z}_{>0}. \label{eq:S07_01_radial_integerization}\] Here \(n_r\) is the “radial integerization exponent” and cannot be chosen after seeing outcomes (post hoc choice is FAIL under G-NT).

9.2.7.3 [D-7.1-7.3] Ideal slot count (volume ratio) \(n_r^3\)

If [eq:S07_01_radial_integerization] is adopted, then the mathematical maximum number of slots for same-scale spherical units inside the core sphere is given by the volume ratio \[\frac{V_{\mathrm{core}}}{v_u}=\left(\frac{R_p}{r_u}\right)^3 = n_r^3 \label{eq:S07_01_slot_count}\] (where \(V_{\mathrm{core}}\propto R_p^3\) and \(v_u\propto r_u^3\)). This value is a mathematical upper bound assuming “full filling”; actual placement can be smaller due to rotation, exclusion, and voids.

9.2.8 7.1.8 Packing–rectification coefficient \(\phi_{\mathrm{pack}}\) and the \(125\rightarrow 82\) closed loop

This section records, as an operational coefficient, the fact that the “mathematical slot count” \(n_r^3\) is not realized as-is. Define the packing–rectification coefficient \(\phi_{\mathrm{pack}}\in(0,1]\) by \[N_{\mathrm{core}}\ :=\ \phi_{\mathrm{pack}}\,n_r^3, \qquad \phi_{\mathrm{pack}}:=\frac{N_{\mathrm{core}}}{n_r^3}. \label{eq:S07_01_pack_rect_def}\] Here \(N_{\mathrm{core}}\) is the core unit count (locked as \(82\) in this document), and \(\phi_{\mathrm{pack}}\) is not a post hoc tuning freedom. When \(N_{\mathrm{core}}\) and \(n_r\) are locked, \(\phi_{\mathrm{pack}}\) is an automatically derived quantity.

9.2.8.1 7.1.8.1 Locking \(n_r=5\) and \(125\rightarrow 82\)

In the core(82) regime of this document, lock the radial integerization exponent as \[n_r \equiv 5. \label{eq:S07_01_nr_lock5}\] This choice cannot be tuned after seeing the outcome (\(82\)); changes are allowed only by a version bump. Substituting [eq:S07_01_nr_lock5] into [eq:S07_01_slot_count], the ideal slot count is \[n_r^3 = 5^3 = 125. \label{eq:S07_01_125_slots}\] With \(N_{\mathrm{core}}=82\) locked, [eq:S07_01_pack_rect_def] immediately yields \[\phi_{\mathrm{pack}}=\frac{82}{125}\approx 0.66 \label{eq:S07_01_phi_pack_value}\] Thus about 34% of the “125 mathematical slots” remain as voids. In the unjamming/influx (Flux) narrative, these voids can be interpreted as channels and elastic margins, but interpretive eligibility must pass a separate Gate (regime/log/reproducibility).

9.2.8.2 7.1.8.2 Qualification for strong claims (inevitability of 82)

To claim that “82 is inevitable from radial integerization alone,” one must independently measure (or seal by simulation) \(\phi_{\mathrm{pack}}\) externally and show that \(n_r^3\phi_{\mathrm{pack}}\) converges to the integer 82. Conversely, in the present document where \(82\) is locked as a structural integer (minimum-variance 3-sector + 82+7 structure), [eq:S07_01_phi_pack_value] should be treated as a back-calculated consistency indicator rather than a prediction.

9.2.8.3 7.1.8.3 Gate (interpretability decision)

A physical interpretation of \(\phi_{\mathrm{pack}}\) (e.g., “jamming packing efficiency”) is permitted only when the following conditions are simultaneously satisfied.

  1. \(n_r\) is pre-registered in analysis_lock/canon_lock and is not changed after seeing outcomes.

  2. The object attribution for \(N_{\mathrm{core}}\) (what is counted as 1 unit) and the inclusion/exclusion conventions are locked in canon_lock.

  3. \(\phi_{\mathrm{pack}}\) is not absorbed into or redefined as another coefficient (\(\delta,\alpha,\phi_{\mathrm{jam}}\), etc.). Symbol overloading is FAIL under G-SYM.

If any of these conditions is violated, \(\phi_{\mathrm{pack}}\) may be reported as a number (CT-LIM), but using its interpretation as a basis for a conclusion is forbidden.

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9.3 7.2 Definition of build times \(T_p, T_n\)

Let the time variable \(t\) denote realization time (unit: s). An event is counted according to the operational definition locked in canon_lock, and event counts are recorded as dimensionless integers. For a time interval \([t_1,t_2)\), define the raw event set and raw event count by \[\mathcal{E}_0[t_1,t_2) :=\{\,e\mid t_1\le t(e)<t_2\,\}, \qquad N_0(t_1,t_2):=\bigl|\mathcal{E}_0[t_1,t_2)\bigr|. \label{eq:S07_02_eventset}\] Define the canonical event rate (canonical turnover rate) \(\nu_{p,\mathrm{can}}\) by the limit \[\nu_{p,\mathrm{can}} := \lim_{T\to\infty}\frac{N_0(t,t+T)}{T}. \label{eq:S07_02_nu_can_def}\] Definition [eq:S07_02_nu_can_def] is the definition of “events per unit time,” and the unit of \(\nu_{p,\mathrm{can}}\) is locked as \([\mathrm{s}^{-1}]\). When \(\nu_{p,\mathrm{can}}\) is treated as a constant within the same regime, it is used so that \[N_0(t_1,t_2)\equiv \nu_{p,\mathrm{can}}\,(t_2-t_1). \label{eq:S07_02_N_equals_nuT}\] Equation [eq:S07_02_N_equals_nuT] is the usage convention of the canonical event rate. If the regime is not locked or Gate judges INCONCLUSIVE, this usage is forbidden.

9.3.2 7.2.2 Definition of structure counts \(N_p, N_n\) (82+7 and 82)

Build time is defined as “the time required to accumulate a prescribed structure count \(N\).” Lock two structure counts as follows.

  1. Total count of the \(p\)-structure: \[N_p:=82+7=89. \label{eq:S07_02_Np_def}\]

  2. Total count of the \(n\)-structure: \[N_n:=82. \label{eq:S07_02_Nn_def}\]

Here \(82\) and \(7\) are integers provided by the definition of the discrete structure (core 82, shell 7). Their object attribution and inclusion/exclusion conventions must be locked in canon_lock. Changing \(N_p,N_n\) after seeing outcomes is forbidden; changes are allowed only by a version bump.

9.3.3 7.2.3 Minimum-variance 3-sector integerization and sector counts \((k_1,k_2,k_3)\)

3-sector integerization decomposes a total count \(N\) into three sector integers while minimizing bias (variance). Divide \(N\) as \[N=3m+r, \qquad m:=\left\lfloor\frac{N}{3}\right\rfloor, \qquad r\in\{0,1,2\}. \label{eq:S07_02_div}\] Lock the minimum-variance condition as choosing an integer triple satisfying \[|k_i-k_j|\le 1\qquad (i,j\in\{1,2,3\}), \qquad k_1+k_2+k_3=N \label{eq:S07_02_minvar}\] The solutions are fully classified up to permutation as \[(k_1,k_2,k_3)= \begin{cases} (m,m,m),& r=0,\\ (m,m,m+1)\ \text{(permuted)},& r=1,\\ (m,m+1,m+1)\ \text{(permuted)},& r=2. \end{cases} \label{eq:S07_02_triplets}\] Because the permutation choice (which sector receives \(m+1\)) cannot be decided after seeing outcomes, pre-register and lock the sector priority permutation \(\pi_{\mathrm{sec}}\) in analysis_lock. Using \(\pi_{\mathrm{sec}}\), assign \(m+1\) to \(r\) sectors and \(m\) to the remainder.

9.3.3.1 Sector decomposition of \(N_p=89\)

From [eq:S07_02_Np_def], \[89=3\cdot 29+2 \quad\Longrightarrow\quad m_p=29,\ r_p=2. \label{eq:S07_02_89}\] Hence the minimum-variance sector counts are \[(k_{p,1},k_{p,2},k_{p,3}) \ \text{is a permutation of } (29,30,30). \label{eq:S07_02_kp}\]

9.3.3.2 Sector decomposition of \(N_n=82\)

From [eq:S07_02_Nn_def], \[82=3\cdot 27+1 \quad\Longrightarrow\quad m_n=27,\ r_n=1. \label{eq:S07_02_82}\] Hence the minimum-variance sector counts are \[(k_{n,1},k_{n,2},k_{n,3}) \ \text{is a permutation of } (27,27,28). \label{eq:S07_02_kn}\]

9.3.4 7.2.4 Definition of build time (canonical event-rate based)

9.3.4.1 [D-7.2-1] Build-time function

In a regime where the canonical event rate \(\nu_{p,\mathrm{can}}\) is locked, define the mean time required to accumulate total count \(N\) as the build time: \[T_{\mathrm{build}}(N) :=\frac{N}{\nu_{p,\mathrm{can}}}. \label{eq:S07_02_Tbuild_def}\] Definition [eq:S07_02_Tbuild_def] is the basic time–count link. It is not an additional assumption; it is equivalent to solving \(N=\nu T\) for \(T=N/\nu\) in [eq:S07_02_N_equals_nuT].

9.3.4.2 \(p\)-structure build time \(T_p\)

From [eq:S07_02_Tbuild_def] and [eq:S07_02_Np_def], \[T_p :=T_{\mathrm{build}}(N_p) =\frac{N_p}{\nu_{p,\mathrm{can}}} =\frac{89}{\nu_{p,\mathrm{can}}}. \label{eq:S07_02_Tp_def}\]

9.3.4.3 \(n\)-structure build time \(T_n\)

From [eq:S07_02_Tbuild_def] and [eq:S07_02_Nn_def], \[T_n :=T_{\mathrm{build}}(N_n) =\frac{N_n}{\nu_{p,\mathrm{can}}} =\frac{82}{\nu_{p,\mathrm{can}}}. \label{eq:S07_02_Tn_def}\]

9.3.5 7.2.5 Time–structure invariants (conclusions where the canonical event rate cancels)

From [eq:S07_02_Tp_def]–[eq:S07_02_Tn_def], the following invariants follow immediately.

9.3.5.1 7.2.5.1 Ratio invariant

\[\frac{T_p}{T_n} = \frac{89/\nu_{p,\mathrm{can}}}{82/\nu_{p,\mathrm{can}}} = \frac{89}{82}. \label{eq:S07_02_ratio_invariant}\] Hence \(T_p/T_n\) is a structural ratio invariant independent of the value of \(\nu_{p,\mathrm{can}}\).

9.3.5.2 7.2.5.2 Difference invariant (time version of the count difference 7)

\[T_p-T_n = \frac{89}{\nu_{p,\mathrm{can}}}-\frac{82}{\nu_{p,\mathrm{can}}} = \frac{7}{\nu_{p,\mathrm{can}}}. \label{eq:S07_02_diff_invariant}\] That is, the build-time difference between the \(p\)-structure and the \(n\)-structure is fixed as the time version of “additional count 7” under the canonical event rate.

9.3.5.3 7.2.5.3 Count–time closed loop (dimensionless coupling)

\[\nu_{p,\mathrm{can}}\,T_p = 89, \qquad \nu_{p,\mathrm{can}}\,T_n = 82. \label{eq:S07_02_closed_loop}\] Equation [eq:S07_02_closed_loop] is a dimensionless closed loop in which “canonical rate \(\times\) build time” returns to the structure count. It is meaningful only under the same locked version.

For the 3-sector integerization outputs [eq:S07_02_kp], [eq:S07_02_kn], define the sector-wise build times by \[T_{p,i}:=\frac{k_{p,i}}{\nu_{p,\mathrm{can}}} \quad (i=1,2,3), \qquad T_{n,i}:=\frac{k_{n,i}}{\nu_{p,\mathrm{can}}} \quad (i=1,2,3). \label{eq:S07_02_sector_times}\] By sum preservation [eq:S07_02_minvar], \[T_{p,1}+T_{p,2}+T_{p,3} = \frac{k_{p,1}+k_{p,2}+k_{p,3}}{\nu_{p,\mathrm{can}}} = \frac{N_p}{\nu_{p,\mathrm{can}}} = T_p, \label{eq:S07_02_sector_sum_p}\] \[T_{n,1}+T_{n,2}+T_{n,3} = \frac{k_{n,1}+k_{n,2}+k_{n,3}}{\nu_{p,\mathrm{can}}} = \frac{N_n}{\nu_{p,\mathrm{can}}} = T_n. \label{eq:S07_02_sector_sum_n}\] Hence build time decomposes fully into the time version of 3-sector structure counts.

9.3.7 7.2.7 Regime conditions and failure handling (undefined cases and violations)

For the definitions in this section to have conclusion status, the following conditions must be locked.

  1. \(\nu_{p,\mathrm{can}}\) must be treatable as a constant within the same regime, and the event logs required by its definition ([eq:S07_02_nu_can_def]) must be complete.

  2. The object attribution and inclusion/exclusion conventions for the structure counts \(N_p,N_n\) must be locked.

  3. In 3-sector integerization, the permutation-selection rule (priority \(\pi_{\mathrm{sec}}\)) must be locked, and the minimum-variance condition [eq:S07_02_minvar] must not be violated.

If these conditions are not satisfied, \(T_p,T_n\) are judged undefined (INCONCLUSIVE) or as regime/structure violations (FAIL), and the outputs do not have conclusion status.

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9.4 7.3 Definitions of \(\Phi\) and \(\chi\) (handoff criteria between observation and analysis)

9.4.1 7.3.1 Purpose

This section (i) fixes observable aggregates \(\Phi\) and \(\chi\) by internal definitions, (ii) specifies the measurability conditions of \(\Phi\) and \(\chi\) (log-based computability), and (iii) defines the Gate that hands \(\Phi\) and \(\chi\) off to the analysis part (closure/gate/estimator stack). The \(\Phi\) and \(\chi\) in this section are defined from event logs and the 3-sector integerization + rectification conventions only, with no appeal to external texts.

9.4.2 7.3.2 Minimal schema for observation input (event log)

Because \(\Phi\) and \(\chi\) are event-aggregation quantities, the event log must include the following fields as mandatory. Missing fields make the quantities undefined and must be judged immediately as INCONCLUSIVE or FAIL.

  1. Event set: \(\mathcal{E}\).

  2. Event index: \(e\in\mathcal{E}\).

  3. Event time (tick): \(n(e)\in\mathbb{Z}\).

  4. Tick boundaries for window definition: \(n_1<n_2\).

  5. Pre/post state identifiers: \(\mathrm{pre}(e)\), \(\mathrm{post}(e)\).

  6. VP subset involved in the event: \(\mathcal{V}(e)\subseteq\mathcal{V}\).

  7. 3-sector integerization input: either total count \(N(e)\) or the sector triple \((k_1(e),k_2(e),k_3(e))\).

  8. (If rectified survival is used) two phase values: \(\theta(e)\in[0,2\pi)\), \(\varphi(e)\in[0,2\pi)\).

The realization time tick \(\Delta t\) used in this section must be locked in realization_lock. Define the window duration by \[\Delta T := (n_2-n_1)\Delta t \label{eq:S07_03_DeltaT}\]

9.4.3 7.3.3 Definition of \(\Phi\) (survival-rectification log-odds)

9.4.3.1 7.3.3.1 Time window and raw event set

Define the raw event set in the tick interval \([n_1,n_2)\) by \[\mathcal{E}_0[n_1,n_2) :=\{\,e\in\mathcal{E}\mid n_1\le n(e)<n_2\,\}. \label{eq:S07_03_E0}\] Define the raw event count by \[N_0 := \bigl|\mathcal{E}_0[n_1,n_2)\bigr|. \label{eq:S07_03_N0}\] If \(N_0=0\), \(\Phi\) is undefined.

9.4.3.2 7.3.3.2 Survival weight and rectified event count

Define the half-wave rectifier by \[_+ := \max(0,x). \label{eq:S07_03_pospart}\] Define the survival weight of an event \(e\) by \[w(e) := [\cos\theta(e)]_+\,[\cos\varphi(e)]_+. \label{eq:S07_03_w}\] Define the rectified event count by \[N_\delta :=\sum_{e\in\mathcal{E}_0[n_1,n_2)} w(e). \label{eq:S07_03_Ndelta}\] By definition \(0\le N_\delta\le N_0\).

9.4.3.3 7.3.3.3 Survival ratio and stabilization constant \(\varepsilon_\Phi\)

Define the survival ratio by \[\rho := \frac{N_\delta}{N_0}. \label{eq:S07_03_rho}\] Because the log-odds diverges at \(\rho=0\) or \(\rho=1\), lock a stabilization constant \(\varepsilon_\Phi\) by \[\varepsilon_\Phi>0, \qquad \varepsilon_\Phi\ \text{is pre-registered in } \texttt{analysis\_lock}. \label{eq:S07_03_epsphi_lock}\] Define the stabilized survival ratio by \[\tilde{\rho} :=\frac{N_\delta+\varepsilon_\Phi}{N_0+2\varepsilon_\Phi}. \label{eq:S07_03_rhotilde}\] Then \(0<\tilde{\rho}<1\) holds.

9.4.3.4 7.3.3.4 Definition of \(\Phi\) (log-odds)

Define \(\Phi\) by \[\Phi :=\log\left(\frac{\tilde{\rho}}{1-\tilde{\rho}}\right). \label{eq:S07_03_Phi_def}\] The type of \(\Phi\) is scalar and its dimension is dimensionless (\([1]\)). \(\Phi\) is an observable that aggregates “is survival dominant or is cancellation dominant” as a log-odds. It exists only as a quantity computed from event logs under a locked protocol.

9.4.4 7.3.4 Definition of \(\chi\) (directional coherence \(\chi_{\mathrm{dir}}\) and internal stability \(\chi_{\mathrm{int}}\))

9.4.4.1 7.3.4.1 3-sector residual vector (integer-based)

Lock one of the following two methods for obtaining 3-sector integerization results on a window \([n_1,n_2)\).

  1. Event-wise integerization: each event \(e\) provides \((k_1(e),k_2(e),k_3(e))\).

  2. Window-aggregate integerization: first define the total count \(N\) over the window, and then compute \((k_1,k_2,k_3)\) by minimum-variance integerization.

If window-aggregate integerization is used, define the total count by \[N := \sum_{e\in\mathcal{E}_0[n_1,n_2)} N(e), \label{eq:S07_03_N_total}\] and compute \((k_1,k_2,k_3)\) by the minimum-variance rule in §7.1. The 3-sector axes \(\{\mathbf{n}_1,\mathbf{n}_2,\mathbf{n}_3\}\) are locked by the 120 conditions: \[\mathbf{n}_i\cdot\mathbf{n}_j= \begin{cases} 1,& i=j,\\ -\dfrac{1}{2},& i\neq j, \end{cases} \qquad \mathbf{n}_1+\mathbf{n}_2+\mathbf{n}_3=\mathbf{0}. \label{eq:S07_03_sector_axes}\] Define the window-aggregate residual vector by \[\mathbf{V} :=k_1\mathbf{n}_1+k_2\mathbf{n}_2+k_3\mathbf{n}_3. \label{eq:S07_03_V_window}\]

If event-wise integerization is used, define the event-wise residual vector by \[\mathbf{V}(e) :=k_1(e)\mathbf{n}_1+k_2(e)\mathbf{n}_2+k_3(e)\mathbf{n}_3 \label{eq:S07_03_V_event}\] and perform the following aggregation event-wise.

9.4.4.2 7.3.4.2 Directional coherence \(\chi_{\mathrm{dir}}\)

Lock the coherence axis (direction reference) \(\mathbf{n}_\chi\) as \[\mathbf{n}_\chi\in \Pi,\qquad \|\mathbf{n}_\chi\|=1, \qquad \mathbf{n}_\chi\ \text{is pre-registered in } \texttt{analysis\_lock}. \label{eq:S07_03_nchi_lock}\]

If event-wise aggregation is used, define the effective event set by \[\mathcal{E}_{\mathrm{eff}} :=\{\,e\in\mathcal{E}_0[n_1,n_2)\mid \|\mathbf{V}(e)\|>0\,\}, \qquad N_{\mathrm{eff}}:=|\mathcal{E}_{\mathrm{eff}}|. \label{eq:S07_03_Eeff}\] Define the unit residual direction by \[\widehat{\mathbf{V}}(e):=\frac{\mathbf{V}(e)}{\|\mathbf{V}(e)\|}\qquad (e\in\mathcal{E}_{\mathrm{eff}}). \label{eq:S07_03_Vhat}\] Define the directional coherence by \[\chi_{\mathrm{dir}} := \left| \frac{1}{N_{\mathrm{eff}}}\sum_{e\in\mathcal{E}_{\mathrm{eff}}}\widehat{\mathbf{V}}(e)\cdot \mathbf{n}_\chi \right|. \label{eq:S07_03_chidir_def}\] By definition \(0\le \chi_{\mathrm{dir}}\le 1\). If \(N_{\mathrm{eff}}=0\), \(\chi_{\mathrm{dir}}\) is undefined.

If only a single window-aggregate residual vector is used, define it by a single projection: \[\chi_{\mathrm{dir}} := \left|\frac{\mathbf{V}}{\|\mathbf{V}\|}\cdot \mathbf{n}_\chi\right| \qquad (\|\mathbf{V}\|>0). \label{eq:S07_03_chidir_window}\]

9.4.4.3 7.3.4.3 Internal stability \(\chi_{\mathrm{int}}\) (multi-window consistency)

Define internal stability using the dependence of \(\Phi\) on window length. Lock a set of multiple window lengths by \[\mathcal{W}:=\{\Delta N_1,\Delta N_2,\ldots,\Delta N_M\}, \qquad \Delta N_m\in\mathbb{Z}_{>0}, \qquad \mathcal{W}\ \text{is pre-registered in } \texttt{analysis\_lock}. \label{eq:S07_03_Wset}\] For each window length \(\Delta N_m\), compute \(\Phi_m\) using the same protocol (same event definition, same survival weights, same \(\varepsilon_\Phi\)): \[\Phi_m := \Phi(\Delta N_m). \label{eq:S07_03_Phim}\] Define the mean and variance by \[\overline{\Phi} :=\frac{1}{M}\sum_{m=1}^{M}\Phi_m, \qquad \sigma_\Phi^2 :=\frac{1}{M}\sum_{m=1}^{M}(\Phi_m-\overline{\Phi})^2, \qquad \sigma_\Phi:=\sqrt{\sigma_\Phi^2}. \label{eq:S07_03_sigmaPhi}\] Lock an internal-stability scale \(\varepsilon_{\chi}>0\) by \[\varepsilon_{\chi}>0, \qquad \varepsilon_{\chi}\ \text{is pre-registered in } \texttt{analysis\_lock}. \label{eq:S07_03_epschi_lock}\] Define \(\chi_{\mathrm{int}}\) by \[\chi_{\mathrm{int}} := \frac{1}{1+\sigma_\Phi/\varepsilon_{\chi}}. \label{eq:S07_03_chiint_def}\] By definition \(0<\chi_{\mathrm{int}}\le 1\), and if \(\sigma_\Phi=0\) then \(\chi_{\mathrm{int}}=1\). If \(M=0\) or any \(\Phi_m\) is undefined, then \(\chi_{\mathrm{int}}\) is undefined.

9.4.4.4 7.3.4.4 Combined coherence \(\chi\)

Define the combined coherence by the product \[\chi := \chi_{\mathrm{dir}}\chi_{\mathrm{int}}. \label{eq:S07_03_chi_def}\] Hence \(0\le \chi\le 1\). \(\chi\) is an observable but is also used as the core input that decides whether the quantity can be elevated to an “analysis-eligible” input.

9.4.5 7.3.5 Measurability conditions (preventing undefined cases)

The measurability of \(\Phi\) and \(\chi\) holds only when the following conditions are satisfied.

  1. Log completeness: all fields listed in §7.3.2 exist and each field schema is locked by protocol_lock.

  2. Sample size: \(N_0\ge N_{\min}\) and \(N_{\mathrm{eff}}\ge N_{\min,\mathrm{eff}}\).

  3. Rectification definition preserved: the definition [eq:S07_03_w] of \(w(e)\) is preserved under the same version.

  4. Integerization rule preserved: the 3-sector axes [eq:S07_03_sector_axes] and the minimum-variance integerization rule (§7.1) are preserved under the same version.

  5. Multi-window set fixed: \(\mathcal{W}\) is locked and each \(\Phi_m\) is computed under the same convention.

\(N_{\min}\) and \(N_{\min,\mathrm{eff}}\) are thresholds pre-registered in gate_lock and cannot be moved after seeing outcomes.

9.4.6 7.3.6 Gate to hand off to the analysis part (definitions, verdicts, and registration convention)

9.4.6.1 7.3.6.1 Gate classes and standard outputs

The handoff Gate for \(\Phi\) and \(\chi\) consists of the following three gates. Outputs are recorded only as PASS/FAIL/INCONCLUSIVE.

  1. G-OBS-PHI: judge definability and stability of \(\Phi\).

  2. G-OBS-CHI: judge definability of \(\chi\) and threshold passage for coherence.

  3. G-HANDOFF: assuming simultaneous PASS of G-OBS-PHI and G-OBS-CHI, register quantities as analysis inputs (OBS-REF).

9.4.6.2 7.3.6.2 Verdict for G-OBS-PHI

Judge G-OBS-PHI=PASS if and only if all of the following conditions hold.

  1. \(N_0\ge N_{\min}\).

  2. \(\Phi\) is computable by [eq:S07_03_Phi_def] (i.e., \(\varepsilon_\Phi\) is locked and \(N_0>0\)).

  3. The multi-window set \(\mathcal{W}\) is locked ([eq:S07_03_Wset]) and all \(\Phi_m\) are definable.

If any required schema item or lock is missing, judge INCONCLUSIVE. If a definition conflict is detected (e.g., symbol-meaning conflict for \(\Phi\), post hoc change of \(\varepsilon_\Phi\), replacement of \(w(e)\)), judge FAIL.

9.4.6.3 7.3.6.3 Verdict for G-OBS-CHI

Judge G-OBS-CHI=PASS if and only if all of the following conditions hold.

  1. \(N_{\mathrm{eff}}\ge N_{\min,\mathrm{eff}}\).

  2. \(\chi_{\mathrm{dir}}\) is definable by [eq:S07_03_chidir_def] or [eq:S07_03_chidir_window] (i.e., \(\mathbf{n}_\chi\) is locked and the \(\|\mathbf{V}\|>0\) condition holds).

  3. \(\chi_{\mathrm{int}}\) is definable by [eq:S07_03_chiint_def] (i.e., \(\mathcal{W}\) and \(\varepsilon_{\chi}\) are locked and \(\Phi_m\) are definable).

  4. Threshold passage: \[\chi_{\mathrm{dir}}\ge \chi_{\mathrm{dir,min}}, \qquad \chi_{\mathrm{int}}\ge \chi_{\mathrm{int,min}}, \qquad \chi\ge \chi_{\min}, \label{eq:S07_03_chi_thresholds}\] where \(\chi_{\mathrm{dir,min}},\chi_{\mathrm{int,min}},\chi_{\min}\) are thresholds pre-registered in gate_lock.

If thresholds are not locked, or \(\mathbf{n}_\chi\) is not locked, or integerization rules are mixed, judge INCONCLUSIVE or FAIL. Moving thresholds after seeing results is a No-Tuning violation and is FAIL.

9.4.6.4 7.3.6.4 G-HANDOFF (registration of analysis inputs)

Define G-HANDOFF by \[\texttt{G-HANDOFF}=\texttt{PASS} \quad\Longleftrightarrow\quad (\texttt{G-OBS-PHI}=\texttt{PASS})\ \wedge\ (\texttt{G-OBS-CHI}=\texttt{PASS}). \label{eq:S07_03_handoff_rule}\] If G-HANDOFF=PASS, then \(\Phi\), \(\chi_{\mathrm{dir}}\), \(\chi_{\mathrm{int}}\), \(\chi\) are registered as analysis inputs (OBS-REF). Lock the registration record format as follows.

obs_ref_records:
  - quantity_id: Q-PHI-001
    value: (Phi)
    window: [n1,n2)
    lock_refs: {canon_lock_id, realization_lock_id, analysis_lock_id}
    gate_refs: {G-OBS-PHI: PASS, G-OBS-CHI: PASS, G-HANDOFF: PASS}
  - quantity_id: Q-CHI-DIR-001
    value: (chi_dir)
    window: [n1,n2)
    lock_refs: { ... }
    gate_refs: { ... }
  - quantity_id: Q-CHI-INT-001
    value: (chi_int)
    window: [n1,n2)
    lock_refs: { ... }
    gate_refs: { ... }
  - quantity_id: Q-CHI-001
    value: (chi)
    window: [n1,n2)
    lock_refs: { ... }
    gate_refs: { ... }

These records must be sealed by inclusion into analysis_lock or registry_snapshot. Unsealed values cannot be used as analysis inputs.

9.4.6.5 7.3.6.5 Allowed scope under FAIL/INCONCLUSIVE

If G-HANDOFF\(\neq\)PASS, then \(\Phi\) and \(\chi\) cannot be used as the basis of a conclusion. Only the following is permitted.

  1. Record as a limit conclusion (CT-LIM) together with labels for “undefined/violated conditions.”

  2. Record the causal labels (missing logs, missing locks, regime violation, threshold failure, No-Tuning violation) in the Gate report.

Sentences that neutralize FAIL/INCONCLUSIVE by interpretation are forbidden.

LOCK/Gate connections for this section (none if empty)

9.5 7.4 Sector/sign (\(\pm\))\(\rightarrow\) electron/positron labels

9.5.1 7.4.1 Purpose

This section fixes, by definition, an “electron label” and a “positron label” using the 3-sector integerization output and the sign of the residual (non-cancelled) vector. This section does not interpret or justify the meaning of the labels. The outputs of this section are (i) the label definition formulas, (ii) the conditions under which labels are definable, and (iii) the handling rule when labels are undefined.

9.5.2 7.4.2 Inputs (LOCK): 3-sector axes, integerization, residual vector

9.5.2.1 7.4.2.1 Locking the 3-sector axes

Lock the 3-sector axes \(\{\mathbf{n}_1,\mathbf{n}_2,\mathbf{n}_3\}\) by the 120 conventions: \[\mathbf{n}_i\cdot\mathbf{n}_j= \begin{cases} 1,& i=j,\\ -\dfrac{1}{2},& i\neq j, \end{cases} \qquad \mathbf{n}_1+\mathbf{n}_2+\mathbf{n}_3=\mathbf{0}. \label{eq:S07_04_axes}\] The axis choice and ordering must be locked in analysis_lock and cannot be swapped after seeing outcomes.

9.5.2.2 7.4.2.2 Sector integerization output

For a total count \(N\), let the sector integerization output be \[(k_1,k_2,k_3)\in\mathbb{Z}_{\ge 0}^3, \qquad k_1+k_2+k_3=N \label{eq:S07_04_k}\] The generation rule of \((k_1,k_2,k_3)\) (minimum-variance + tie-break) must be locked in analysis_lock.

9.5.2.3 7.4.2.3 Definition of the residual vector

Define the residual (non-cancelled) vector by \[\mathbf{V}:=k_1\mathbf{n}_1+k_2\mathbf{n}_2+k_3\mathbf{n}_3. \label{eq:S07_04_V}\] Given the integerization output and the locked axes, \(\mathbf{V}\) is uniquely determined.

9.5.3 7.4.3 Locking the reference axis for the label (charge axis) \(\mathbf{n}_Q\)

Define electron/positron labels by judging the sign of \(\mathbf{V}\) along a single reference axis. Let the reference axis (label axis) be \(\mathbf{n}_Q\) and lock \[\mathbf{n}_Q\in\Pi,\qquad \|\mathbf{n}_Q\|=1, \qquad \mathbf{n}_Q\ \text{is pre-registered in } \texttt{analysis\_lock}. \label{eq:S07_04_nQ_lock}\] Here \(\Pi\) is the sector plane on which the 3-sector axes lie. \(\mathbf{n}_Q\) cannot be chosen after seeing outcomes; changing \(\mathbf{n}_Q\) is allowed only by a version bump.

9.5.4 7.4.4 Sign verdict function and label definition

9.5.4.1 7.4.4.1 Sign verdict function

Define the sign verdict function by \[s_Q(\mathbf{V}) := \mathrm{sgn}\!\left(\mathbf{V}\cdot\mathbf{n}_Q\right), \qquad \mathrm{sgn}(x)= \begin{cases} +1,& x>0,\\ 0,& x=0,\\ -1,& x<0. \end{cases} \label{eq:S07_04_sign}\] \(s_Q(\mathbf{V})\in\{+1,0,-1\}\), where \(0\) is a reserved value indicating a boundary (neutral) or an undefined verdict.

9.5.4.2 7.4.4.2 Non-degeneracy (verdict-eligible) threshold

For stability of sign verdicts, lock a non-degeneracy threshold \(V_{\min}>0\) by \[V_{\min}>0, \qquad V_{\min}\ \text{is pre-registered in } \texttt{gate\_lock}. \label{eq:S07_04_Vmin}\] Define the non-degeneracy condition by \[\|\mathbf{V}\|\ge V_{\min}. \label{eq:S07_04_nondeg}\] If [eq:S07_04_nondeg] does not hold, the label is classified as undefined (or treated as neutral).

9.5.4.3 7.4.4.3 Definition of electron/positron labels (definition only)

Define the electron label and positron label by \[\mathcal{L}_{e}(\mathbf{V}) := \begin{cases} \texttt{ELECTRON}, & \|\mathbf{V}\|\ge V_{\min}\ \wedge\ s_Q(\mathbf{V})=+1,\\ \texttt{UNDEFINED}, & \text{otherwise}, \end{cases} \label{eq:S07_04_label_e}\] \[\mathcal{L}_{\bar e}(\mathbf{V}) := \begin{cases} \texttt{POSITRON}, & \|\mathbf{V}\|\ge V_{\min}\ \wedge\ s_Q(\mathbf{V})=-1,\\ \texttt{UNDEFINED}, & \text{otherwise}. \end{cases} \label{eq:S07_04_label_ebar}\] Also define a combined single label function by \[\mathcal{L}(\mathbf{V}) := \begin{cases} \texttt{ELECTRON}, & \|\mathbf{V}\|\ge V_{\min}\ \wedge\ s_Q(\mathbf{V})=+1,\\ \texttt{POSITRON}, & \|\mathbf{V}\|\ge V_{\min}\ \wedge\ s_Q(\mathbf{V})=-1,\\ \texttt{NEUTRAL}, & \|\mathbf{V}\|<V_{\min}\ \vee\ s_Q(\mathbf{V})=0. \end{cases} \label{eq:S07_04_label_combined}\] In [eq:S07_04_label_combined], NEUTRAL is a reserved output meaning “label neutral/undefined.” Whether this output is permissible in conclusion sentences (e.g., only as CT-LIM) must be locked by PASS.rules.

When the 3-sector axes are locked by [eq:S07_04_axes], minimum-variance integerization yields a residual direction determined by \(N\ (\mathrm{mod}\ 3)\). This section records that fact only as a definitional link.

  1. If \(N\equiv 0\ (\mathrm{mod}\ 3)\) and minimum-variance integerization gives \((m,m,m)\), then \(\mathbf{V}=\mathbf{0}\) and [eq:S07_04_label_combined] yields NEUTRAL (it fails the non-degeneracy threshold).

  2. If \(N\equiv 1\ (\mathrm{mod}\ 3)\) and the output is a permutation of \((m,m,m+1)\), then \(\mathbf{V}\) points along the sector axis that receives “\(+1\).”

  3. If \(N\equiv 2\ (\mathrm{mod}\ 3)\) and the output is a permutation of \((m,m+1,m+1)\), then \(\mathbf{V}\) points opposite to the sector axis that receives \(m\) (the “missing 1”).

These items are not an interpretation of labels. They are structural facts following from the definition [eq:S07_04_V] of \(\mathbf{V}\) and the axis-sum condition [eq:S07_04_axes]. The label decision is made only from the sign of \(\mathbf{V}\cdot\mathbf{n}_Q\).

9.5.6 7.4.6 Label prohibition rules (no interpretation, no redefinition)

Fix the following prohibition rules for the labels defined in this section.

  1. No interpretation: do not describe the physical meaning of the outputs ELECTRON/POSITRON/NEUTRAL in this section. This section provides definitions only.

  2. No post hoc axis selection: \(\mathbf{n}_Q\) cannot be chosen after seeing outcomes; axis choice must be pre-registered in analysis_lock.

  3. No post hoc threshold shift: post hoc movement of \(V_{\min}\) is forbidden; changes are allowed only by a version bump.

  4. No bypassing integerization: directly assigning labels while bypassing \((k_1,k_2,k_3)\) or \(\mathbf{V}\) is forbidden.

  5. No mixing: mixing results from different lock_id combinations (axes/integerization/thresholds) into a single label conclusion is forbidden.

LOCK/Gate connections for this section (none if empty)

10 8. Discrete Proton Structure (82+7)

Purpose of the chapter (locking definitions and outputs)

This chapter defines the proton structure as a discrete coordinate set and a contact graph, and locks a verification frame that checks compatibility with the continuum-core results (Chapter 6) and with the rectification/integerization conventions (Chapters 5–7). The outputs of this chapter are locked into the following three bundles.

  1. Coordinate outputs: the core-82 coordinate set \(\mathcal{X}_{82}\), the shell-7 coordinate (or vector) set \(\mathcal{S}_{7}\), the center \(\mathbf{x}_c\), and the coordinate-system locks.

  2. Graph outputs: node sets (82 or 89), edge sets (contact/adjacency), and the definitions of structural indicators such as contact degree/path/backbone.

  3. Verification outputs: structural invariants, radius/cross-section consistency, the cancellation–survival convention, regime/lock integrity, a reproducibility package, and Gate decision logs.

In this chapter, “82+7” is a definition-locked integer structure; changing these integers after seeing outcomes is forbidden. Any change of the integers is allowed only by a version bump.

Declaration of the coordinate system (locking coordinates/units/reference point)

Coordinate system and units

Discrete coordinates are defined on the 3D Euclidean space \(\mathbb{R}^3\). The coordinate unit is fixed to an internal length unit, and a dimensionless coordinate system may be used with the normalization length \(L_q\) as the reference. The dimensionless coordinate system is defined by the following transformation. \[\tilde{\mathbf{x}} := \frac{\mathbf{x}}{L_q}, \qquad \tilde{\mathbf{s}} := \frac{\mathbf{s}}{L_q}. \label{eq:S08_x_tilde}\] Here \(L_q\) is the selection length locked in §6.1, and [eq:S08_x_tilde] is used only in versions where the identification \(L_q=\lambda_C\) is locked.

Locking the center \(\mathbf{x}_c\)

Lock the center \(\mathbf{x}_c\) of the core structure as follows. \[\mathbf{x}_c\in\mathbb{R}^3, \qquad \mathbf{x}_c\ \text{is determined by a pre-registered selection convention in}\ \texttt{analysis\_lock}. \label{eq:S08_center_lock}\] The center \(\mathbf{x}_c\) cannot be moved after seeing outcomes. Re-selecting the center is allowed only by a version bump.

Definition of the core-82 coordinate set \(\mathcal{X}_{82}\)

Define the core 82 as a set of 82 coordinate points (or representative points). \[\mathcal{X}_{82}:=\{\mathbf{x}_i\}_{i=1}^{82}, \qquad \mathbf{x}_i\in\mathbb{R}^3. \label{eq:S08_X82_def}\] The meaning of each coordinate (what a representative point stands for: VP representative point/contact center/lattice node, etc.) is locked by the object definition in canon_lock. The coordinate set \(\mathcal{X}_{82}\) is stored as a single source of truth (SSOT), and it must not be redefined as a different coordinate set elsewhere in the main text.

Definition of the shell-7 set \(\mathcal{S}_{7}\)

Define the shell 7 as a set of 7 coordinates (or vectors). \[\mathcal{S}_{7}:=\{\mathbf{s}_k\}_{k=1}^{7}, \qquad \mathbf{s}_k\in\mathbb{R}^3. \label{eq:S08_S7_def}\] The meaning of \(\mathbf{s}_k\) (whether it is a shell coordinate, a shell direction vector, or an event marker point) is locked by the object definition in canon_lock. The cancellation–survival convention for the shell is locked later as a separate definition; in this overview we include only the requirement that “the shell 7 attaches near the core boundary” in the verification frame (linked to the conditions in §6.4).

Declaration of the graph (contact/adjacency) (locking nodes/edges/weights)

Node sets

The node set used in this chapter is locked to one of the following two options depending on the regime and purpose. \[\mathcal{V}_{82}:=\{1,2,\ldots,82\}, \qquad \mathcal{V}_{89}:=\{1,2,\ldots,82,\,82+1,\ldots,82+7\}. \label{eq:S08_nodes}\] \(\mathcal{V}_{82}\) is used for validating the internal core structure, while \(\mathcal{V}_{89}\) is used for validating the core–shell coupling and the cancellation–survival convention. Within a single output, the two node sets must not be mixed.

Contact predicate and edge set

Define the contact predicate \(C(i,j)\) for nodes \(i,j\) as follows. \[C(i,j)\in\{0,1\}, \qquad C(i,j)=1\ \Longleftrightarrow\ \text{satisfies the pre-registered contact convention}. \label{eq:S08_contact_pred}\] The contact convention (distance-based/surface-based/persistent-contact-after-relaxation, etc.) is locked in analysis_lock. Define the edge set as \[\mathcal{E}_c := \{(i,j)\mid i\neq j,\ C(i,j)=1\}. \label{eq:S08_edges}\] Therefore the contact graph is defined by \[\mathcal{G}_c := (\mathcal{V},\mathcal{E}_c) \label{eq:S08_graph}\] where \(\mathcal{V}\) is the node set selected in [eq:S08_nodes].

Adjacency matrix and degree

Define the adjacency matrix as \[A_{ij}:= \begin{cases} 1,& (i,j)\in\mathcal{E}_c,\\ 0,& \text{otherwise}. \end{cases} \label{eq:S08_adj_matrix}\] Define the contact degree (degree) of each node as \[z_i := \sum_{j\in\mathcal{V}} A_{ij}. \label{eq:S08_degree}\] The contact degree is an input for derived indicators such as deficits/throats/paths; if the contact convention is not locked, then \(A_{ij}\) and \(z_i\) are undefined.

Weights (optional) and weighted graphs

If a weighted graph is used, define a weight function \(W(i,j)\ge 0\) by \[W:\mathcal{E}_c\to\mathbb{R}_{\ge 0}, \qquad (i,j)\mapsto W(i,j). \label{eq:S08_weight}\] The definition of the weight (distance/angle/throat thickness/path resistance, etc.) is locked in analysis_lock. Conclusions that use weights must not be mixed with conclusions that do not use weights.

Declaration of the verification frame (structural invariants + Gate stack)

Categories of structural invariants

The verification in this chapter targets the following four categories of structural invariants.

  1. Geometric-consistency invariants: core-radius consistency (\(R_{82}\equiv R_p\)), cross-section invariants (\(\sigma_{82}/L_q^2\equiv 4/\pi\)), etc. (linked to the condition list in §6.4).

  2. Graph invariants: connectivity, bottleneck indicators, min-cut/alternate-path sensitivity, regime consistency of the degree distribution, etc.

  3. Cancellation–survival invariants: the shell-7 cancellation structure (“6 cancel + 1 survive”) and survival-vector non-degeneracy (\(\|\mathbf{V}\|\ge V_{\min}\)), etc.

  4. Integer/sector invariants: sum preservation and minimum-variance conditions of the 3-sector integerization, and the definability conditions of residual-direction labels (linked to Chapter 7).

Standard composition of the Gate stack

Conclusions in this chapter require PASS of the following Gate stack as necessary conditions.

  1. G-SYM: no conflicts in coordinate/unit/diameter–radius/cell-geometry meanings.

  2. G-LOCK: coordinate sets/contact conventions/weight definitions/thresholds belong to the same lock_id.

  3. G-REG: regime coordinate axes (dimension/driving/spanning/bottleneck/initial conditions/observation axes) are consistent.

  4. G-STR: structural invariants (integers, symmetry, cancellation convention) of core 82 and shell 7 are preserved.

  5. G-NUM: numerical stability of procedures (contact predicate, projected area, aggregation operations) and repeatability.

  6. G-REP: re-running with the same package reproduces the same results and the same verdict.

  7. G-NT: no post hoc changes or selection bias are detected for coordinates/conventions/thresholds/selection rules.

The detailed thresholds and report formats of the Gate stack must be pre-registered in gate_lock and protocol_lock, and must not be modified after seeing outcomes.

Package sealing (coordinates/graph/verdict logs)

The coordinate outputs \(\mathcal{X}_{82},\mathcal{S}_{7}\), the graph output \(\mathcal{G}_c\), and verdict logs must be sealed within the same release by being included in manifest and checksums. Unsealed outputs are not granted conclusion status.

LOCK/Gate connections for this section (none if empty)

10.1 8.1 Core-82 coordinates / hierarchy (d2) + contact graph

10.1.1 8.1.1 Outputs (coordinate file + hierarchy labels + graph)

The outputs of this section are locked to the following three bundles.

  1. Coordinate set: \[\mathcal{X}_{82}:=\{\mathbf{x}_i\}_{i=1}^{82},\qquad \mathbf{x}_i\in\mathbb{R}^3, \label{eq:S08_01_X82}\] and the center \(\mathbf{x}_c\).

  2. Hierarchy (distance) label: the shortest-path hierarchy function on the contact graph, \[d:\{1,\ldots,82\}\to\mathbb{Z}_{\ge 0}, \qquad d(i):=\mathrm{dist}_{\mathcal{G}_c}(i,\mathcal{R}_0), \label{eq:S08_01_layer_d}\] and the \(d=2\) layer set \[\mathcal{L}_2:=\{\,i\mid d(i)=2\,\}. \label{eq:S08_01_L2}\]

  3. Contact graph: \[\mathcal{G}_c:=(\mathcal{V}_{82},\mathcal{E}_c), \qquad \mathcal{V}_{82}:=\{1,2,\ldots,82\}. \label{eq:S08_01_Gc}\]

Coordinates, hierarchy, and graph belong to the same analysis_lock version, and different conventions are not mixed within a single output.

10.1.2 8.1.2 Inputs (LOCK): length/radius/procedure parameters

The construction procedure in this section is defined only when the following inputs are locked.

  1. Core node count: \[N_{\mathrm{core}}:=82. \label{eq:S08_01_Ncore}\]

  2. Normalization length: \[L_q\ \text{(locked)},\qquad \tilde{\mathbf{x}}:=\mathbf{x}/L_q. \label{eq:S08_01_Lq_norm}\]

  3. Core radius: \[R_p\ \text{(locked)}. \label{eq:S08_01_Rp_lock}\]

  4. Minimum separation length (impenetrability / duplicate prevention): \[d_{\min}>0\ \text{(locked)}. \label{eq:S08_01_dmin}\]

  5. Contact-judgement length: \[d_c:=\gamma_c\,d_{\min}, \qquad \gamma_c>1\ \text{(locked)}. \label{eq:S08_01_dc}\]

  6. Candidate-point grid step and boundary: \[h_{\mathrm{grid}}>0\ \text{(locked)},\qquad B\in\mathbb{Z}_{>0}\ \text{(locked)}. \label{eq:S08_01_grid_params}\]

  7. Stabilization (relaxation) tolerance and maximum iterations: \[\varepsilon_{\mathrm{pos}}>0\ \text{(locked)},\qquad K_{\max}\in\mathbb{Z}_{>0}\ \text{(locked)}. \label{eq:S08_01_relax_params}\]

  8. Hierarchy root-set size: \[N_0\in\{1,2,\ldots,82\}\ \text{(locked)}. \label{eq:S08_01_N0}\]

  9. Tie-break (deterministic rule): lock a rule TB that decides selection order under exact ties (e.g., lexicographic sorting, index-priority, etc.).

If any of the above items are not locked, the outputs of this section are undefined and are judged INCONCLUSIVE.

10.1.3 8.1.3 Generating the candidate set \(\mathcal{C}\) (fully deterministic procedure)

Candidate points are generated by mapping an integer grid-index set into real space.

10.1.3.1 8.1.3.1 Integer index set

Define the integer index set as follows. \[\mathcal{I}:=\{(u,v,w)\in\mathbb{Z}^3\mid -B\le u,v,w\le B\}. \label{eq:S08_01_Iset}\]

10.1.3.2 8.1.3.2 Grid map and candidate points

Define the map as follows. \[\mathbf{y}(u,v,w):=h_{\mathrm{grid}}\,(u,v,w). \label{eq:S08_01_grid_map}\] Define the candidate set \(\mathcal{C}\) with the following filter. \[\mathcal{C} := \left\{ \mathbf{y}(u,v,w)\ \middle|\ (u,v,w)\in\mathcal{I},\ 0<\|\mathbf{y}(u,v,w)\|\le R_p \right\}. \label{eq:S08_01_candidates}\] The point \(\|\mathbf{y}\|=0\) (the center point) is excluded by default in this section. A version that must include the center is allowed only by the include_center flag in analysis_lock; if the flag is false, the center is always excluded.

10.1.3.3 8.1.3.3 Candidate sorting (deterministic order)

Fix the deterministic order of candidates by sorting with the following key. \[\mathrm{key}(\mathbf{y}) := \Bigl(\|\mathbf{y}\|,\ y_x,\ y_y,\ y_z\Bigr), \label{eq:S08_01_candidate_key}\] where \(y_x,y_y,y_z\) are the components of \(\mathbf{y}\). Sorting is fixed as (1) increasing radius, then (2) lexicographic increasing component order.

10.1.4 8.1.4 Rule for selecting 82 points (deterministic Poisson-gap sampling)

The coordinate set \(\mathcal{X}_{82}\) is constructed by selecting 82 points from the candidate set \(\mathcal{C}\). The selection rule is fixed to satisfy both the “minimum separation length” and “suppression of maximal variance (uniform distribution).”

10.1.4.1 8.1.4.1 Minimum-separation condition

The selected points \(\mathbf{x}_i\) must satisfy \[\|\mathbf{x}_i-\mathbf{x}_j\|\ge d_{\min} \qquad (1\le i<j\le 82). \label{eq:S08_01_separation}\]

10.1.4.2 8.1.4.2 Score function (maximize the minimum distance)

Given a partial subset \(\mathcal{S}\subset\mathcal{C}\), define the score of a candidate point \(\mathbf{y}\in\mathcal{C}\setminus\mathcal{S}\) as \[\mathrm{score}(\mathbf{y};\mathcal{S}) := \min_{\mathbf{z}\in\mathcal{S}}\|\mathbf{y}-\mathbf{z}\|. \label{eq:S08_01_score}\] The score is the “minimum distance to the already-selected points,” and a candidate with a larger score is selected in a direction that produces a more uniform distribution.

10.1.4.3 8.1.4.3 Selection algorithm (fully fixed procedure)

Selection is locked by the following deterministic procedure.

ALG-CORE82-SELECT (inputs: C, d_min, N_core=82, TB)

S := empty set
1) Initial selection:
   - sort C by key(y)
   - add the first element y0 to S (smallest radius; if tie, lexicographic)
2) Iterative selection (|S| < 82):
   - build the candidate set C_ok that does not violate the separation constraint:
       C_ok := { y in C \ S | for all z in S: ||y-z|| >= d_min }
   - if C_ok is empty: FAIL-CORE82-NOTENOUGH
   - for each y in C_ok, compute score(y;S)
   - choose y* with the maximal score
   - if tied, choose one by TB (tie-break)
   - add y* to S
3) Output:
   - index elements of S as {x_i}_{i=1..82} and form X82

This algorithm is deterministic and uses no randomness. If the algorithm cannot fill 82 points, construction fails and does not proceed to downstream steps (hierarchy/graph).

10.1.5 8.1.5 Scale consistency (normalization to the boundary radius \(R_p\))

The selected coordinate set \(\mathcal{X}_{82}\) is normalized by a single scale factor to ensure radius consistency.

10.1.5.1 8.1.5.1 Boundary-candidate set and aggregated radius

Define the radius of each point as \(r_i:=\|\mathbf{x}_i-\mathbf{x}_c\|\). The center is locked by default as \[\mathbf{x}_c:=\mathbf{0} \label{eq:S08_01_center_zero}\] and any alternative center selection is allowed only by a version bump.

Define the boundary-candidate set as follows, where \(K_b\) is a locked integer. \[K_b\in\{1,\ldots,82\}\ \text{(locked)},\qquad \mathcal{B}_{82}:=\text{the set of indices of the top $K_b$ largest radii $r_i$}. \label{eq:S08_01_boundary_set}\] Define the aggregated radius as \[R_{82}:=\mathrm{Agg}\bigl(\{r_i\}_{i\in\mathcal{B}_{82}}\bigr), \label{eq:S08_01_R82_agg}\] where \(\mathrm{Agg}\) is an aggregation operator locked in analysis_lock (e.g., median, mean).

10.1.5.2 8.1.5.2 Scale factor and coordinate redefinition

Define the scale factor as \[s:=\frac{R_p}{R_{82}}. \label{eq:S08_01_scale_factor}\] Redefine coordinates as \[\mathbf{x}_i \leftarrow s\,\mathbf{x}_i \qquad (i=1,\ldots,82). \label{eq:S08_01_rescale}\] After rescaling, \(R_{82}\equiv R_p\) holds by definition (definitional consistency). However, this consistency is meaningful only when \(\mathrm{Agg}\) and \(\mathcal{B}_{82}\) are locked.

10.1.6 8.1.6 Stabilization (relaxation): simultaneous satisfaction of minimum separation and boundary constraint

Even after rescaling, to enforce [eq:S08_01_separation] and \(\|\mathbf{x}_i\|\le R_p\), perform the following deterministic relaxation procedure.

10.1.6.1 8.1.6.1 Defining violation amounts

Define the pairwise violation amount as \[\Delta_{ij}:=\max\!\bigl(0,\ d_{\min}-\|\mathbf{x}_i-\mathbf{x}_j\|\bigr), \qquad (i<j). \label{eq:S08_01_pair_violation}\] Define the boundary violation amount as \[\Delta_i^{(R)}:=\max\!\bigl(0,\ \|\mathbf{x}_i\|-R_p\bigr). \label{eq:S08_01_rad_violation}\] Define the overall maximal violation as \[\Delta_{\max} := \max\left( \max_{i<j}\Delta_{ij}, \max_i \Delta_i^{(R)} \right). \label{eq:S08_01_Dmax}\]

10.1.6.2 8.1.6.2 Deterministic projection–separation relaxation algorithm

ALG-CORE82-RELAX (inputs: X82, d_min, R_p, eps_pos, K_max)

for iter = 1..K_max:
  # (A) pair-separation projection
  for i = 1..82:
    for j = i+1..82:
      d = ||x_i - x_j||
      if d < d_min:
        u = (x_i - x_j) / max(d, tiny)   # tiny>0 is a fixed numerical safeguard constant
        shift = 0.5 * (d_min - d)
        x_i := x_i + shift * u
        x_j := x_j - shift * u

  # (B) radial-boundary projection
  for i = 1..82:
    r = ||x_i||
    if r > R_p:
      x_i := (R_p / r) * x_i

  # (C) stopping rule
  compute D_max
  if D_max <= eps_pos:
    break

if D_max > eps_pos:
  FAIL-CORE82-NOCONVERGE

This algorithm uses a fixed update order (in increasing indices), and tiny is a fixed numerical safeguard constant. After termination, the following must be guaranteed: \[\|\mathbf{x}_i-\mathbf{x}_j\|\ge d_{\min}-\varepsilon_{\mathrm{pos}}, \qquad \|\mathbf{x}_i\|\le R_p, \label{eq:S08_01_relax_guarantee}\] otherwise the verdict is FAIL.

10.1.7 8.1.7 Constructing the contact graph (single distance-based convention)

Fix the contact predicate as a single convention determined from the coordinates.

10.1.7.1 8.1.7.1 Contact predicate function

Define the contact predicate \(C(i,j)\) as \[C(i,j):= \begin{cases} 1,& \|\mathbf{x}_i-\mathbf{x}_j\|\le d_c,\\ 0,& \text{otherwise}, \end{cases} \qquad (i\neq j). \label{eq:S08_01_contact_pred}\] where \(d_c\) is the contact-judgement length locked by [eq:S08_01_dc] and must not be adjusted after seeing outcomes.

10.1.7.2 8.1.7.2 Edge set and adjacency matrix

Define the edge set as \[\mathcal{E}_c:=\{(i,j)\mid 1\le i<j\le 82,\ C(i,j)=1\}. \label{eq:S08_01_edges}\] Define the adjacency matrix as \[A_{ij}:= \begin{cases} 1,& (i,j)\in\mathcal{E}_c\ \text{or}\ (j,i)\in\mathcal{E}_c,\\ 0,& \text{otherwise}, \end{cases} \qquad A_{ii}:=0. \label{eq:S08_01_adj}\] Define the degree as \[z_i:=\sum_{j=1}^{82}A_{ij}. \label{eq:S08_01_degree}\]

10.1.7.3 8.1.7.3 Connectivity condition (minimal requirement)

For the graph to function as a single core, it must be connected. Define the connectivity indicator as \[\chi_{\mathrm{conn}}:= \begin{cases} 1,& \mathcal{G}_c\ \text{is connected},\\ 0,& \text{otherwise}. \end{cases} \label{eq:S08_01_conn}\] If \(\chi_{\mathrm{conn}}=0\), the verdict is FAIL-CORE82-DISCONNECTED.

10.1.8 8.1.8 Hierarchy (d2): root set \(\mathcal{R}_0\) and BFS distance

Define the hierarchy by the shortest-path distance on the contact graph.

10.1.8.1 8.1.8.1 Determining the root set \(\mathcal{R}_0\)

The size \(N_0\) of the root set is locked by [eq:S08_01_N0]. Determine the root set by the “smallest radius” criterion. \[r_i:=\|\mathbf{x}_i\|, \qquad \mathcal{R}_0:=\text{the set of indices of the top $N_0$ smallest radii $r_i$}. \label{eq:S08_01_rootset}\] Ties are resolved by the pre-registered tie-break rule (TB).

10.1.8.2 8.1.8.2 Hierarchy function \(d(i)\)

On the contact graph \(\mathcal{G}_c\), define the shortest distance (number of edges) to the root set as \[d(i):=\min_{r\in\mathcal{R}_0}\mathrm{dist}_{\mathcal{G}_c}(i,r), \qquad d(r)=0\ (r\in\mathcal{R}_0). \label{eq:S08_01_di_def}\] Define the layer sets as \[\mathcal{L}_\ell:=\{\,i\mid d(i)=\ell\,\}, \qquad \ell=0,1,2,\ldots \label{eq:S08_01_Lell}\] where the “d2” of this section is locked to \(\mathcal{L}_2\) ([eq:S08_01_L2]).

10.1.8.3 8.1.8.3 Hierarchy completeness condition

Every node must be reachable from the root; hence \[\max_{1\le i\le 82} d(i) < \infty \label{eq:S08_01_layer_finite}\] is required. This is equivalent to connectivity [eq:S08_01_conn], and violation triggers FAIL-CORE82-DISCONNECTED.

10.1.9 8.1.9 Storage format (single source of truth for coordinates/hierarchy/graph)

The outputs of this section are stored in the following files and must be sealed by inclusion in the same release’s manifest and checksums.

  1. X82.csv: \((i,x_i,y_i,z_i,r_i)\).

  2. G82.edgelist: edge list \((i,j)\).

  3. layers82.csv: \((i,d(i))\) and counts \(|\mathcal{L}_\ell|\) by \(\ell\).

  4. params82.yaml: \(R_p,L_q,d_{\min},\gamma_c,h_{\mathrm{grid}},B,\varepsilon_{\mathrm{pos}},K_{\max},N_0,K_b,\mathrm{Agg},\texttt{TB}\).

Unsealed outputs are not granted conclusion status.

LOCK/Gate connections for this section (none if empty)

10.2 8.2 Basic 7-shell structure

Geometric minimality of the shell count (7) + residual match

In this section, \(N_{\mathrm{shell}}=7\) is not an arbitrary choice but is fixed as the minimum integer that satisfies simultaneously (i) the 3D cancellation (nulling) convention and (ii) the existence of a single surviving vector (label input). This minimality provides the prior justification for the definitional statement "2 pairs + 4-quad + 1 survive" in §8.2.1.

10.2.0.1 (I) Geometric minimality of cancellation elements

The shell must "cancel as much as possible" (stability) while "not becoming identically zero" (label). In 3D, bias-reducing cancellation naturally comes in two types.

  1. Pair cancellation (2-element): \((\mathbf{s},-\mathbf{s})\) cancels perfectly in any coordinate system.

  2. Quad cancellation (4-element): Sum-zero cancellation of three vectors always lies in a plane and thus cannot enforce 3D isotropic cancellation. In contrast, the four unit vectors in tetrahedral directions \(\{\mathbf{q}_i\}_{i=1}^4\) satisfy \[\mathbf{q}_1+\mathbf{q}_2+\mathbf{q}_3+\mathbf{q}_4=\mathbf{0}, \qquad \mathbf{q}_i\cdot\mathbf{q}_j=-\frac{1}{3}\ (i\neq j) \label{eq:S08_02_tetra_cancel}\] and provide a representative (isotropic) cancellation element.

Therefore, to maximize cancellation while including both "one pair" and "one quad", the shell requires at least \(2+4=6\) vectors.

10.2.0.2 (II) One surviving vector

To carry the charge/electron label, at least one non-cancelled surviving vector is required (definition in §8.2.1). Hence \[\boxed{N_{\mathrm{shell}}=2+4+1=7.} \label{eq:S08_02_shell7_minimal}\]

10.2.0.3 (III) Residual match (arithmetic check)

Separately, when the total structural count \(N_{\mathrm{str}}\) and the core count \(N_{\mathrm{core}}\) are locked independently, the shell count is also forced arithmetically by \[N_{\mathrm{shell}}=N_{\mathrm{str}}-N_{\mathrm{core}} \label{eq:S08_02_shell_residual}\] In this version, \(N_{\mathrm{str}}=89\) (structure count in §7.2) and \(N_{\mathrm{core}}=82\) (core construction in §8.1) are locked, so the residual is \(7\). This matches the minimality result [eq:S08_02_shell7_minimal].

10.2.1 8.2.1 Purpose

This section fixes, by definition, the internal structure of the 7-shell as the composite "2 pairs + 4-quad + 1 survive" structure. This section does not interpret; it locks (i) the shell vector (or coordinate) set, (ii) the partitioning rule, (iii) the cancellation predicate, and (iv) the definition of the surviving term. If the definitions of this section are not locked, then the "surviving vector" used as an input in downstream charge/electron labels and in event-rate/mass derivations becomes undefined.

10.2.2 8.2.2 Inputs (LOCK): the shell-7 vector set and coordinate system

Define the shell 7 as the following set. \[\mathcal{S}_7:=\{\mathbf{s}_k\}_{k=1}^{7}, \qquad \mathbf{s}_k\in\mathbb{R}^3. \label{eq:S08_02_S7}\] The meaning of \(\mathbf{s}_k\) must be locked to one of the following two modes (exclusive lock).

  1. Point mode (SHELL-POINT): \(\mathbf{s}_k\) is a representative shell point (coordinate) relative to the core center \(\mathbf{x}_c\).

  2. Vector mode (SHELL-VEC): \(\mathbf{s}_k\) is a vector representing a shell direction/contribution.

Regardless of the adopted mode, the coordinate system (reference axes, units, normalization) must be locked in analysis_lock, and cannot be replaced after seeing outcomes.

10.2.3 8.2.3 Standard definition of cancellation judgment (cancellation operators)

"Cancellation" in the shell structure means that the sum of vectors becomes sufficiently small, and cancellation judgment is defined by a threshold.

10.2.3.1 8.2.3.1 Cancellation threshold

Lock the cancellation threshold \(s_{\min}>0\). \[s_{\min}>0, \qquad s_{\min}\ \text{is pre-registered in}\ \texttt{gate\_lock}. \label{eq:S08_02_smin}\]

10.2.3.2 8.2.3.2 Pair cancellation judgment

For two distinct indices \(a\neq b\), define the pair-cancellation predicate as \[\mathrm{CancelPair}(a,b) := \begin{cases} 1,& \|\mathbf{s}_a+\mathbf{s}_b\|\le s_{\min},\\ 0,& \text{otherwise}. \end{cases} \label{eq:S08_02_cancel_pair}\] This is an operational definition of cancellation and must not be changed after seeing outcomes.

10.2.3.3 8.2.3.3 Quad cancellation judgment

For four distinct indices \((a,b,c,d)\), define the quad-cancellation predicate as \[\mathrm{CancelQuad}(a,b,c,d) := \begin{cases} 1,& \|\mathbf{s}_a+\mathbf{s}_b+\mathbf{s}_c+\mathbf{s}_d\|\le s_{\min},\\ 0,& \text{otherwise}. \end{cases} \label{eq:S08_02_cancel_quad}\] Quad cancellation means that the sum of the four terms lies within the cancellation threshold \(s_{\min}\).

10.2.4 8.2.4 Structural definition of "2 pairs + 4-quad + 1 survive"

The core of this section is to define a rule that partitions the index set \(\{1,\ldots,7\}\) into the following three parts. \[\{1,2,\ldots,7\} = P_1 \,\dot\cup\, P_2 \,\dot\cup\, Q \,\dot\cup\, \{u\}. \label{eq:S08_02_partition}\] Here \[|P_1|=2,\quad |P_2|=2,\quad |Q|=4,\quad u\in\{1,\ldots,7\}, \label{eq:S08_02_sizes}\] and \(\dot\cup\) denotes disjoint union (no overlap). \(P_1,P_2\) are two pairs, \(Q\) is one quad, and \(\{u\}\) is the surviving index.

10.2.4.1 8.2.4.1 Definition of the "two pairs"

Define the two pairs as index pairs satisfying \[P_1=\{a_1,b_1\},\qquad P_2=\{a_2,b_2\}, \qquad \mathrm{CancelPair}(a_1,b_1)=1,\quad \mathrm{CancelPair}(a_2,b_2)=1. \label{eq:S08_02_two_pairs}\] That is, each pair must cancel within the threshold \(s_{\min}\).

10.2.4.2 8.2.4.2 Definition of the "quad"

Define the quad as the remaining four-index set \(Q\) satisfying \[Q=\{c_1,c_2,c_3,c_4\}, \qquad \mathrm{CancelQuad}(c_1,c_2,c_3,c_4)=1. \label{eq:S08_02_quad}\] That is, the sum of the four terms must cancel within \(s_{\min}\).

10.2.4.3 8.2.4.3 Definition of "one survive"

Define the surviving index \(u\) as the unique index not included in the pairs and the quad. \[u\notin P_1\cup P_2\cup Q, \qquad \{u\}=\{1,\ldots,7\}\setminus(P_1\cup P_2\cup Q). \label{eq:S08_02_survivor_index}\] Define the surviving vector (or surviving coordinate) as \[\mathbf{V}_{\mathrm{surv}} := \mathbf{s}_u. \label{eq:S08_02_Vsurv_simple}\] However, in an operational procedure, pair/quad residuals can be nonzero (within the threshold but not exactly zero). Therefore, extend and lock the standard definition of the surviving vector as \[\mathbf{V}_{\mathrm{surv}} := \mathbf{s}_u + (\mathbf{s}_{a_1}+\mathbf{s}_{b_1}) + (\mathbf{s}_{a_2}+\mathbf{s}_{b_2}) + (\mathbf{s}_{c_1}+\mathbf{s}_{c_2}+\mathbf{s}_{c_3}+\mathbf{s}_{c_4}). \label{eq:S08_02_Vsurv_full}\] Definition [eq:S08_02_Vsurv_full] is the definition of the "surviving residual including cancellation residuals", and is used later as the input to charge/electron labels. [eq:S08_02_Vsurv_simple] is the special case under ideal perfect cancellation (zero residual).

10.2.5 8.2.5 Partition selection rule (closure to remove non-uniqueness)

The conditions [eq:S08_02_two_pairs] and [eq:S08_02_quad] can admit multiple solutions. Therefore, the choice of the partition \((P_1,P_2,Q,u)\) must be locked as a closure. This section fixes only the slot of the selection rule and declares that the concrete selection rule is locked as CL-G-SHELL7-PARTITION in analysis_lock.

10.2.5.1 8.2.5.1 Valid partition set

Define the set of valid partitions by \[\mathcal{P}_{\mathrm{valid}} := \left\{ (P_1,P_2,Q,u)\ \middle|\ \eqref{eq:S08_02_partition},\ \eqref{eq:S08_02_sizes},\ \eqref{eq:S08_02_two_pairs},\ \eqref{eq:S08_02_quad}\ \text{hold} \right\}. \label{eq:S08_02_Pvalid}\] If \(\mathcal{P}_{\mathrm{valid}}=\varnothing\), then the structure definition of this section is not applicable, and the shell-7 structure does not hold in the given regime.

10.2.5.2 8.2.5.2 Selection function (locked as closure)

Define the selection function \(\mathrm{Select}(\cdot)\) by \[(P_1^\ast,P_2^\ast,Q^\ast,u^\ast) := \mathrm{Select}\bigl(\mathcal{P}_{\mathrm{valid}}\bigr). \label{eq:S08_02_select}\] \(\mathrm{Select}\) must be locked in analysis_lock including the following items.

  1. The optimization objective (choose one: maximize/minimize \(\|\mathbf{V}_{\mathrm{surv}}\|\), minimize total residual, maximize projection to a specified axis, etc.).

  2. Tie handling (tie-break).

  3. Failure modes (no valid partition, unresolved multiple solutions, numerical instability, etc.).

  4. Required Gate stack (G-SYM, G-LOCK, G-REG, G-STR, and if needed G-NUM, G-NT).

Changing \(\mathrm{Select}\) after seeing outcomes violates No-Tuning.

10.2.6 8.2.6 Summary of the structural requirements (definitional conclusion)

In this section, the "basic 7-shell structure" is defined by the following single statement.

The shell 7 is partitioned, for the index set \(\{1,\ldots,7\}\), into two cancelling pairs \((P_1,P_2)\), one cancelling quad \(Q\), and one surviving index \(u\); the pairs and the quad cancel within the cancellation threshold \(s_{\min}\); and the surviving vector is defined by [eq:S08_02_Vsurv_full].

The above is a definition (not an interpretation), and downstream sections refer to it without restating it.

LOCK/Gate connections for this section (none if empty)

10.3 8.3 Verification frame for “6 cancel + 1 survive”

10.3.1 8.3.1 Definition of the verification target

In this section, “6 cancel + 1 survive” is the verification problem of judging whether the shell-7 partition structure defined in §8.2 holds for actual data (or for the actually produced coordinates/vectors). The verification targets the following three items.

  1. Existence: under the cancellation threshold \(s_{\min}\), does a valid partition \((P_1,P_2,Q,u)\) exist.

  2. Consistency: does the selected partition satisfy the cancellation conditions (two pairs, one quad).

  3. Nondegeneracy: does the surviving vector (including cancellation residuals) pass the non-degeneracy threshold so that it can be used as an input for labels/events/downstream derivations.

Verification returns only PASS/FAIL/INCONCLUSIVE; interpretation cannot nullify the verdict.

10.3.2 8.3.2 Inputs (LOCK) and prior definitions

10.3.2.1 8.3.2.1 Shell vector set

The shell 7 is input as the following set. \[\mathcal{S}_7:=\{\mathbf{s}_k\}_{k=1}^{7}, \qquad \mathbf{s}_k\in\mathbb{R}^3. \label{eq:S08_03_S7}\] The meaning of \(\mathbf{s}_k\) (point mode / vector mode), the coordinate system, and the normalization conventions must be locked in analysis_lock.

10.3.2.2 8.3.2.2 Cancellation threshold and non-degeneracy threshold

Input the cancellation threshold \(s_{\min}\) and the non-degeneracy threshold \(V_{\min}\) as \[s_{\min}>0,\qquad V_{\min}>0, \label{eq:S08_03_thresholds}\] where \(s_{\min},V_{\min}\) must be pre-registered in gate_lock and must not be moved after seeing outcomes.

10.3.2.3 8.3.2.3 Cancellation predicates (refer to the definitions in §8.2)

Pair and quad cancellation are defined by \[\mathrm{CancelPair}(a,b)=1 \Longleftrightarrow \|\mathbf{s}_a+\mathbf{s}_b\|\le s_{\min}, \label{eq:S08_03_CancelPair}\] \[\mathrm{CancelQuad}(a,b,c,d)=1 \Longleftrightarrow \|\mathbf{s}_a+\mathbf{s}_b+\mathbf{s}_c+\mathbf{s}_d\|\le s_{\min}. \label{eq:S08_03_CancelQuad}\] These predicate definitions are not modified within the same version.

10.3.2.4 8.3.2.4 Partition-selection closure (locking the selection rule)

Define the valid-partition set and the selection function as \[\mathcal{P}_{\mathrm{valid}} := \left\{ (P_1,P_2,Q,u)\ \middle|\ \{1,\ldots,7\}=P_1\dot\cup P_2\dot\cup Q\dot\cup\{u\},\ |P_1|=|P_2|=2,\ |Q|=4, \ \mathrm{CancelPair}(P_1)=\mathrm{CancelPair}(P_2)=\mathrm{CancelQuad}(Q)=1 \right\}, \label{eq:S08_03_Pvalid}\] \[(P_1^\ast,P_2^\ast,Q^\ast,u^\ast) := \mathrm{Select}\bigl(\mathcal{P}_{\mathrm{valid}}\bigr). \label{eq:S08_03_Select}\] The concrete rule of \(\mathrm{Select}\) (objective function / tie-break / failure mode / required Gate stack) must be locked in analysis_lock; otherwise the verdict is INCONCLUSIVE.

10.3.2.5 8.3.2.5 Surviving vector definition (including cancellation residuals)

For the selected partition, define the surviving vector by \[\mathbf{V}_{\mathrm{surv}} := \mathbf{s}_{u^\ast} + \sum_{k\in P_1^\ast}\mathbf{s}_{k} + \sum_{k\in P_2^\ast}\mathbf{s}_{k} + \sum_{k\in Q^\ast}\mathbf{s}_{k}. \label{eq:S08_03_Vsurv}\] Equation [eq:S08_03_Vsurv] is the standard residual definition of “6 cancel residuals + 1 survive,” and is used downstream as an input for charge/electron labels.

10.3.3 8.3.3 Verification items (checklist) definition

The verification items of this section are locked to the following six items. Each item produces a cause label for PASS/FAIL/INCONCLUSIVE.

10.3.4 (V1) Input lock completeness

The following items must all be locked.

  1. The meaning of \(\mathcal{S}_7\) (point/vector mode), coordinate system, and unit/normalization convention.

  2. Threshold values of \(s_{\min}\) and \(V_{\min}\).

  3. Objective/tie-break/failure mode/required Gate of the \(\mathrm{Select}\) closure.

If any item is missing, the verdict is INCONCLUSIVE; if post hoc change evidence is detected, it is FAIL.

10.3.5 (V2) Existence of a valid partition

\(\mathcal{P}_{\mathrm{valid}}\) must be nonempty. \[\mathcal{P}_{\mathrm{valid}}=\varnothing \quad\Longrightarrow\quad \texttt{FAIL-SHELL7-NOPART}. \label{eq:S08_03_fail_nopart}\]

10.3.6 (V3) Cancellation consistency of the selected partition (two pairs + one quad)

For the selected result \((P_1^\ast,P_2^\ast,Q^\ast,u^\ast)\), require \[\mathrm{CancelPair}(P_1^\ast)=1,\quad \mathrm{CancelPair}(P_2^\ast)=1,\quad \mathrm{CancelQuad}(Q^\ast)=1. \label{eq:S08_03_cancel_consistency}\] Violation of any condition yields FAIL-SHELL7-CANCEL.

10.3.7 (V4) Quantitative recording of cancellation residuals (consistency indicators)

Define the cancellation residual norms as \[r_{P_1}:=\left\|\sum_{k\in P_1^\ast}\mathbf{s}_k\right\|, \qquad r_{P_2}:=\left\|\sum_{k\in P_2^\ast}\mathbf{s}_k\right\|, \qquad r_{Q}:=\left\|\sum_{k\in Q^\ast}\mathbf{s}_k\right\|. \label{eq:S08_03_residuals}\] By definition, if [eq:S08_03_cancel_consistency] holds then \(r_{P_1},r_{P_2},r_Q\le s_{\min}\) must hold. This item is not an “extra decision” but a mandatory log indicator; missing indicators yield INCONCLUSIVE.

10.3.8 (V5) Surviving non-degeneracy (eligibility as a label/event input)

Define the non-degeneracy condition for the surviving vector by \[\|\mathbf{V}_{\mathrm{surv}}\|\ge V_{\min}. \label{eq:S08_03_surv_nondeg}\] If [eq:S08_03_surv_nondeg] is violated, the verdict is FAIL-SHELL7-DEGEN. If this item is FAIL, then this window’s shell data cannot be used as an input for downstream charge/electron labels (§7.4) and event-rate/mass derivations (only limitation conclusions are allowed).

10.3.9 (V6) Robustness (consistency under the defined replay stack)

This item requires verdict consistency over a pre-registered “replay set” under the same analysis_lock. The replay set must be locked as \[\mathcal{R}_{\mathrm{replay}}:=\{r_1,r_2,\ldots,r_M\}, \qquad M\in\mathbb{Z}_{>0}, \label{eq:S08_03_replayset}\] where each \(r_m\) means (i) a different time window from the same input log, or (ii) a different subsample (by a pre-registered selection rule), or (iii) re-computation of the same data (same code/environment). If the construction rule of the replay set is not locked, the verdict is INCONCLUSIVE.

Let the surviving vector in each replay be \(\mathbf{V}_{\mathrm{surv}}^{(m)}\) and define the directional-consistency indicator by \[c_{mn} :=\frac{\mathbf{V}_{\mathrm{surv}}^{(m)}\cdot \mathbf{V}_{\mathrm{surv}}^{(n)}}{\|\mathbf{V}_{\mathrm{surv}}^{(m)}\|\,\|\mathbf{V}_{\mathrm{surv}}^{(n)}\|} \qquad (m\neq n). \label{eq:S08_03_coscons}\] Lock the threshold \(c_{\min}\) in gate_lock (\(-1\le c_{\min}\le 1\)). Define the robustness rule as \[\min_{m\neq n} c_{mn} \ge c_{\min}. \label{eq:S08_03_robust_rule}\] Violation of [eq:S08_03_robust_rule] is treated as FAIL-SHELL7-ROBUST. This item applies only when the replay set is locked; otherwise robustness verification is not executable (INCONCLUSIVE).

10.3.10 8.3.4 Log specification (mandatory records)

This section grants conclusion status only when the following logs are complete. The format (JSON/YAML/CSV, etc.) is locked in protocol_lock, but the field meanings are fixed as follows.

10.3.10.1 8.3.4.1 Input log

  1. shell_id: shell data identifier.

  2. s_k: coordinates/components of \(\mathbf{s}_k\) for \(k=1..7\).

  3. mode: SHELL-POINT or SHELL-VEC.

  4. locks: canon_lock_id, realization_lock_id, analysis_lock_id.

  5. thresholds: \(s_{\min}\), \(V_{\min}\), (optional) \(c_{\min}\).

10.3.10.2 8.3.4.2 Partition/cancellation log

  1. P1, P2, Q, u: the selected partition \((P_1^\ast,P_2^\ast,Q^\ast,u^\ast)\).

  2. residuals: \((r_{P_1},r_{P_2},r_Q)\).

  3. CancelPair(P1), CancelPair(P2), CancelQuad(Q): cancellation predicate values (0/1).

  4. V_surv: components of \(\mathbf{V}_{\mathrm{surv}}\) and the norm \(\|\mathbf{V}_{\mathrm{surv}}\|\).

10.3.10.3 8.3.4.3 Robustness log (if applicable)

  1. replay_set: replay set identifier and construction rule.

  2. V_surv[m]: \(\mathbf{V}_{\mathrm{surv}}^{(m)}\) for each replay \(m\).

  3. c_min_pair: \(\min_{m\neq n}c_{mn}\).

  4. pass_robust: true/false.

10.3.11 8.3.5 Decision Gate (stack) definition

The Gates in this section are locked to the following stack. Each Gate outputs one of PASS/FAIL/INCONCLUSIVE.

10.3.11.1 8.3.5.1 G-SHELL7-LOCK (input lock Gate)

10.3.11.2 8.3.5.2 G-SHELL7-PART (partition existence Gate)

10.3.11.3 8.3.5.3 G-SHELL7-CANCEL (cancellation consistency Gate)

10.3.11.4 8.3.5.4 G-SHELL7-SURV (surviving non-degeneracy Gate)

10.3.11.5 8.3.5.5 G-SHELL7-ROBUST (robustness Gate; optional)

10.3.11.6 8.3.5.6 Final Gate: G-SHELL7-6C1S

Define the final verdict as \[\texttt{G-SHELL7-6C1S}=\texttt{PASS} \Longleftrightarrow (\texttt{G-SHELL7-LOCK}=\texttt{PASS}) \wedge (\texttt{G-SHELL7-PART}=\texttt{PASS}) \wedge (\texttt{G-SHELL7-CANCEL}=\texttt{PASS}) \wedge (\texttt{G-SHELL7-SURV}=\texttt{PASS}) \wedge \bigl(\texttt{G-SHELL7-ROBUST}\in\{\texttt{PASS},\texttt{INCONCLUSIVE}\}\bigr), \label{eq:S08_03_final_gate}\] that is, robustness requires PASS when it is locked, but may remain INCONCLUSIVE when it is not locked; in that case, the sentence “robustness verification passed” is forbidden (restricted by PASS.rules).

Only when G-SHELL7-6C1S=PASS are conclusion sentences of the following types allowed.

  1. “This shell data satisfies two-pair + one-quad cancellation and non-degenerate survival under the threshold \(s_{\min}\).”

  2. “The surviving vector \(\mathbf{V}_{\mathrm{surv}}\) is defined and \(\|\mathbf{V}_{\mathrm{surv}}\|\ge V_{\min}\).”

If the verdict is FAIL or INCONCLUSIVE, only limitation conclusions (CT-LIM) are allowed, and the FAIL label and violated items (V1–V6) must be recorded together.

LOCK/Gate connections for this section (none if empty)

10.4 8.4 Electron-generation mechanism + charge-label definition

10.4.1 8.4.1 Definitions (surviving vector \(\rightarrow\) emission event \(\rightarrow\) label)

10.4.1.1 8.4.1.1 Completed core/shell state and completion event

Define the completed state with core 82 and shell 7 coupled as the following object. \[\mathcal{P} := (\mathcal{X}_{82},\ \mathcal{S}_{7},\ \mathbf{x}_c,\ R_p), \label{eq:S08_04_proton_state}\] where \(\mathcal{X}_{82}\) is the core-82 coordinate set, \(\mathcal{S}_{7}\) is the shell-7 vector (or coordinate) set, \(\mathbf{x}_c\) is the center, and \(R_p\) is the core radius. The meaning/unit/coordinate system of each item must be locked in canon_lock/analysis_lock.

Define the completion event as \[E_{\mathrm{build}}:\ \text{the event that the state }\mathcal{P}\text{ is generated and recorded}. \label{eq:S08_04_Ebuild_def}\] The occurrence time \(t_{\mathrm{build}}\) of the completion event is defined by the build-time convention. In regimes where the canonical event rate \(\nu_{p,\mathrm{can}}\) is locked, \[T_p := \frac{N_p}{\nu_{p,\mathrm{can}}}, \qquad N_p:=89, \label{eq:S08_04_Tp_def}\] and given a build-start time \(t_0\) we define \[t_{\mathrm{build}} := t_0 + T_p \label{eq:S08_04_tbuild}\] If \(\nu_{p,\mathrm{can}}\) is undefined, then \(t_{\mathrm{build}}\) is replaced by a window/tick-based definition; the replacement convention must be locked in analysis_lock.

10.4.1.2 8.4.1.2 Surviving vector \(\mathbf{V}_{\mathrm{surv}}\)

Assume the shell-7 partition structure (2 pairs + 1 quad + 1 survive) and the cancellation threshold \(s_{\min}\). Valid-partition selection is locked as closure, and \[(P_1^\ast,P_2^\ast,Q^\ast,u^\ast) := \mathrm{Select}\bigl(\mathcal{P}_{\mathrm{valid}}\bigr) \label{eq:S08_04_select_partition}\] defines the selected partition. Define the surviving vector (including cancellation residuals) as \[\mathbf{V}_{\mathrm{surv}} := \mathbf{s}_{u^\ast} + \sum_{k\in P_1^\ast}\mathbf{s}_{k} + \sum_{k\in P_2^\ast}\mathbf{s}_{k} + \sum_{k\in Q^\ast}\mathbf{s}_{k}. \label{eq:S08_04_Vsurv}\] Definition [eq:S08_04_Vsurv] is the unique standard survival definition of the “6 cancel + 1 survive” structure; it cannot be replaced by another convention (e.g. assuming perfect cancellation, absolute-sum rules, alternative weighted sums) within the same version.

10.4.1.3 8.4.1.3 Emission direction (unit direction) and emission point

Define the survival (emission) direction as \[\widehat{\mathbf{n}}_{\mathrm{emit}} := \frac{\mathbf{V}_{\mathrm{surv}}}{\|\mathbf{V}_{\mathrm{surv}}\|}, \qquad \text{provided }\|\mathbf{V}_{\mathrm{surv}}\|>0. \label{eq:S08_04_nemit}\] If \(\|\mathbf{V}_{\mathrm{surv}}\|=0\), the emission direction is undefined.

Lock the emission length \(\ell_{\mathrm{emit}}>0\) as \[\ell_{\mathrm{emit}}>0, \qquad \ell_{\mathrm{emit}}\ \text{is pre-registered in}\ \texttt{analysis\_lock}. \label{eq:S08_04_lemit_lock}\] Define the emission point (emission location) as \[\mathbf{x}_{\mathrm{emit}} := \mathbf{x}_c + (R_p+\ell_{\mathrm{emit}})\,\widehat{\mathbf{n}}_{\mathrm{emit}}. \label{eq:S08_04_xemit}\] Definition [eq:S08_04_xemit] is the geometric convention “move outward by \(\ell_{\mathrm{emit}}\) beyond the core boundary (\(R_p\))”, and it is not adjusted after seeing outcomes.

10.4.1.4 8.4.1.4 Charge axis \(\mathbf{n}_Q\) and charge unit \(q_0\)

The charge label is defined by judging the sign of the surviving vector relative to a single reference axis. Lock the charge axis as \[\mathbf{n}_Q\in\mathbb{R}^3, \qquad \|\mathbf{n}_Q\|=1, \qquad \mathbf{n}_Q\ \text{is pre-registered in}\ \texttt{analysis\_lock}. \label{eq:S08_04_nQ_lock}\] Lock the charge unit as \[q_0:=1, \qquad q_0\ \text{is locked in}\ \texttt{canon\_lock}\ \text{as the base unit of charge dimension }[Q]. \label{eq:S08_04_q0_lock}\]

10.4.1.5 8.4.1.5 Charge-sign function and label functions

Define the charge-sign function as \[q(\mathbf{V}_{\mathrm{surv}}) := \mathrm{sgn}\!\left(\mathbf{V}_{\mathrm{surv}}\cdot\mathbf{n}_Q\right), \qquad \mathrm{sgn}(x)= \begin{cases} +1,& x>0,\\ 0,& x=0,\\ -1,& x<0. \end{cases} \label{eq:S08_04_q_sign}\] Define the charge label as \[Q_{\mathrm{label}} := q_0\,q(\mathbf{V}_{\mathrm{surv}}). \label{eq:S08_04_Qlabel}\] Define the electron/anti-electron label as \[\mathcal{L}_{e}(\mathbf{V}_{\mathrm{surv}}) := \begin{cases} \texttt{ELECTRON}, & q(\mathbf{V}_{\mathrm{surv}})=+1,\\ \texttt{ANTI\_ELECTRON}, & q(\mathbf{V}_{\mathrm{surv}})=-1,\\ \texttt{NEUTRAL}, & q(\mathbf{V}_{\mathrm{surv}})=0. \end{cases} \label{eq:S08_04_e_label}\] Definition [eq:S08_04_e_label] is the formal definition of the label format; it does not include an interpretation of the label meanings.

10.4.2 8.4.2 Rules (emission conditions and label assignment rules)

10.4.2.1 8.4.2.1 Emission-validity conditions (mandatory Gate premises)

The emission event is defined only when all of the following conditions hold.

10.4.2.2 (R1) Pass of the “6 cancel + 1 survive” verification

The shell verification Gate must be PASS. \[\texttt{G-SHELL7-6C1S}=\texttt{PASS}. \label{eq:S08_04_rule_gate_shell}\] If it is not PASS, emission is not defined and the label cannot be used as an input for conclusions.

10.4.2.3 (R2) Survival non-degeneracy condition

Lock the non-degeneracy threshold \(V_{\min}>0\) as \[V_{\min}>0, \qquad V_{\min}\ \text{is pre-registered in}\ \texttt{gate\_lock}. \label{eq:S08_04_Vmin_lock}\] Define the emission condition as \[\|\mathbf{V}_{\mathrm{surv}}\|\ge V_{\min}. \label{eq:S08_04_emit_condition_V}\] If [eq:S08_04_emit_condition_V] fails, then \(\widehat{\mathbf{n}}_{\mathrm{emit}}\) is unstable/undefined and emission is not defined.

10.4.2.4 (R3) Lock-integrity of axes/thresholds/conventions

The following items must all belong to the same lock_id combination. \[(\mathbf{n}_Q,\ q_0,\ s_{\min},\ V_{\min},\ \ell_{\mathrm{emit}},\ \mathrm{Select}\ \text{rules}) \ \text{have consistent lock\_ids}. \label{eq:S08_04_lock_integrity}\] If inconsistency or mixing is detected, emission/labels are immediate FAIL.

10.4.2.5 8.4.2.2 Definition of the emission event (time/location/direction/state)

Define the emission event as \[E_{\mathrm{emit}}:\ \text{the event that records }(t_{\mathrm{emit}},\ \mathbf{x}_{\mathrm{emit}},\ \widehat{\mathbf{n}}_{\mathrm{emit}},\ \mathbf{V}_{\mathrm{surv}}). \label{eq:S08_04_Eemit_def}\] Lock the emission time by \[t_{\mathrm{emit}} := t_{\mathrm{build}} \label{eq:S08_04_temit}\] that is, emission is defined as an event recorded at the same time as the core–shell completion event. (To allow a different relative timing between completion and emission requires a separate version bump; it cannot be adjusted within the same version.)

The emission event’s location and direction are locked by [eq:S08_04_nemit] and [eq:S08_04_xemit]. The event’s state labels are locked by [eq:S08_04_Qlabel] and [eq:S08_04_e_label].

10.4.2.6 8.4.2.3 Label assignment rules (no interpretation; definitional assignment)

When emission is valid, assign labels by the following rules.

  1. Charge sign: \(q:=q(\mathbf{V}_{\mathrm{surv}})\).

  2. Charge label: \(Q_{\mathrm{label}}:=q_0 q\).

  3. Electron/anti-electron label: \(\mathcal{L}_{e}:=\mathcal{L}_{e}(\mathbf{V}_{\mathrm{surv}})\).

This label assignment is definitional (not interpretive). Changing the basis (e.g. using a different axis, a different threshold, or a different coupling rule) is forbidden within the same version.

10.4.2.7 8.4.2.4 Undefinedness handling rules

In the following cases, emission and labels are treated as undefined, and only limitation conclusions (CT-LIM) are allowed.

  1. G-SHELL7-6C1S\(\neq\)PASS.

  2. \(\|\mathbf{V}_{\mathrm{surv}}\|<V_{\min}\).

  3. \(\mathbf{V}_{\mathrm{surv}}\cdot\mathbf{n}_Q=0\) (zero sign).

  4. Lock_id inconsistency or missing lock registration (thresholds/axis/selection rules not registered).

Patching undefinedness by interpretation, or making it “defined” by post hoc changes of thresholds/axes/conventions, violates No-Tuning and is forbidden.

10.4.3 8.4.3 Output forms (emission record, charge label, electron/anti-electron label)

10.4.3.1 8.4.3.1 Output form: emission records (must be sealed)

When emission is valid, record and seal an emission record with the following fields. The record follows the schema in protocol_lock, while the meanings of the fields are fixed as below.

emit_records:
  - emit_id: (unique)
    t_emit: (t_build)
    x_emit: [x, y, z]
    n_emit: [nx, ny, nz]
    V_surv: [Vx, Vy, Vz]
    V_norm: ||V_surv||
    q_sign: -1 | 0 | +1
    Q_label: q0 * q_sign
    e_label: ELECTRON | ANTI_ELECTRON | NEUTRAL
    refs:
      core_state_id: (X82,S7,xc,Rp snapshot id)
      shell_partition: (P1*,P2*,Q*,u*)
    lock_refs:
      canon_lock_id: (...)
      realization_lock_id: (...)
      analysis_lock_id: (...)
    gate_refs:
      G-SHELL7-6C1S: PASS
      (optional) G-REG: PASS
      (optional) G-REP: PASS

The emission record must be sealed by inclusion in manifest and checksums; unsealed records cannot be used as inputs for downstream derivations.

10.4.3.2 8.4.3.2 Output form: charge label

When emission is valid, the charge label is locked by [eq:S08_04_Qlabel]. \[Q_{\mathrm{label}}=q_0\,\mathrm{sgn}(\mathbf{V}_{\mathrm{surv}}\cdot\mathbf{n}_Q). \label{eq:S08_04_Qlabel_result}\] If \(q_0=1\) is locked, then \(Q_{\mathrm{label}}\in\{-1,0,+1\}\).

10.4.3.3 8.4.3.3 Output form: electron/anti-electron label

When emission is valid, the electron/anti-electron label is locked by [eq:S08_04_e_label]. \[\mathcal{L}_{e}= \begin{cases} \texttt{ELECTRON}, & \mathbf{V}_{\mathrm{surv}}\cdot\mathbf{n}_Q>0,\\ \texttt{ANTI\_ELECTRON}, & \mathbf{V}_{\mathrm{surv}}\cdot\mathbf{n}_Q<0,\\ \texttt{NEUTRAL}, & \mathbf{V}_{\mathrm{surv}}\cdot\mathbf{n}_Q=0. \end{cases} \label{eq:S08_04_label_result}\] Whether the NEUTRAL output is allowed as a conclusion sentence is locked by PASS.rules; this section does not decide allow/forbid (it fixes definitions only).

LOCK/Gate connections for this section (none if empty)

11 9. Event (Quantum) Definition and Canonical Event Rate

Purpose of the chapter (locking operational definitions)

This chapter fixes “event (quantum event)” as a computable/loggable operational definition, and defines the “canonical event rate” by coupling event occurrences to time (ticks). The outputs of this chapter are locked into the following four bundles.

  1. Operational definition of state: what must be recorded as a state (required fields).

  2. Operational definition of event: which form of state change is counted as an event (trigger rules).

  3. Operational definition of annihilation: rules by which an event terminates/cancels under specified conditions (including label rules).

  4. Definition of the canonical event rate: based on the locked event definition, define a long-time averaged event rate, and fix the handoff point to later chapters (build time / mass / force / unit realization).

The definitions in this chapter rely only on internal premises—axioms (infinite rigidity / plenum / local rule), registries (LOCK/SSOT), rectification constants (\(\alpha,\delta\)), and integerization rules (3-sector, 82+7 structure)—without external-text justification.

Higher-level preconditions (regime/locks/symbol conventions)

The definitions of this chapter are valid only when the following preconditions are locked.

  1. Stone/plenum/local-rule regime: under the VP axiom set, events arise only as compositions of local updates.

  2. Canonical cell and coordinates: the canonical cell (CELL-CUBE) and the center/boundary/cut conventions must be locked.

  3. Rectification constants and survival convention: the definitions \(\alpha=2/\pi\) and \(\delta=1/\pi^2\) are locked, and the operational definition of the survival weight \(w(e)\) is locked (when rectified event rates are used).

  4. Integer structure and labels: the 3-sector integerization and the shell-7 cancellation–survival convention are locked; if charge/electron labels are used, the label axis and thresholds are locked.

If the above preconditions are not locked, the event/state/annihilation definitions in this chapter are undefined and cannot be handed off to subsequent chapters.

Status of operational definitions (fixation as definitions)

In this chapter, an “operational definition” means the following.

  1. A definition must be mechanically decidable: from the log fields alone, the existence/non-existence of an event must be decidable automatically.

  2. A definition must be fixed to a single source of truth (SSOT): the same event must not be counted by different rules in different sections.

  3. A definition is not modifiable after the fact: changing a definition changes inputs to event rates, build time, and mass/force derivations; therefore any change is permitted only by a version bump.

Accordingly, every item in this chapter is presented only as “definitions/rules”, not as interpretive prose.

Connection skeleton: state / event / annihilation

The core skeleton of this chapter is fixed in the following order. \[\text{State logging} \;\Longrightarrow\; \text{event-trigger decision} \;\Longrightarrow\; \text{event aggregation (count/weight)} \;\Longrightarrow\; \text{annihilation decision} \;\Longrightarrow\; \text{canonical event rate}. \label{eq:S09_chain}\] Here

This overview only declares the skeleton; the concrete definitions are completed in §9.1 and onward.

11.1 9.1 Operational definitions for quantum/event/state/annihilation

11.1.1 9.1.1 Purpose

This section fixes quantum, event, state, and annihilation as loggable operational definitions. The definitions include (i) required log fields, (ii) a decidable trigger function, (iii) prohibition of post hoc tuning (No-Tuning), and (iv) a single-source-of-truth (SSOT) convention; interpretive prose is not included.

11.1.2 9.1.2 Common time/window definitions (ticks and realized time)

11.1.2.1 [D-9.1-1] Tick index

Define the tick index as \(n\in\mathbb{Z}\). In event logs, every record must include the integer tick \(n\) as a required field.

11.1.2.2 [D-9.1-2] Realized time

When the realized-time tick \(\Delta t\) is locked in realization_lock, define realized time as \[t := n\,\Delta t. \label{eq:S09_01_time_real}\]

11.1.2.3 [D-9.1-3] Time window

For two ticks \(n_1<n_2\), define the time window as follows. \[W[n_1,n_2) := \{\,n\in\mathbb{Z}\mid n_1\le n<n_2\,\}, \qquad \Delta N := n_2-n_1, \qquad \Delta T := \Delta N\,\Delta t. \label{eq:S09_01_window}\] If \(\Delta t\) is not definable, then \(\Delta T\) cannot be used and all event rates must be recorded only on the tick basis.

11.1.3 9.1.3 Operational definition of state: log record (required fields)

11.1.3.1 [D-9.1-4] State record

Define a state as a “fixed snapshot at one tick”, and denote the state record by \(S[n]\). The record \(S[n]\) must include the following fields (if any is missing, the state is undefined).

  1. state_id: unique state identifier (string).

  2. tick: \(n\) (integer).

  3. regime_id: regime identifier (locked in 4.3).

  4. lock_refs: canon_lock_id, realization_lock_id, analysis_lock_id.

  5. core_state_ref: reference to the core state (e.g., snapshot key or hash for X82.csv).

  6. shell_state_ref: reference to the shell state (e.g., snapshot key or hash for S7).

  7. center_ref: reference to (or value of) the center \(\mathbf{x}_c\).

  8. geometry_ref: references to the length scales used, such as \(R_p\), \(L_q\), \(D_{\mathrm{anch}}\) (item name + lock_id).

  9. graph_ref: reference to the contact graph (e.g., hash or snapshot key of G82.edgelist).

11.1.3.2 [D-9.1-5] Completeness condition for a state

A state record \(S[n]\) is defined as a “complete state” iff the following holds. \[\mathrm{Complete}(S[n])=1 \Longleftrightarrow \text{all required fields in 9.1.3 exist and \texttt{lock\_refs} are sealed in the snapshot}. \label{eq:S09_01_complete_state}\] If \(\mathrm{Complete}(S[n])=0\), that tick cannot be used as an input for event decisions.

11.1.4 9.1.4 Operational definition of event: transition and trigger

11.1.4.1 [D-9.1-6] Transition

When two consecutive states \(S[n-1],S[n]\) are both complete, define the transition as \[\Delta S[n] := (S[n-1]\rightarrow S[n]). \label{eq:S09_01_transition}\]

11.1.4.2 [D-9.1-7] Event trigger function

Define that an event occurs for a transition \(\Delta S[n]\) iff the following trigger function returns 1. \[\mathrm{Trig}(\Delta S[n])\in\{0,1\}, \qquad \mathrm{Trig}(\Delta S[n])=1\ \Longleftrightarrow\ \Delta S[n]\ \text{is counted as an event}. \label{eq:S09_01_trig_def}\] The concrete rule of \(\mathrm{Trig}\) must be locked in analysis_lock, and it must not be replaced after seeing results.

11.1.4.3 [D-9.1-8] Event record

If \(\mathrm{Trig}(\Delta S[n])=1\), generate an event record \(e[n]\). The required fields of an event record are as follows.

  1. event_id: unique event identifier (string).

  2. tick: \(n\).

  3. pre_state_id: \(S[n-1].\mathrm{state\_id}\).

  4. post_state_id: \(S[n].\mathrm{state\_id}\).

  5. trigger_id: which trigger fired (enumeration).

  6. lock_refs: canon_lock_id, realization_lock_id, analysis_lock_id.

  7. regime_id: \(S[n].\mathrm{regime\_id}\).

  8. event_payload: trigger-specific required data (9.1.4.1–9.1.4.3).

11.1.4.4 9.1.4.1 Trigger set (standard trigger IDs)

In this white paper, trigger IDs are fixed to the following enumeration (additional triggers are permitted only by a version bump).

  1. TRIG-GRAPH: trigger for contact-graph changes.

  2. TRIG-SHELL: trigger for shell split / cancellation–survival changes.

  3. TRIG-LABEL: trigger for charge/electron label changes.

  4. TRIG-EMIT: trigger for generating an emission record.

  5. TRIG-ANN: trigger for annihilation decision (9.1.6).

For each trigger, the concrete decision rule is composed only of “log comparisons”, and its thresholds are locked in gate_lock.

11.1.4.5 9.1.4.2 Definition of TRIG-GRAPH (contact-graph change)

Denote the contact graph by \(\mathcal{G}_c[n]\) and its edge set by \(\mathcal{E}_c[n]\). Define the graph-change magnitude by \[\Delta E[n] := |\mathcal{E}_c[n]\ \triangle\ \mathcal{E}_c[n-1]|, \label{eq:S09_01_graph_delta}\] where \(\triangle\) denotes the symmetric difference. Lock the threshold \(E_{\min}\in\mathbb{Z}_{\ge 0}\) in gate_lock. Define TRIG-GRAPH as \[\mathrm{Trig}_{\mathrm{GRAPH}}(\Delta S[n])=1 \Longleftrightarrow \Delta E[n]\ge E_{\min}. \label{eq:S09_01_trig_graph}\] The event_payload of a TRIG-GRAPH event must include references to \((\Delta E[n],E_{\min},\mathcal{E}_c[n-1],\mathcal{E}_c[n])\).

11.1.4.6 9.1.4.3 Definition of TRIG-LABEL (label change)

Denote the electron/positron/neutral label by \(\mathcal{L}[n]\in\{\texttt{ELECTRON},\texttt{ANTI\_ELECTRON},\texttt{NEUTRAL}\}\) (see the definitions in 7.4 and 8.4). Define TRIG-LABEL as \[\mathrm{Trig}_{\mathrm{LABEL}}(\Delta S[n])=1 \Longleftrightarrow \mathcal{L}[n]\neq \mathcal{L}[n-1]. \label{eq:S09_01_trig_label}\] The event_payload of a TRIG-LABEL event must include \((\mathcal{L}[n-1],\mathcal{L}[n])\) and a reference (or hash) to \(\mathbf{V}_{\mathrm{surv}}\) used in the label computation.

11.1.5 9.1.5 Operational definition of quantum: minimal unit of an event

11.1.5.1 [D-9.1-9] Quantum

In this document, the reserved word “quantum” means one event record \(e[n]\). \[\text{Quantum } q[n]\ \equiv\ e[n]\quad (\mathrm{Trig}(\Delta S[n])=1\ \text{when}). \label{eq:S09_01_quantum_def}\] Thus a quantum is the minimal loggable unit, and any discussion of a quantum must be reducible to the required fields of \(e[n]\). If a “size” or “weight” of a quantum is needed, it may refer only to items locked in analysis_lock, such as the survival weight \(w(e)\) or a count like \(N_0\). (Introducing an arbitrary size function for a quantum is forbidden.)

11.1.6 9.1.6 Operational definition of annihilation: rule of cancellation events

Define annihilation as a rule by which “the simultaneous existence of opposite labels cancels within a specified window and is removed”.

11.1.6.1 9.1.6.1 Definition of the annihilation windows (time/space)

Because annihilation decisions require a time window and a spatial window, lock the following thresholds. \[\Delta n_{\mathrm{ann}}\in\mathbb{Z}_{>0}\ \text{(locked)}, \qquad \rho_{\mathrm{ann}}>0\ \text{(locked)}. \label{eq:S09_01_ann_windows}\] Define the time window on the tick basis by \[W_{\mathrm{ann}}[n] := W[n-\Delta n_{\mathrm{ann}}+1,\ n+1). \label{eq:S09_01_ann_time_window}\] Define the spatial window by the distance between emission points (8.4) \(\mathbf{x}_{\mathrm{emit}}\).

11.1.6.2 9.1.6.2 Definition of annihilation candidates (opposite-label pairs)

Within the time window \(W_{\mathrm{ann}}[n]\), define the set of emission-event records as \[\mathcal{E}_{\mathrm{emit}}(W_{\mathrm{ann}}[n]) := \{\, e[m]\mid m\in W_{\mathrm{ann}}[n],\ \texttt{trigger\_id}=\texttt{TRIG-EMIT}\,\}. \label{eq:S09_01_emit_set}\] Each emission event carries references (by the definition in 8.4) to the emission point \(\mathbf{x}_{\mathrm{emit}}(e)\), the label \(\mathcal{L}(e)\), and the survival vector \(\mathbf{V}_{\mathrm{surv}}(e)\).

Define two distinct emission events \(e_a,e_b\) as an annihilation-candidate pair if they satisfy \[\mathcal{L}(e_a)=\texttt{ELECTRON}, \qquad \mathcal{L}(e_b)=\texttt{ANTI\_ELECTRON}. \label{eq:S09_01_opposite_labels}\]

11.1.6.3 9.1.6.3 Spatial proximity condition

Define the spatial proximity condition by \[\|\mathbf{x}_{\mathrm{emit}}(e_a)-\mathbf{x}_{\mathrm{emit}}(e_b)\|\le \rho_{\mathrm{ann}}. \label{eq:S09_01_ann_spatial}\]

11.1.6.4 9.1.6.4 Cancellation condition (collapse of the sum of survival vectors)

Lock the cancellation threshold \(V_{\mathrm{ann}}>0\). Define the annihilation-cancellation condition by \[\left\|\mathbf{V}_{\mathrm{surv}}(e_a)+\mathbf{V}_{\mathrm{surv}}(e_b)\right\|\le V_{\mathrm{ann}}. \label{eq:S09_01_ann_cancel}\]

11.1.6.5 9.1.6.5 TRIG-ANN

Define the annihilation trigger by \[\mathrm{Trig}_{\mathrm{ANN}}(\Delta S[n])=1 \Longleftrightarrow \exists (e_a,e_b)\subset \mathcal{E}_{\mathrm{emit}}(W_{\mathrm{ann}}[n])\ \text{s.t.}\ \eqref{eq:S09_01_opposite_labels},\ \eqref{eq:S09_01_ann_spatial},\ \eqref{eq:S09_01_ann_cancel}\ \text{all hold}. \label{eq:S09_01_trig_ann}\] The event_payload of a TRIG-ANN event must include references to \((e_a,e_b)\), the distance value, the norm of the vector sum, and the thresholds \((\rho_{\mathrm{ann}},V_{\mathrm{ann}},\Delta n_{\mathrm{ann}})\).

11.1.6.6 9.1.6.6 State-update rule for annihilation (label erasure)

When a TRIG-ANN event occurs, define the state-update rule by \[\mathcal{L}[n]\leftarrow \texttt{NEUTRAL} \quad\text{and}\quad \text{the target emission records }(e_a,e_b)\text{ are labeled as annihilated}. \label{eq:S09_01_ann_state_update}\] This rule is an operational definition: “annihilation is recorded as label erasure”; it assigns no additional meaning (interpretation).

11.1.7 9.1.7 Forbidden ambiguous statements (definition violation / Gate invalidation / post hoc tuning)

Under the operational-definition system of this section, the following statement types are forbidden. The criterion for prohibition is “not decidable from logs”, or “allows post hoc tuning”, or “invalidates Gate decisions”.

  1. Missing-definition type: “A state changed, so it is an event.” (which field change triggers an event is unspecified)

  2. Undefined-threshold type: “If they are close enough, it is annihilation.” (\(\rho_{\mathrm{ann}}\) or \(V_{\mathrm{ann}}\) not locked)

  3. Regime-ignoring type: “The same event rate holds in any regime.” (regime coordinate axes / allowed stack unspecified)

  4. Gate-bypass type: “Even if Gate is FAIL, interpret it as an event.” (decision invalidation)

  5. Post hoc tuning type: “If the result does not match, adjust the trigger threshold.” (No-Tuning violation)

  6. Lock-mixing type: “This time, re-define the label using a different reference axis.” (lock_id mixing)

  7. Justifying incomplete logs: “There is no log, but it probably happened.” (violates log-based decision)

11.2 9.2 Canonical event-rate law \(\nu_{\mathrm{can}}=s\cdot\delta\)

11.2.1 9.2.1 Fixing the classification (definition/axiom/theorem)

Fix the status of the key statements used in this section as follows.

  1. [D] definitions: \(s\) (attempt rate), \(\delta\) (survival rectification coefficient), \(\nu_{\mathrm{can}}\) (canonical event rate), and the counts/weights/time windows/limit operations that constitute them.

  2. [A] axioms: (i) existence and convergence of long-time averages (canonical stationarity), and (ii) applicability conditions for the universality of \(\delta\) (uniform phases, dual constraints, product measure) and its scope.

  3. [T] theorem: the derivation of \(\nu_{\mathrm{can}}=s\cdot\delta\) from the above definitions and axioms.

Accordingly, the identity \(\nu_{\mathrm{can}}=s\cdot\delta\) in this section is fixed as a theorem [T], while the conditions under which the theorem holds are fixed as axioms [A]. The meanings of the symbols \(s,\delta,\nu_{\mathrm{can}}\) are fixed only by definitions [D].

11.2.2 9.2.2 [D] Attempt-event set and attempt rate \(s\)

11.2.2.1 9.2.2.1 Time window and attempt-event set

Let the realized time window be \([t,t+T)\) (\(T>0\)). Define the set of “attempt events” occurring in this window by \[\mathcal{E}_0(t;T) :=\{\, e\mid t\le t(e)<t+T,\ \mathrm{Trig}_0(e)=1\,\}. \label{eq:S09_02_E0}\] Here \(\mathrm{Trig}_0(e)\in\{0,1\}\) is the “attempt-event trigger” and is locked in analysis_lock so that it satisfies:

  1. if \(\mathrm{Trig}_0(e)=1\), the minimum conditions for producing an event record are satisfied (including log completeness);

  2. \(\mathrm{Trig}_0\) is the event-count rule before applying the survival constraint (the \(w(e)\) below).

If the definition of \(\mathrm{Trig}_0\) is not locked, [eq:S09_02_E0] is undefined.

Define the number of attempt events (raw event count) by \[N_0(t;T):=\bigl|\mathcal{E}_0(t;T)\bigr|. \label{eq:S09_02_N0}\]

11.2.2.2 9.2.2.2 Definition of the attempt rate \(s\) (long-time average)

Define the attempt rate \(s\) by the following long-time average. \[s := \lim_{T\to\infty}\frac{N_0(t;T)}{T}. \label{eq:S09_02_s_def}\] The conditions under which the limit in [eq:S09_02_s_def] exists and converges to the same value independent of \(t\) are fixed as axiom [A-9.2-S1] (§9.2.5). Thus \(s\) is, as a definition [D], the “time density of attempt-event counts”; it is not justified by external texts.

11.2.3 9.2.3 [D] Survival weight and canonical event rate \(\nu_{\mathrm{can}}\)

11.2.3.1 9.2.3.1 Dual-constrained phases and half-wave rectification

For each attempt event \(e\in\mathcal{E}_0(t;T)\), define two phase variables by \[\theta(e)\in[0,2\pi), \qquad \varphi(e)\in[0,2\pi). \label{eq:S09_02_theta_phi}\] The production rules for \(\theta(e)\) and \(\varphi(e)\) (from which log fields and by which computation) are locked in analysis_lock. Define the half-wave rectification operator by \[_+ := \max(0,x). \label{eq:S09_02_pospart}\]

11.2.3.2 9.2.3.2 Definition of the survival weight \(w(e)\)

Define the survival weight of an attempt event \(e\) by \[w(e) := [\cos\theta(e)]_{+}\,[\cos\varphi(e)]_{+}. \label{eq:S09_02_w_def}\] Definition [eq:S09_02_w_def] is the unique source of the survival convention; within the same version it must not be replaced by \(|\cdot|\) or any other nonlinear function.

11.2.3.3 9.2.3.3 Definition of the canonical event count \(N_{\mathrm{can}}\)

Define the canonical event count (rectified count) in the time window \([t,t+T)\) by \[N_{\mathrm{can}}(t;T) := \sum_{e\in\mathcal{E}_0(t;T)} w(e). \label{eq:S09_02_Ncan_def}\] By definition, \(0\le N_{\mathrm{can}}(t;T)\le N_0(t;T)\).

11.2.3.4 9.2.3.4 Definition of the canonical event rate \(\nu_{\mathrm{can}}\) (long-time average)

Define the canonical event rate by the following long-time average. \[\nu_{\mathrm{can}} := \lim_{T\to\infty}\frac{N_{\mathrm{can}}(t;T)}{T}. \label{eq:S09_02_nucan_def}\] The conditions under which the limit in [eq:S09_02_nucan_def] exists and converges independent of \(t\) are fixed as axiom [A-9.2-S1] (§9.2.5).

11.2.4 9.2.4 [D] Definition of \(\delta\) and its locked value (lock location)

11.2.4.1 9.2.4.1 Definition of \(\delta\) (mean survival coefficient over attempt events)

Define \(\delta\) as the following mean survival coefficient. \[\delta := \lim_{T\to\infty} \frac{1}{N_0(t;T)} \sum_{e\in\mathcal{E}_0(t;T)} w(e), \qquad (\text{with }N_0(t;T)>0). \label{eq:S09_02_delta_def}\] For [eq:S09_02_delta_def] to be meaningful, it must hold that \(N_0(t;T)\to\infty\) for long times; this condition is included in axiom [A-9.2-S1].

11.2.4.2 9.2.4.2 Locking the value of \(\delta\) (regimes where the universality axiom applies)

In regimes where the universality axiom [A-5.2-U] applies, the value of \(\delta\) is locked as \[\delta=\frac{1}{\pi^2}. \label{eq:S09_02_delta_value_lock}\] Equation [eq:S09_02_delta_value_lock] is locked in its unique source location in §5.2; it is not re-derived here. In this section, only referencing [eq:S09_02_delta_value_lock] is permitted.

11.2.5 9.2.5 [A] Axioms: canonical stationarity and applicability of \(\delta\) universality

11.2.6 [A-9.2-S1] Canonical stationarity axiom (existence and convergence of long-time averages)

Fix as an axiom that the following limits exist and converge to the same value independent of the start time \(t\). \[\lim_{T\to\infty}\frac{N_0(t;T)}{T}=s, \qquad \lim_{T\to\infty}\frac{N_{\mathrm{can}}(t;T)}{T}=\nu_{\mathrm{can}}, \qquad \lim_{T\to\infty}\frac{1}{N_0(t;T)}\sum_{e\in\mathcal{E}_0(t;T)}w(e)=\delta. \label{eq:S09_02_stationarity_axiom}\] Axiom [eq:S09_02_stationarity_axiom] means that the “canonical event rate”, the “attempt rate”, and the “survival coefficient” can be treated as constants within a regime. If this axiom fails, the premise of the theorem \(\nu_{\mathrm{can}}=s\delta\) collapses.

11.2.7 [A-9.2-S2] \(\delta\)-universality axiom (value fixation in applicable regimes)

Use \(\delta=\frac{1}{\pi^2}\) only in regimes where the following conditions are locked.

  1. phase uniformity: \(\theta,\varphi\) each follow the uniform measure on the full cycle \([0,2\pi)\);

  2. dual constraints: the survival weight is fixed to the product form [eq:S09_02_w_def];

  3. product measure (separated measure): the joint measure of \((\theta,\varphi)\) factorizes as a product measure.

This axiom is a restatement, in this chapter, of the regime restriction of the universality axiom set [A-5.2-U] in §5.2; the value fixation is attributed to canon_lock.

11.2.8 9.2.6 [T] Canonical event-rate law \(\nu_{\mathrm{can}}=s\cdot\delta\)

In regimes where axiom [A-9.2-S1] holds, derive the following theorem from definitions [eq:S09_02_s_def], [eq:S09_02_nucan_def], and [eq:S09_02_delta_def].

11.2.9 [T-9.2-1] Theorem (canonical event-rate law)

The following holds. \[\nu_{\mathrm{can}} = s\cdot \delta. \label{eq:S09_02_theorem_goal}\]

11.2.9.1 Proof

Using the definition of the canonical event count [eq:S09_02_Ncan_def], \[\frac{N_{\mathrm{can}}(t;T)}{T} = \frac{1}{T}\sum_{e\in\mathcal{E}_0(t;T)} w(e). \label{eq:S09_02_step1}\] Multiply and divide the right-hand side by \(N_0(t;T)\) to obtain (for \(T\) such that \(N_0(t;T)>0\)) \[\begin{aligned} \frac{N_{\mathrm{can}}(t;T)}{T} &= \left(\frac{N_0(t;T)}{T}\right) \left(\frac{1}{N_0(t;T)}\sum_{e\in\mathcal{E}_0(t;T)} w(e)\right). \label{eq:S09_02_step2}\end{aligned}\] Now take the limit \(T\to\infty\). By axiom [A-9.2-S1], \[\lim_{T\to\infty}\frac{N_0(t;T)}{T}=s, \qquad \lim_{T\to\infty}\frac{1}{N_0(t;T)}\sum_{e\in\mathcal{E}_0(t;T)} w(e)=\delta, \label{eq:S09_02_step3}\] so from [eq:S09_02_step2] we obtain \[\lim_{T\to\infty}\frac{N_{\mathrm{can}}(t;T)}{T} = s\cdot\delta. \label{eq:S09_02_step4}\] The left-hand side is \(\nu_{\mathrm{can}}\) by definition [eq:S09_02_nucan_def], hence \[\nu_{\mathrm{can}}=s\cdot\delta. \label{eq:S09_02_step5}\] Therefore [eq:S09_02_theorem_goal] holds. \(\square\)

11.2.10 [T-9.2-2] Special form in the universal regime

In regimes where axiom [A-9.2-S2] holds so that [eq:S09_02_delta_value_lock] can be used, theorem [eq:S09_02_theorem_goal] is fixed in the following special form. \[\nu_{\mathrm{can}} = s\cdot\frac{1}{\pi^2}. \label{eq:S09_02_nucan_special}\] Equation [eq:S09_02_nucan_special] is a special case of the theorem and is used only in regimes where value fixation of \(\delta\) is permitted.

11.2.11 9.2.7 Standard conclusion format for connecting to PASS.rules

When using theorem [eq:S09_02_theorem_goal] or [eq:S09_02_nucan_special] as a conclusion sentence, the following items must be stated together (if omitted, the conclusion has no validity).

  1. identifiers of the referenced definitions of \(\mathrm{Trig}_0\) and \(w(e)\) (via the analysis_lock reference);

  2. the regime condition for \(\delta\) value fixation (whether the regime is universal) and the Gate decision related to \(\delta\);

  3. sealed references (manifest/checksums/registry_snapshot) of the time window and log snapshots used to compute \(s,\delta,\nu_{\mathrm{can}}\).

These items are not “narrative style”; they are required fields that constitute the validity of the conclusion sentence, enforced by PASS.rules.

11.3 9.3 Electron canonical: \(\nu_{e,\mathrm{can}}=1\), \(r_e\)

11.3.1 9.3.1 [D] Defining the electron canonical event rate (fixing the unit)

Define and lock the electron canonical event rate \(\nu_{e,\mathrm{can}}\) by the following identity. \[\nu_{e,\mathrm{can}} := 1. \label{eq:S09_03_nue_can_def}\] Equation [eq:S09_03_nue_can_def] is the definition of the “electron canonical” and is not tuned within the same version. The unit of \(\nu_{e,\mathrm{can}}\) is fixed to the canonical event-rate unit used in this document (conversion to realized units is treated only in the unit-realization chapter).

11.3.2 9.3.2 [D] Electron attempt rate \(s_e\) and survival coefficient \(\delta\)

Specialize the general canonical-event-rate components defined in §9.2 to the electron case as follows.

11.3.2.1 9.3.2.1 [D] Electron attempt-event set and attempt rate

Denote the electron attempt-event (raw event) set by \(\mathcal{E}_{0,e}(t;T)\), and define the raw count by \[N_{0,e}(t;T):=\left|\mathcal{E}_{0,e}(t;T)\right| \label{eq:S09_03_N0e}\] Define the electron attempt rate \(s_e\) by the long-time average \[s_e := \lim_{T\to\infty}\frac{N_{0,e}(t;T)}{T}. \label{eq:S09_03_se_def}\] Existence and settlement of the limit are attributed to the stationarity axiom (the axiom items of §9.2) and are not restated here.

11.3.2.2 9.3.2.2 [D] Electron survival coefficient \(\delta\)

Assume the survival weight \(w(e)\) is defined for electron attempt events \(e\in\mathcal{E}_{0,e}(t;T)\), and define the electron survival coefficient \(\delta\) by the average \[\delta := \lim_{T\to\infty}\frac{1}{N_{0,e}(t;T)}\sum_{e\in\mathcal{E}_{0,e}(t;T)} w(e), \qquad ( N_{0,e}(t;T)>0 ). \label{eq:S09_03_delta_def}\] The symbol \(\delta\) is used by referring to the rectification-constant chapter (Chapter 5) and is not redefined within the same version.

11.3.3 9.3.3 [T] Applying the theorem: \(\nu_{e,\mathrm{can}}=s_e\cdot\delta\)

Applying the canonical-event-rate theorem of §9.2 to the electron yields \[\nu_{e,\mathrm{can}}=s_e\cdot\delta. \label{eq:S09_03_nue_factor}\] Substitute [eq:S09_03_nue_can_def] into [eq:S09_03_nue_factor]. \[\begin{aligned} \nu_{e,\mathrm{can}}=s_e\cdot\delta &\Longrightarrow 1=s_e\cdot\delta \label{eq:S09_03_step1} \\ &\Longrightarrow s_e=\frac{1}{\delta}. \label{eq:S09_03_step2}\end{aligned}\] Therefore the electron attempt rate \(s_e\) is fixed as the reciprocal of the survival coefficient \(\delta\). This relation holds even without assuming the universal value of \(\delta\) (e.g., \(\delta=1/\pi^2\)).

11.3.4 9.3.4 [D] Defining the electron length \(r_e\) (geometric implementation of the attempt rate)

This section locks the electron length \(r_e\) by definition as a “geometric implementation of the electron attempt rate \(s_e\).”

11.3.4.1 9.3.4.1 [D] Anchor length and half-length

Assume the canonical-cell representative length \(D_{\mathrm{anch}}\) is locked, and fix the half-length \(r_0\) by the following derived definition. \[r_0:=\frac{D_{\mathrm{anch}}}{2}. \label{eq:S09_03_r0}\] The geometric meaning of \(D_{\mathrm{anch}}\) (edge length in the canonical cell) is locked in §3.3; this section only refers to it.

11.3.4.2 9.3.4.2 [D] Geometric definition of the attempt rate

Define the electron attempt rate \(s_e\) by the following geometric ratio. \[s_e := \frac{r_0}{r_e}. \label{eq:S09_03_se_geom}\] Definition [eq:S09_03_se_geom] adopts the convention that the attempt rate is “how many times the half-length \(r_0\) contains the electron radius \(r_e\),” and is not modified within the same version. Equation [eq:S09_03_se_geom] is admissible only when the meanings of the symbols (radius/half-length) are locked; any radius/diameter confusion or cell-geometry confusion is an immediate FAIL (see the convention in §2.4).

11.3.5 9.3.5 Step-by-step derivation of \(r_e\)

Combine [eq:S09_03_step2] and [eq:S09_03_se_geom] to derive \(r_e\). \[\begin{aligned} s_e=\frac{1}{\delta} \ \ \text{and}\ \ s_e=\frac{r_0}{r_e} &\Longrightarrow \frac{r_0}{r_e}=\frac{1}{\delta} \label{eq:S09_03_re_step1} \\ &\Longrightarrow r_e=r_0\,\delta. \label{eq:S09_03_re_step2}\end{aligned}\] Substitute [eq:S09_03_r0] into [eq:S09_03_re_step2]. \[\begin{aligned} r_e=r_0\,\delta &\Longrightarrow r_e=\left(\frac{D_{\mathrm{anch}}}{2}\right)\delta. \label{eq:S09_03_re_step3}\end{aligned}\] Therefore the final form of the electron radius is fixed as \[\boxed{ r_e=\frac{D_{\mathrm{anch}}}{2}\,\delta } \label{eq:S09_03_re_final}\] Equation [eq:S09_03_re_final] is derived only by combining the electron canonical definition \(\nu_{e,\mathrm{can}}=1\), the geometric attempt-rate definition [eq:S09_03_se_geom], and the survival-coefficient definition [eq:S09_03_delta_def].

11.3.6 9.3.6 Special form in the universal regime (value substitution)

If \(\delta\) is locked to \[\delta=\frac{1}{\pi^2} \label{eq:S09_03_delta_univ}\] in a regime where the universality axiom applies, then substituting [eq:S09_03_delta_univ] into [eq:S09_03_re_final] yields the special form \[\begin{aligned} r_e=\frac{D_{\mathrm{anch}}}{2}\,\delta &\Longrightarrow r_e=\frac{D_{\mathrm{anch}}}{2}\cdot\frac{1}{\pi^2} =\frac{D_{\mathrm{anch}}}{2\pi^2}. \label{eq:S09_03_re_univ}\end{aligned}\] Special form [eq:S09_03_re_univ] is used only in the universal regime; in regimes where universality triggers are broken, only the general form [eq:S09_03_re_final] (regime-dependent \(\delta\)) is allowed.

11.3.7 9.3.7 Derived quantity (diameter)

Define the electron diameter (length) by the derived definition \[\ell_e := 2r_e. \label{eq:S09_03_le_def}\] Therefore, from [eq:S09_03_re_final], \[\ell_e = 2\left(\frac{D_{\mathrm{anch}}}{2}\delta\right)=D_{\mathrm{anch}}\delta, \label{eq:S09_03_le_final}\] and in the universal regime, from [eq:S09_03_re_univ], \[\ell_e=\frac{D_{\mathrm{anch}}}{\pi^2} \label{eq:S09_03_le_univ}\] is fixed. The symbol \(\ell_e\) is subject to the diameter/radius disambiguation convention (§2.4); using \(\ell_e\) as a radius is an immediate FAIL.

11.4 9.4 \(292.339978\ldots\)

11.4.1 9.4.1 LOCK inputs and reference formula (starting point)

This section assumes that the following inputs are fixed (LOCK) by canon_lock. \[\begin{aligned} &D_{\mathrm{anch}}=\Danchm, \label{eq:S09_04_Danch_lock}\\ &r_p=\rprotonm, \label{eq:S09_04_rp_lock}\\ &\pi\ \text{(dimensionless constant, locked)}, \qquad \delta=\frac{1}{\pi^2}. \label{eq:S09_04_delta_def}\end{aligned}\] It also applies the canonical event-rate law (theorem) of §9.2 to the proton and uses \[\nu_{p,\mathrm{can}}=s_p\cdot \delta. \label{eq:S09_04_nu_sp_delta}\]

11.4.2 9.4.2 Definition: proton scale factor \(s_p\)

Fix the canonical-cell half-length \(r_0\) by the derived definition \[r_0:=\frac{D_{\mathrm{anch}}}{2}. \label{eq:S09_04_r0_def}\] Define the proton scale factor \(s_p\) as the following dimensionless ratio. \[s_p:=\frac{r_0}{r_p}=\frac{D_{\mathrm{anch}}}{2r_p}. \label{eq:S09_04_sp_def}\] In definition [eq:S09_04_sp_def], the geometric meanings of \(D_{\mathrm{anch}}\) and \(r_p\) (diameter/radius/cell geometry) must already be locked; any confusion (overloading) is an immediate FAIL.

11.4.2.1 Note: why \(s_p\) is defined as a linear ratio (dimensionality of the event definition)

The \(s_p\) in [eq:S09_04_sp_def] is a scale factor used in the canonical event-rate law of §9.2; it is not “the number of volume slots” such as \((r_0/r_p)^3\). In this white paper, an event is not defined as “volume filling” but as turnover on a propagation/inflow backbone path. Effective reduction due to 3D geometry/angle cancellation/phase overlap is already absorbed into the rectification coefficient \(\delta\) (§5.2 and §9.4.3.1). Therefore, interpreting \(s_p\) again as a 3D volume ratio would double-count the same geometric factor.

11.4.3 9.4.3 Computation: \(\delta\), \(s_p\), \(\nu_{p,\mathrm{can}}\)

11.4.3.1 9.4.3.1 Computing the rectification coefficient \(\delta\)

The rectification coefficient is locked from the unique source of §5.2 as \[\delta=\frac{1}{\pi^2} =0.10132118364233778\ldots \qquad (\text{dimensionless}). \label{eq:S09_04_delta_numeric}\]

11.4.3.2 9.4.3.2 Computing the scale factor \(s_p\)

Substitute [eq:S09_04_Danch_lock] and [eq:S09_04_rp_lock] into definition [eq:S09_04_sp_def]. \[\begin{aligned} s_p &=\frac{D_{\mathrm{anch}}}{2r_p} =\frac{4.854194962126561\times 10^{-12}}{2\times 0.8412\times 10^{-15}} \notag\\ &=\frac{4.854194962126561}{1.6824}\times 10^{3} \notag\\ &=2885.27993469244\ldots \qquad (\text{dimensionless}). \label{eq:S09_04_sp_numeric}\end{aligned}\]

11.4.3.3 9.4.3.3 Computing the canonical event rate \(\nu_{p,\mathrm{can}}\)

Substitute [eq:S09_04_sp_numeric] and [eq:S09_04_delta_numeric] into [eq:S09_04_nu_sp_delta]. \[\begin{aligned} \nu_{p,\mathrm{can}} &=s_p\cdot\delta \notag\\ &=\left(2885.27993469244\ldots\right)\times \left(0.10132118364233778\ldots\right) \notag\\ &=292.33997812252504182604913619807\ldots\ \mathrm{s^{-1}}. \label{eq:S09_04_nup_numeric}\end{aligned}\] Fix the effective digits (limited by input precision) as \[\nu_{p,\mathrm{can}}\approx 292.339978123\ \mathrm{s^{-1}}. \label{eq:S09_04_nup_round}\] Or, changing only the unit label, \[\nu_{p,\mathrm{can}}\approx 292.3399781\ \mathrm{Hz} \label{eq:S09_04_nup_hz}\] may be recorded. Here “\(\mathrm{Hz}=\mathrm{s^{-1}}\)” reads the canonical second as equivalent to the SI second; the equivalence judgment is performed by the unit-realization (cross-validation) Gate.

11.4.4 9.4.4 Equivalent forms (a fully closed single expression)

Combining [eq:S09_04_nu_sp_delta], [eq:S09_04_sp_def], and [eq:S09_04_delta_def] yields \[\boxed{ \nu_{p,\mathrm{can}} =\left(\frac{D_{\mathrm{anch}}}{2r_p}\right)\left(\frac{1}{\pi^2}\right) } \label{eq:S09_04_closed_form}\] The numeric derivation in this section is completed by direct substitution into [eq:S09_04_closed_form]; no additional assumptions enter.

11.4.4.1 Equivalent form (reduction to the electron canonical)

In the universal regime, using the electron-radius definition \(r_e=(D_{\mathrm{anch}}/2)\delta\) from §9.3, [eq:S09_04_closed_form] can be written equivalently as \[\nu_{p,\mathrm{can}}=\frac{r_e}{r_p} \qquad (\delta=1/\pi^2\ \text{universal regime}). \label{eq:S09_04_equiv_re_over_rp}\] That is, “the form derived via the electron” and “the form derived via the anchor-to-proton ratio” are two notations of the same equation; no new assumptions are introduced here.

11.4.5 9.4.5 Sensitivity / error budget (LOCK linkage)

11.4.5.1 9.4.5.1 Differential sensitivity (directly from the definition)

From [eq:S09_04_closed_form], \(\nu_{p,\mathrm{can}}\) is proportional to \(D_{\mathrm{anch}}\) and inversely proportional to \(r_p\). Differentiating gives \[\frac{\partial \nu_{p,\mathrm{can}}}{\partial D_{\mathrm{anch}}} =\frac{1}{2r_p}\cdot\frac{1}{\pi^2}, \qquad \frac{\partial \nu_{p,\mathrm{can}}}{\partial r_p} =-\frac{D_{\mathrm{anch}}}{2r_p^2}\cdot\frac{1}{\pi^2} =-\frac{\nu_{p,\mathrm{can}}}{r_p} \label{eq:S09_04_sensitivity}\] The relative sensitivity is summarized as \[\frac{\Delta \nu_{p,\mathrm{can}}}{\nu_{p,\mathrm{can}}} \approx \frac{\Delta D_{\mathrm{anch}}}{D_{\mathrm{anch}}} -\frac{\Delta r_p}{r_p}, \label{eq:S09_04_rel_error}\] where the value of \(\delta=1/\pi^2\) is locked as a rectification constant (within the same version); therefore the only degrees of freedom in the error budget are the input precisions of \(D_{\mathrm{anch}}\) and \(r_p\).

11.4.5.2 9.4.5.2 Example of input variation (importance of LOCK values)

If \(r_p\) deviates from the LOCK value, \(\nu_{p,\mathrm{can}}\) changes immediately. For example, as a reference comparison, \[r_p=0.84\times 10^{-15}\ \mathrm{m} \label{eq:S09_04_rp_alt}\] would give \[\begin{aligned} \nu_{p}(0.84) &=\left(\frac{D_{\mathrm{anch}}}{2\times 0.84\times 10^{-15}}\right)\left(\frac{1}{\pi^2}\right) \notag\\ &=292.7576066627001\ldots\ \mathrm{s^{-1}}. \label{eq:S09_04_nu_alt}\end{aligned}\] Therefore the difference is \[\begin{aligned} \Delta \nu &:=\nu_{p}(0.84)-\nu_{p,\mathrm{can}} \notag\\ &=292.7576066627001\ldots-292.3399781225250\ldots \notag\\ &=0.4176285401750\ldots\ \mathrm{s^{-1}}, \label{eq:S09_04_diff}\end{aligned}\] with relative difference \[\frac{\Delta \nu}{\nu_{p,\mathrm{can}}} \approx \frac{0.4176285}{292.3400} \approx 0.00143 \qquad (\approx 0.143\%). \label{eq:S09_04_rel_diff}\] Hence, locking \(r_p\) to [eq:S09_04_rp_lock] simultaneously fixes the corresponding \(\nu_{p,\mathrm{can}}\) to the single value [eq:S09_04_nup_numeric].

12 10. Implementing the speed of light (clock-free \(c\)) and lattice propagation

Purpose of this chapter (declared linkage)

This chapter defines a “propagating / non-propagating” switch using the jamming lattice (Point-J) and a percolation (critical-throat) structure, and then fixes the linkage so that this switch becomes an input to a clock-free realization of the speed of light \(c\). It also defines the amplification coefficient \(A\) that emerges when global propagation exists, and binds both \(c\) and \(A\) to a verifiable numerical package (reproducible code/logs/Gates).

This chapter does not introduce \(c\) by appealing to external texts. In this document, \(c\) is treated only as a realization of an internally defined propagation indicator, or as an implemented quantity obtained by combining that internal indicator with an operational anchor.

Global skeleton: switch \(\rightarrow\) percolation \(\rightarrow\) amplification \(A\) \(\rightarrow\) numerical package

The skeleton of this chapter is fixed in the following order. \[\text{jamming/Point-J switch} \;\Longrightarrow\; \text{critical-throat (percolation) definition} \;\Longrightarrow\; \text{propagation path extraction (backbone/stream)} \;\Longrightarrow\; \text{define amplification coefficient }A \;\Longrightarrow\; \text{propagation-speed indicator (internal $c$)} \;\Longrightarrow\; \text{numerical package (reproducibility + Gate)}. \label{eq:S10_chain}\] Each arrow requires closure and a Gate stack; any linkage whose definition/judgment is not locked is not allowed.

Declaration of the \(c\) switch (propagating / non-propagating)

In this chapter, the “\(c\) switch” is not a continuous change of a value; it is fixed as a defined regime transition. The switch is declared by the following indicator. \[\chi_c := \begin{cases} 0,& \text{global propagation is not definable (non-rigid regime)},\\ 1,& \text{global propagation is definable (rigid regime)}. \end{cases} \label{eq:S10_chic_def}\] The decision of \(\chi_c\) is determined by the jamming lattice \(\mathfrak{J}\) (defined in §3.2), the rigidity indicator \(\chi_{\mathrm{ST}}\), and the bottleneck metric \(\kappa_{\min}\); the concrete decision rule is locked in analysis_lock.

In the regime \(\chi_c=0\), any attempt to assign a numerical value to \(c\) is forbidden. The only allowed record is the boundary statement “not definable” or “0-regime” (CT-LIM).

Declaration of the percolation (critical-throat) linkage

In the regime where global propagation is definable (\(\chi_c=1\)), propagation is implemented via connectivity in a critical-throat network. This chapter declares that the critical-throat items are linked in the following structure.

  1. Definition of the throat object OBJ-THROAT (gap/thickness/bottleneck).

  2. Definition of the representative effective critical throat \(\delta_{\mathrm{eff}}\) (aggregation convention).

  3. Definition of the critical-throat graph/path (node/edge and weight conventions).

  4. Extraction of a backbone or stream-tube (minimum cut / maximum flow convention).

Each item must be locked as a closure, and closure dependencies are fixed as a DAG (no cycles; see §4.2).

12.1 10.5 Declaration of amplification coefficient \(A\) (definition slot)

In the global-propagation regime, “amplification” is defined as an indicator of how strongly propagation paths concentrate on the critical-throat network. We declare that the amplification coefficient \(A\) is defined in the following slot. \[A := \mathrm{Amp}\bigl(\mathcal{G}_{\mathrm{throat}},\ \mathcal{B},\ \mathcal{F}\bigr), \label{eq:S10_A_slot}\] where

The concrete definition of \(\mathrm{Amp}(\cdot)\) (e.g., concentration, entropy-type measures, max/mean ratio, etc.) must be locked in analysis_lock. Replacing \(\mathrm{Amp}\) or moving thresholds after seeing results is forbidden.

12.2 10.6 Declaration of clock-free \(c\) realization (internal propagation indicator \(\rightarrow\) realization)

In this chapter, \(c\) denotes a procedure that makes internal propagation comparable without assuming an external clock. “Clock-free” is declared as the combination of the following two components.

  1. Internal propagation indicator \(\tilde{c}\): a propagation-speed indicator defined through critical throats/backbone (tick-based or event-based).

  2. Realization map: a mapping from the internal indicator to realized units using \(a,\Delta t\) and an operational anchor (\(c_{\mathrm{ref}}\) or a cross-channel).

Accordingly, \(c\) is fixed to have a realization formula of the form \[c := \frac{a}{\Delta t}\,\tilde{c} \label{eq:S10_c_realize_form}\] where the lock of \(a,\Delta t\) belongs to realization_lock in §2.3.

The definition of \(\tilde{c}\) is provided by the percolation/backbone/amplification definitions of this chapter. If the definition of \(\tilde{c}\) fails, then \(c\) is also not definable.

12.3 10.7 Numerical package (reproducibility) and Gate stack declaration

All conclusions in this chapter (\(\chi_c\), \(\delta_{\mathrm{eff}}\), backbone, \(A\), \(\tilde{c}\), \(c\)) must be sealed as a numerical package. The package must include:

  1. registry snapshot (registry_snapshot) and the lock_id combination,

  2. input data / initial conditions / boundary conditions / seeds / drive-condition logs,

  3. code and logs for critical-throat estimation and graph construction,

  4. code and logs for backbone/path extraction,

  5. code and logs for \(A\) and \(\tilde{c}\) computation,

  6. Gate report (PASS/FAIL/INCONCLUSIVE) and FAIL labels,

  7. sealing via manifest and checksums.

The Gate stack of this chapter must include at least:

  1. G-SYM: no conflicts in symbols/units/cell geometry/diameter–radius meanings,

  2. G-LOCK: lock_id consistency and snapshot sealing,

  3. G-REG: regime suitability (rigid/non-rigid, drive conditions, etc.),

  4. G-STR: structural invariants of graph/backbone/path definitions,

  5. G-NUM: numerical stability (convergence/sensitivity/repeatability),

  6. G-RCROSS (if applicable): cross-channel consistency,

  7. G-REP: reproducibility (re-running the same package yields the same verdict),

  8. G-NT: detection of post hoc tuning.

Any result that does not achieve PASS has no status as a conclusion and cannot be used as support in later sections.

12.4 10.1 Operational definition of \(c\) (\(B_{\mathrm{eff}}/\rho_{\mathrm{eff}}\)) + switch

12.4.1 10.1.1 Premise: define a propagation switch (regime switch)

Let the canonical domain (a cell or a union of cells) be \(\mathcal{D}\), with locked boundary sets \(\partial\mathcal{D}^{-},\partial\mathcal{D}^{+}\). Given a locked contact (or adjacency) graph \(\mathcal{G}_c=(\mathcal{V},\mathcal{E}_c)\), define the global-spanning indicator as \[\chi_{\mathrm{span}} := \begin{cases} 1, & \exists\, i\in\mathcal{V}^{-},\exists\, j\in\mathcal{V}^{+}\ \text{s.t.}\ i\leadsto j\ \text{in }\mathcal{G}_c,\\ 0, & \text{otherwise}, \end{cases} \label{eq:S10_01_chi_span}\] where \(\mathcal{V}^{\pm}\) are the sets of boundary-contact nodes, and \(i\leadsto j\) denotes the existence of a path.

Define a bottleneck-based rigidity switch via a minimum cut: \[\kappa_{\min} :=\min\left\{|\mathcal{C}|\ \middle|\ \mathcal{C}\subseteq\mathcal{E}_c,\ \mathcal{V}^{-}\ \text{and }\ \mathcal{V}^{+}\ \text{are separated in }\ \mathcal{E}_c\setminus\mathcal{C}\right\}, \label{eq:S10_01_kappa_min}\] and let the integer threshold \(\kappa_{\mathrm{ST}}\in\mathbb{Z}_{\ge 1}\) be a locked value. Define the rigidity indicator as \[\chi_{\mathrm{ST}} := \begin{cases} 1, & \chi_{\mathrm{span}}=1\ \wedge\ \kappa_{\min}\ge \kappa_{\mathrm{ST}},\\ 0, & \text{otherwise}. \end{cases} \label{eq:S10_01_chi_ST}\]

Define the \(c\) switch as \[\chi_{c}:=\chi_{\mathrm{ST}}\in\{0,1\}. \label{eq:S10_01_chi_c}\] Therefore, in the regime \(\chi_c=0\) we do not numerically define \(c\). The only allowed record is the limit statement “propagation not definable (out of regime)”.

12.4.2 10.1.2 Common structure for the operational definition: domain, deformation, relaxation

In the regime where the Stone/plenum/local rules apply, denote by \(\mathcal{A}(\mathcal{D})\) the set of admissible configurations inside the domain \(\mathcal{D}\) (satisfying non-penetration/plenum/local rules). An admissible configuration \(\mathcal{C}\in\mathcal{A}(\mathcal{D})\) consists of the set of VP-occupied regions (or their representative coordinates/graph representation).

Define an external manipulation (probe) applied to the domain as a “deformation operator”, and define the procedure that restores admissibility after deformation as a “relaxation operator”. \[\mathcal{T}_{\varepsilon}:\mathcal{A}(\mathcal{D})\to \mathcal{A}_{\varepsilon}(\mathcal{D}), \qquad \mathcal{R}:\mathcal{A}_{\varepsilon}(\mathcal{D})\to \mathcal{A}(\mathcal{D}), \label{eq:S10_01_TR_ops}\] where \(\varepsilon\) is a dimensionless probe size and \(\mathcal{A}_{\varepsilon}(\mathcal{D})\) is an intermediate state space after deformation (mixing admissible and inadmissible states). Relaxation is defined only as a composition of local updates (local-rule axiom). If relaxation fails, the quantity is recorded as “not definable”.

Define the relaxation cost (energy unit) as follows. The unit energy \(U_{\mathrm{lat}}\) is a locked reference energy unit via realization_lock, and the update weight \(\omega_{\mathrm{upd}}(k)\) is locked via analysis_lock. \[W(\varepsilon) := U_{\mathrm{lat}}\, \min_{\mathcal{R}} \left(\sum_{k=1}^{N_{\mathrm{upd}}(\varepsilon)} \omega_{\mathrm{upd}}(k)\right), \qquad W(0)=0, \label{eq:S10_01_Work_def}\] where the minimization is performed under the constraint “return to an admissible configuration”, and the search/termination rules must be locked in analysis_lock.

12.4.3 10.1.3 Operational definition of \(B_{\mathrm{eff}}\) (static isotropic-compression curvature)

12.4.3.1 10.1.3.1 Isotropic compression probe and volumetric strain

Let the isotropic compression parameter \(\varepsilon\in(0,\varepsilon_{\max})\) be in a locked range. Define an isotropic scaling deformation on domain coordinates by \[\mathbf{x}\ \mapsto\ \mathbf{x}^{(\varepsilon)}:=(1-\varepsilon)\mathbf{x}, \label{eq:S10_01_iso_map}\] and define the corresponding domain volume by \[V(\varepsilon):=V_0(1-\varepsilon)^3, \qquad V_0:=V(0), \label{eq:S10_01_Veps}\] Define the volumetric strain as \[\eta(\varepsilon):=\frac{V_0-V(\varepsilon)}{V_0}=1-(1-\varepsilon)^3. \label{eq:S10_01_eta_def}\] Thus \(\eta(0)=0\) and, for small \(\varepsilon\), \(\eta(\varepsilon)=3\varepsilon+O(\varepsilon^2)\).

12.4.3.2 10.1.3.2 Isotropic compression cost function

For an admissible configuration \(\mathcal{C}\in\mathcal{A}(\mathcal{D})\), define the relaxation cost after isotropic compression as \[W_{\mathrm{iso}}(\varepsilon):=W(\varepsilon) \label{eq:S10_01_Wiso}\] where \(W_{\mathrm{iso}}(\varepsilon)\) is the cost obtained from [eq:S10_01_Work_def]. For the definition to hold, relaxation must return to an admissible configuration for all \(\varepsilon\); if return fails at any \(\varepsilon\), the definition is not valid.

12.4.3.3 10.1.3.3 Definition of \(B_{\mathrm{eff}}\) (curvature with respect to volumetric strain)

Define the effective stiffness (curvature) \(B_{\mathrm{eff}}\) by \[B_{\mathrm{eff}} := \left.\frac{1}{V_0}\frac{d^2 W_{\mathrm{iso}}}{d\eta^2}\right|_{\eta=0}. \label{eq:S10_01_Beff_def}\] Definition [eq:S10_01_Beff_def] is an operational definition of “the cost curvature under isotropic compression” and uses no external justification. The dimension of \(B_{\mathrm{eff}}\) is locked as energy/volume.

In actual computation, small \(\eta\) samples \(\{\eta_j\}_{j=1}^{J}\) must be preregistered (analysis_lock) and the quadratic curvature is estimated by a discrete approximation. For example, with a three-point symmetric difference, \[\widehat{B}_{\mathrm{eff}} := \frac{1}{V_0}\frac{W_{\mathrm{iso}}(\eta_+)-2W_{\mathrm{iso}}(0)+W_{\mathrm{iso}}(\eta_-)}{\eta_+^2}, \qquad \eta_-=-\eta_+, \label{eq:S10_01_Beff_est}\] where the sample values and the choice of estimator must be locked in analysis_lock. (If a protocol does not allow negative \(\eta\), an asymmetric estimator is used and that selection rule must also be locked.)

12.4.4 10.1.4 Operational definition of \(\rho_{\mathrm{eff}}\) (dynamic drift curvature)

To define a propagation indicator of the form \(B_{\mathrm{eff}}/\rho_{\mathrm{eff}}\), we operationally define \(\rho_{\mathrm{eff}}\) as a “cost curvature under a dynamic drift probe”.

12.4.4.1 10.1.4.1 Drift probe and velocity-like parameter

Define the drift parameter \(u\in(-u_{\max},u_{\max})\) as a dimensionless probe. Define the boundary-driven displacement over one tick by \[\Delta x(u):=u\,L_q \label{eq:S10_01_dx_u}\] where \(L_q\) is a locked canonical length (or a canonically identified length), and \(u\) is a dimensionless coefficient of “displacement per tick”.

Define the drift deformation operator \(\mathcal{T}_u\) as \[\mathcal{T}_u:\ \mathbf{x}\ \mapsto\ \mathbf{x}^{(u)}:=\mathbf{x}+\Delta x(u)\,\mathbf{n}_u, \qquad \|\mathbf{n}_u\|=1, \label{eq:S10_01_drift_map}\] where \(\mathbf{n}_u\) is a unit vector giving the drift direction, and must be locked in analysis_lock.

12.4.4.2 10.1.4.2 Drift cost function

Define the relaxation cost required to return to an admissible configuration after drift as \[W_{\mathrm{drift}}(u) := U_{\mathrm{lat}}\, \min_{\mathcal{R}} \left(\sum_{k=1}^{N_{\mathrm{upd}}(u)} \omega_{\mathrm{upd}}(k)\right), \qquad W_{\mathrm{drift}}(0)=0. \label{eq:S10_01_Wdrift}\] If relaxation fails in [eq:S10_01_Wdrift], then \(W_{\mathrm{drift}}(u)\) is not definable; in that regime, \(\rho_{\mathrm{eff}}\) cannot be defined.

12.4.4.3 10.1.4.3 Definition of \(\rho_{\mathrm{eff}}\) (drift curvature)

Define the effective density \(\rho_{\mathrm{eff}}\) by \[\rho_{\mathrm{eff}} := \left.\frac{1}{V_0}\frac{d^2 W_{\mathrm{drift}}}{du^2}\right|_{u=0}. \label{eq:S10_01_rhoeff_def}\] Definition [eq:S10_01_rhoeff_def] is an operational definition of “the cost curvature under a drift probe” and uses no external justification. The dimension of \(\rho_{\mathrm{eff}}\) is locked as energy\(\cdot\)time\(^2\)/length\(^5\), and is later rearranged into a velocity unit via \(a,\Delta t\) in the Realization chapter.

In computation, the \(u\) samples \(\{u_j\}_{j=1}^{J}\) and the curvature estimator (central difference, etc.) must be locked in analysis_lock. For example, with a central difference, \[\widehat{\rho}_{\mathrm{eff}} := \frac{1}{V_0}\frac{W_{\mathrm{drift}}(u_+)-2W_{\mathrm{drift}}(0)+W_{\mathrm{drift}}(u_-)}{u_+^2}, \qquad u_-=-u_+. \label{eq:S10_01_rhoeff_est}\]

12.4.5 10.1.5 Operational definition of \(c\) (internal propagation indicator) and realization

12.4.5.1 10.1.5.1 Internal propagation indicator \(\tilde{c}\)

When \(\chi_c=1\) and both \(B_{\mathrm{eff}}\) and \(\rho_{\mathrm{eff}}\) are definable and positive, define the internal propagation indicator \(\tilde{c}\) by \[\tilde{c}^2 := \frac{B_{\mathrm{eff}}}{\rho_{\mathrm{eff}}}, \qquad \tilde{c}:=\sqrt{\frac{B_{\mathrm{eff}}}{\rho_{\mathrm{eff}}}}. \label{eq:S10_01_ctilde_def}\] \(\tilde{c}\) is a propagation indicator in internal units (tick / normalized length) and is not defined when \(\chi_c=0\).

12.4.5.2 10.1.5.2 Realized \(c\)

When the realization length scale \(a\) and the realization time tick \(\Delta t\) are locked via realization_lock, define the realized \(c\) by \[c := \frac{a}{\Delta t}\,\tilde{c}. \label{eq:S10_01_c_real_def}\] Definition [eq:S10_01_c_real_def] is a realization map; post hoc changes of \(a\) or \(\Delta t\) are forbidden (changes are allowed only by versioning).

12.4.6 10.1.6 Regime conditions (conditions for definability)

For the \(c\) definition in this section to hold, all of the following regime conditions must be satisfied.

  1. (R-c1) Rigid regime: \(\chi_c=\chi_{\mathrm{ST}}=1\).

  2. (R-c2) Return to admissibility: in the isotropic-compression probe and the drift probe, relaxation returns to an admissible configuration so that \(W_{\mathrm{iso}}(\varepsilon)\) and \(W_{\mathrm{drift}}(u)\) are defined.

  3. (R-c3) Curvature definable: \(W_{\mathrm{iso}}\) has a second-order curvature at \(\eta=0\), and \(W_{\mathrm{drift}}\) has a second-order curvature at \(u=0\) (the discrete estimator is locked).

  4. (R-c4) Positive curvatures: \(B_{\mathrm{eff}}>0\) and \(\rho_{\mathrm{eff}}>0\).

  5. (R-c5) Lock consistency: the domain/boundary/contact convention/probe samples/estimators/thresholds all belong to the same lock_id combination.

If any of (R-c1)–(R-c5) is violated, then \(c\) is not definable and cannot be used as a conclusion.

12.4.7 10.1.7 FAIL conditions (immediate failure and labels)

This section fixes the failures related to the \(c\) definition by the following FAIL labels (multiple labels allowed).

12.4.7.1 10.1.7.1 Switch failure (out of regime)

\[\chi_c=0 \quad\Longrightarrow\quad \texttt{FAIL-C-SWITCH}. \label{eq:S10_01_fail_switch}\] In this case \(c\) is not defined and \(\tilde{c}\) is also not defined.

12.4.7.2 10.1.7.2 Relaxation failure (not definable)

If an admissible configuration cannot be recovered in isotropic compression or drift, it is an immediate failure. \[\exists\,\varepsilon\ \text{s.t.}\ W_{\mathrm{iso}}(\varepsilon)\ \text{not definable} \quad\Longrightarrow\quad \texttt{FAIL-C-RELAX-ISO}, \label{eq:S10_01_fail_relax_iso}\] \[\exists\,u\ \text{s.t.}\ W_{\mathrm{drift}}(u)\ \text{not definable} \quad\Longrightarrow\quad \texttt{FAIL-C-RELAX-DRIFT}. \label{eq:S10_01_fail_relax_drift}\]

12.4.7.3 10.1.7.3 Curvature-estimation failure (numerical instability)

If curvature estimation does not satisfy preregistered stability conditions, treat it as failure. \[\widehat{B}_{\mathrm{eff}}\ \text{fails to converge or has unstable sign} \quad\Longrightarrow\quad \texttt{FAIL-C-B-NUM}, \label{eq:S10_01_fail_Bnum}\] \[\widehat{\rho}_{\mathrm{eff}}\ \text{fails to converge or has unstable sign} \quad\Longrightarrow\quad \texttt{FAIL-C-RHO-NUM}. \label{eq:S10_01_fail_RHOnum}\]

12.4.7.4 10.1.7.4 Positivity failure (definition violation)

\[B_{\mathrm{eff}}\le 0 \quad\Longrightarrow\quad \texttt{FAIL-C-B-NONPOS}, \label{eq:S10_01_fail_Bnonpos}\] \[\rho_{\mathrm{eff}}\le 0 \quad\Longrightarrow\quad \texttt{FAIL-C-RHO-NONPOS}. \label{eq:S10_01_fail_RHOnonpos}\] A positivity failure collapses the square-root definition [eq:S10_01_ctilde_def] itself, so it is an immediate failure.

12.4.7.5 10.1.7.5 Lock/meaning conflicts and post hoc tuning

The following are immediate failures.

  1. Symbol-meaning conflicts (cell geometry, diameter/radius, unit-dimension mismatch): FAIL-C-SYM.

  2. Mixing different lock_id combinations: FAIL-C-LOCK-MIX.

  3. Post hoc changes of probe samples/estimators/thresholds/relaxation rules: FAIL-C-NT.

12.5 10.2 \(\delta_{\mathrm{eff}}\) and percolation closure

12.5.1 10.2.1 Premises (regime and input locks)

The definitions and procedures of this section apply only when all of the following premises are simultaneously satisfied.

  1. Propagation switch: \(\chi_c=\chi_{\mathrm{ST}}=1\).

  2. Domain \(\mathcal{D}\), opposing boundary sets \(\partial\mathcal{D}^{-},\partial\mathcal{D}^{+}\), and boundary node sets \(\mathcal{V}^{-},\mathcal{V}^{+}\) are locked.

  3. Node coordinates \(\{\mathbf{x}_i\}\), center \(\mathbf{x}_c\), and normalized length \(L_q\) are locked.

  4. Adjacency/contact graph \(\mathcal{G}_c=(\mathcal{V},\mathcal{E}_c)\) and contact-judgment conventions are locked.

  5. Minimum separation length (or reference length) \(d_0>0\) is locked (including diameter/radius meaning).

If any item above is not locked, then the definition of \(\delta_{\mathrm{eff}}\) and the percolation procedure are not valid.

12.5.2 10.2.2 Definition of throats and gaps

In this section, a “throat” is a candidate edge that can constitute a global propagation path, and a “gap” is a length-type scalar weight assigned to a candidate edge.

12.5.2.1 10.2.2.1 Candidate edge set (throat candidates)

Define the set of candidate throat edges by \[\mathcal{E}_{\mathrm{th}} :=\mathcal{E}_c. \label{eq:S10_02_Eth_def}\] That is, in this section, the contact graph edges are the single source of truth (SSOT) for throat candidates. Extending the candidate set (e.g., by adding a \(k\)-nearest-neighbor graph) requires a separate closure definition and is not allowed here.

12.5.2.2 10.2.2.2 Gap (throat weight) \(g_{ij}\)

For each edge \((i,j)\in\mathcal{E}_{\mathrm{th}}\), define the distance and the gap by \[d_{ij}:=\|\mathbf{x}_i-\mathbf{x}_j\|, \qquad g_{ij}:=\max\!\left(0,\ d_{ij}-d_0\right). \label{eq:S10_02_gap_def}\] Here \(d_0\) is a locked reference length. The dimension of \(g_{ij}\) is length and \(g_{ij}\ge 0\). The definition of \(g_{ij}\) cannot be replaced within the same version by an alternative (e.g., squared distance, normalization, nonlinear transforms).

12.5.2.3 10.2.2.3 Normalized gap (optional derived quantity)

Define the normalized gap as \[\tilde{g}_{ij}:=\frac{g_{ij}}{L_q}. \label{eq:S10_02_gap_norm}\] \(\tilde{g}_{ij}\) is dimensionless and can be used only when \(L_q\) is locked.

12.5.3 10.2.3 Percolation (global propagation) definition: threshold \(g\) and open graph

Percolation is defined as the procedure that selects “open edges” according to a gap threshold \(g\ge 0\) and judges whether global connectivity holds.

12.5.3.1 10.2.3.1 Threshold \(g\) and open edges

For \(g\in\mathbb{R}_{\ge 0}\), define the set of open edges as \[\mathcal{E}_{\mathrm{open}}(g) := \left\{(i,j)\in\mathcal{E}_{\mathrm{th}}\ \middle|\ g_{ij}\le g\right\}. \label{eq:S10_02_Eopen_def}\] Define the open graph as \[\mathcal{G}_{\mathrm{open}}(g) := \bigl(\mathcal{V},\mathcal{E}_{\mathrm{open}}(g)\bigr). \label{eq:S10_02_Gopen_def}\] By definition, as \(g\) increases, \(\mathcal{E}_{\mathrm{open}}(g)\) grows monotonically.

12.5.3.2 10.2.3.2 Global-percolation indicator \(\chi_{\mathrm{perc}}(g)\)

For opposing boundary node sets \(\mathcal{V}^{-},\mathcal{V}^{+}\), define the percolation indicator as \[\chi_{\mathrm{perc}}(g) := \begin{cases} 1, & \exists\, i\in\mathcal{V}^{-},\exists\, j\in\mathcal{V}^{+}\ \text{s.t.}\ i\leadsto j\ \text{in }\mathcal{G}_{\mathrm{open}}(g),\\ 0, & \text{otherwise}. \end{cases} \label{eq:S10_02_chi_perc}\] If \(\chi_{\mathrm{perc}}(g)=1\), then at threshold \(g\) global propagation (existence of a path) holds.

12.5.4 10.2.4 Critical-throat threshold \(g_c\) and definition of \(\delta_{\mathrm{eff}}\)

Define the critical-throat threshold \(g_c\) as the minimum threshold at which global propagation holds for the first time.

12.5.4.1 10.2.4.1 Definition of the critical threshold \(g_c\)

\[g_c := \inf\{\, g\ge 0\mid \chi_{\mathrm{perc}}(g)=1\,\}. \label{eq:S10_02_gc_def}\] Since edge weights form a finite set (finite nodes/finite edges), the following discrete definition is locked in actual implementations. \[\mathcal{G}:=\{g_{ij}\mid (i,j)\in\mathcal{E}_{\mathrm{th}}\}\ \text{(sorted after deduplication)}, \qquad g_c:=\min\{\, g\in\mathcal{G}\mid \chi_{\mathrm{perc}}(g)=1\,\}. \label{eq:S10_02_gc_discrete}\]

12.5.4.2 10.2.4.2 Definition of \(\delta_{\mathrm{eff}}\) (length-type)

Define the effective critical-throat thickness (gap) as \[\delta_{\mathrm{eff}} := g_c. \label{eq:S10_02_deltaeff_def}\] Thus \(\delta_{\mathrm{eff}}\) has dimension of length and equals the percolation critical threshold.

12.5.4.3 10.2.4.3 Dimensionless effective critical throat (optional derived quantity)

Define the dimensionless effective critical throat as \[\tilde{\delta}_{\mathrm{eff}}:=\frac{\delta_{\mathrm{eff}}}{L_q}=\frac{g_c}{L_q}. \label{eq:S10_02_deltaeff_tilde}\] \(\tilde{\delta}_{\mathrm{eff}}\) can be used only when \(L_q\) is locked.

12.5.5 10.2.5 Computation procedure: global percolation/critical throat (Union-Find) + backbone extraction

This section fixes the computation procedure as a deterministic algorithm. Randomness, arbitrary choices, and result-dependent choices are forbidden.

12.5.5.1 10.2.5.1 Preprocessing: edge list and weight table

  1. Fix the edge list as \(\mathcal{E}_{\mathrm{th}}\) ([eq:S10_02_Eth_def]).

  2. For each edge compute \((i,j,g_{ij})\) ([eq:S10_02_gap_def]).

  3. Sort the edge list by the following key to lock a deterministic order. \[\mathrm{key}(i,j):=\bigl(g_{ij},\ \min(i,j),\ \max(i,j)\bigr). \label{eq:S10_02_edge_key}\]

12.5.5.2 10.2.5.2 Union-Find

Compute \(g_c\) by the following deterministic procedure.

ALG-PERC-GC (inputs: V, E_th with weights g_ij, boundary sets V-, V+)

1) initialize Union-Find structure UF over nodes V
2) mark boundary membership:
     tag_minus(i)=1 if i in V-, else 0
     tag_plus(i)=1 if i in V+, else 0
   store for each UF component:
     has_minus(component), has_plus(component)
3) sort edges (i,j) by key(i,j)=(g_ij, min(i,j), max(i,j))

4) for each edge (i,j) in sorted order:
     UF.union(i,j)
     update has_minus/has_plus for the merged component
     if exists a component with has_minus=1 and has_plus=1:
         g_c := g_ij of the current edge
         STOP

5) if loop ends without connection:
     FAIL-PERC-NOSPAN

The \(g_c\) at the stopping time is equivalent to the discrete definition [eq:S10_02_gc_discrete], and it connects directly to the \(\delta_{\mathrm{eff}}\) definition [eq:S10_02_deltaeff_def].

12.5.5.3 10.2.5.3 Definition of the percolation backbone

Let the open graph at the critical threshold be \(\mathcal{G}_{\mathrm{open}}(g_c)\), and define the backbone as the set of edges that actually contribute to boundary-to-boundary connectivity.

12.5.5.4 (1) Boundary-spanning component

Define the connected component that contains both \(\mathcal{V}^{-}\) and \(\mathcal{V}^{+}\) at the critical threshold as \[\mathcal{C}_{\mathrm{span}}(g_c)\subseteq \mathcal{V} \label{eq:S10_02_Cspan_def}\] (which can be determined from the final Union-Find component).

12.5.5.5 (2) Two-terminal contribution of an edge (deterministic judgment)

Define the subgraph of \(\mathcal{G}_{\mathrm{open}}(g_c)\) restricted to \(\mathcal{C}_{\mathrm{span}}(g_c)\) as \[\mathcal{G}_{\mathrm{span}}(g_c) := \mathcal{G}_{\mathrm{open}}(g_c)\big|_{\mathcal{C}_{\mathrm{span}}(g_c)} \label{eq:S10_02_Gspan_def}\]

An edge \(e=(u,v)\) “contributes to boundary-to-boundary connectivity” if removing \(e\) destroys all paths from \(\mathcal{V}^{-}\) to \(\mathcal{V}^{+}\). For this purpose define the deterministic decision function \[\mathrm{BridgeTT}(e) := \begin{cases} 1,& \chi_{\mathrm{perc}}^{(-e)}(g_c)=0,\\ 0,& \chi_{\mathrm{perc}}^{(-e)}(g_c)=1, \end{cases} \label{eq:S10_02_bridgeTT_def}\] where \(\chi_{\mathrm{perc}}^{(-e)}(g_c)\) is the value of [eq:S10_02_chi_perc] computed on the graph obtained from \(\mathcal{G}_{\mathrm{span}}(g_c)\) by removing \(e\).

Because this can be expensive, in practical implementation the following deterministic algorithm (dominant-edge extraction) is locked to produce the same result.

12.5.5.6 10.2.5.4 Backbone extraction algorithm (two-terminal mandatory edge set)

Define the backbone edge set \(\mathcal{E}_{\mathrm{bb}}\) by the following algorithm.

ALG-BACKBONE-TT (inputs: G_span(g_c), boundary sets V-, V+)

1) choose deterministic start node s in V-:
     s := min index in (V- CAP C_span)
2) BFS from s in G_span to compute parent tree and levels.
3) choose deterministic target node t in V+ reachable:
     t := min index in (V+ CAP C_span) among reachable nodes
4) extract one canonical path P0 from s to t using parent pointers.
   E_bb := edges of P0

5) augment backbone by mandatory two-terminal edges:
   For each edge e in E_open(g_c) within G_span, in sorted key order:
      Temporarily remove e
      Check reachability from any node in V- to any node in V+ (BFS):
         if disconnected: mark e as mandatory and add to E_bb
      Restore e

6) output E_bb

The algorithm is locked to satisfy:

  1. Determinism: start node / target node / edge order are all locked by convention.

  2. Definitional minimality: contains at least one boundary-to-boundary path (\(P_0\)).

  3. Mandatory-edge augmentation: includes edges whose removal breaks connectivity.

Define the backbone node set \(\mathcal{V}_{\mathrm{bb}}\) as \[\mathcal{V}_{\mathrm{bb}} := \{\, v\in\mathcal{V}\mid \exists\, e=(u,v)\in\mathcal{E}_{\mathrm{bb}}\ \text{or}\ e=(v,w)\in\mathcal{E}_{\mathrm{bb}}\,\}. \label{eq:S10_02_Vbb_def}\]

12.5.6 10.2.6 Global-propagation outputs (definition-result form)

The output artifacts of this section are fixed to the following four items.

  1. critical threshold \(g_c\) (length),

  2. effective critical throat \(\delta_{\mathrm{eff}}:=g_c\) (length) and optional dimensionless \(\tilde{\delta}_{\mathrm{eff}}\),

  3. critical open graph \(\mathcal{G}_{\mathrm{open}}(g_c)\) (open-edge set),

  4. backbone edge set \(\mathcal{E}_{\mathrm{bb}}\) and backbone node set \(\mathcal{V}_{\mathrm{bb}}\).

These artifacts must belong to the same analysis_lock and the same regime_id, and must be sealed by manifest+checksums to be used as inputs for later sections (amplification \(A\), propagation indicator \(\tilde{c}\), and realized \(c\)).

12.5.7 10.2.7 Failure modes and FAIL conditions

If any of the following occurs, the artifacts of this section lose their conclusion status.

  1. Regime violation: \(\chi_c=0\) or boundary/node sets are not locked.

  2. Spanning failure: for no \(g\) does \(\chi_{\mathrm{perc}}(g)=1\). \[\forall g\ge 0,\ \chi_{\mathrm{perc}}(g)=0 \quad\Longrightarrow\quad \texttt{FAIL-PERC-NOSPAN}. \label{eq:S10_02_fail_nospan}\]

  3. Meaning conflict: ambiguity in \(g_{ij}\) due to conflicts in the meaning of \(d_0\) (diameter/radius) or coordinate/unit conventions.

  4. Procedure not locked: any of boundary-node definitions, edge-sorting key, tiebreaking, or backbone-extraction rules is not locked.

  5. Post hoc tuning: result-dependent changes to the candidate edge set, changes to \(d_0\) or the definition of \(g_{ij}\), changes to threshold-selection rules, or changes to backbone-selection rules.

12.6 10.3 SOC percolation and amplification \(A\)

12.6.1 10.3.1 Purpose

This section (i) fixes SOC (self-organized criticality) events as an operational definition on top of a percolation (critical-throat) structure, (ii) defines the amplification coefficient \(A\) as the degree to which global transmission of SOC events concentrates on a path, (iii) fixes estimators for \(A\), and (iv) fixes the Gates (steady state / pinning / robustness) required for \(A\) to have conclusion status.

The \(A\) of this section is a measurable quantity computed from the critical-throat graph/backbone/event logs, without external justification.

12.6.2 10.3.2 Inputs (LOCK): critical-throat graph, backbone, event logs

Assume the artifacts of §10.2 are qualified by PASS and are given as inputs.

  1. Critical threshold and effective critical throat: \[\delta_{\mathrm{eff}}:=g_c, \label{eq:S10_03_deltaeff_in}\]

  2. Critical open graph: \[\mathcal{G}_{\mathrm{open}}(g_c)=(\mathcal{V},\mathcal{E}_{\mathrm{open}}(g_c)), \label{eq:S10_03_Gopen_in}\]

  3. Backbone (two-terminal contributing edge set): \[\mathcal{E}_{\mathrm{bb}}\subseteq \mathcal{E}_{\mathrm{open}}(g_c), \qquad \mathcal{V}_{\mathrm{bb}}\ \text{(backbone node set)}. \label{eq:S10_03_backbone_in}\]

In addition, to define SOC events, an event log \(\mathcal{E}\) must be provided under a locked schema (see the definitions in Chapter 9). Since SOC events are defined as subsets of the event log, missing logs make the definition invalid.

12.6.3 10.3.3 Operational definition of SOC events

Define an SOC event as a “bundle of events in which consecutive local updates are connected within a time window and are aggregated into an avalanche.” This section defines SOC events by the following procedure.

12.6.3.1 10.3.3.1 Event set and event graph

Let the event set on the tick axis be \(\mathcal{E}\). Each event \(e\) is assumed to have the following fields: \[e \mapsto \bigl(n(e),\ \mathcal{V}(e)\bigr), \label{eq:S10_03_event_fields}\] where \(n(e)\) is the tick, and \(\mathcal{V}(e)\) is the set of nodes (or VPs) involved in the event. The meaning of \(\mathcal{V}(e)\) (e.g., throat nodes vs core nodes) must be locked in analysis_lock.

To decide whether two events \(e,e'\) belong to the same SOC cluster, define “event adjacency.” Lock a time threshold \(\Delta n_{\mathrm{soc}}\in\mathbb{Z}_{\ge 0}\) and a spatial threshold (node-intersection convention) \(\tau_{\mathrm{soc}}\in\mathbb{Z}_{\ge 1}\). \[\Delta n_{\mathrm{soc}}\ \text{(locked)}, \qquad \tau_{\mathrm{soc}}\ \text{(locked)}. \label{eq:S10_03_soc_thresholds}\] Define the event-adjacency decision function as \[\mathrm{Adj}_{\mathrm{soc}}(e,e') := \begin{cases} 1,& |n(e)-n(e')|\le \Delta n_{\mathrm{soc}}\ \wedge\ |\mathcal{V}(e)\cap \mathcal{V}(e')|\ge \tau_{\mathrm{soc}},\\ 0,& \text{otherwise}. \end{cases} \label{eq:S10_03_soc_adj}\] The definition of \(\mathrm{Adj}_{\mathrm{soc}}\) cannot be replaced after seeing results.

Define the event graph as \[\mathcal{G}_{\mathrm{soc}}:=(\mathcal{E},\mathcal{E}_{\mathrm{soc}}), \qquad \mathcal{E}_{\mathrm{soc}}:=\{(e,e')\mid e\neq e',\ \mathrm{Adj}_{\mathrm{soc}}(e,e')=1\}. \label{eq:S10_03_Gsoc}\]

12.6.3.2 10.3.3.2 SOC clusters (avalanches)

Define an SOC cluster (avalanche) \(C\) as a connected component of the event graph \(\mathcal{G}_{\mathrm{soc}}\). \[\mathcal{C}_{\mathrm{soc}}:=\{\text{connected components of }\mathcal{G}_{\mathrm{soc}}\}, \qquad C\in\mathcal{C}_{\mathrm{soc}}. \label{eq:S10_03_clusters}\] Define the cluster size (number of events) and duration by \[S(C):=|C|, \qquad T(C):=\bigl(\max_{e\in C}n(e)-\min_{e\in C}n(e)+1\bigr)\Delta t. \label{eq:S10_03_cluster_size_duration}\] Here \(\Delta t\) is the realized time tick locked in realization_lock. If \(\Delta t\) is not definable, then \(T(C)\) is recorded only in tick units.

12.6.4 10.3.4 Preparing to define a flux-concentration (amplification) observable

Amplification \(A\) is defined as a scalar quantifying “how much SOC clusters concentrate on the backbone within the critical-throat network.” For this purpose define the “active node set” of a cluster \(C\).

12.6.4.1 10.3.4.1 Cluster active node set

Define the active node set of cluster \(C\) by \[\mathcal{V}_C := \bigcup_{e\in C}\mathcal{V}(e). \label{eq:S10_03_active_nodes}\] If \(\mathcal{V}(e)\) refers to objects other than nodes (e.g., edges or throats), then the definition of \(\mathcal{V}_C\) must be transformed by a locked mapping from those objects to nodes.

12.6.4.2 10.3.4.2 Backbone occupancy fraction of a cluster (concentration ratio)

For cluster \(C\), define the backbone occupancy fraction as \[p_{\mathrm{bb}}(C) := \frac{|\mathcal{V}_C\cap \mathcal{V}_{\mathrm{bb}}|}{|\mathcal{V}_C|}, \qquad \text{provided }|\mathcal{V}_C|>0. \label{eq:S10_03_pbb}\] By definition, \(0\le p_{\mathrm{bb}}(C)\le 1\).

Also define the “geometric backbone fraction” (baseline fraction) of the critical-throat network by \[p_{\mathrm{bb}}^{(0)} := \frac{|\mathcal{V}_{\mathrm{bb}}|}{|\mathcal{V}|}. \label{eq:S10_03_pbb0}\] The baseline \(p_{\mathrm{bb}}^{(0)}\) is the fraction of nodes occupied by the backbone and is used as a reference for concentration.

12.6.5 10.3.5 Definition of amplification \(A\) (definition-result form)

12.6.5.1 10.3.5.1 Cluster-wise amplification

Define the amplification of cluster \(C\) by \[A(C) := \frac{p_{\mathrm{bb}}(C)}{p_{\mathrm{bb}}^{(0)}}. \label{eq:S10_03_A_of_C}\] Definition [eq:S10_03_A_of_C] is a pure ratio meaning “how many times the backbone occupancy exceeds the baseline fraction” and uses no external justification.

By definition, \(p_{\mathrm{bb}}^{(0)}>0\) is required. If \(p_{\mathrm{bb}}^{(0)}=0\), then the backbone is empty and \(A\) is not definable.

12.6.5.2 10.3.5.2 Window-aggregated amplification

Let the SOC cluster set within a time window (or a sample set) be \(\mathcal{C}_{\mathrm{soc}}\). Lock one of the following two estimators (exclusive choice).

  1. Mean-type estimator: \[A_{\mathrm{mean}} := \frac{1}{|\mathcal{C}_{\mathrm{soc}}|} \sum_{C\in\mathcal{C}_{\mathrm{soc}}} A(C), \qquad (|\mathcal{C}_{\mathrm{soc}}|>0). \label{eq:S10_03_Amean}\]

  2. Weighted-mean estimator (weighted by event count): \[A_{\mathrm{w}} := \frac{\sum_{C\in\mathcal{C}_{\mathrm{soc}}} S(C)\,A(C)}{\sum_{C\in\mathcal{C}_{\mathrm{soc}}} S(C)}, \qquad \left(\sum_{C}S(C)>0\right). \label{eq:S10_03_Aw}\]

Which estimator is adopted (and whether weights other than \(S(C)\) are allowed) must be locked in analysis_lock and cannot be changed after seeing results.

In this section, “amplification coefficient \(A\)” is defined as the output of the selected estimator (e.g., \(A_{\mathrm{mean}}\) or \(A_{\mathrm{w}}\)).

12.6.6 10.3.6 \(G\)-SS-STAT

Amplification \(A\) has conclusion status only on a steady-state interval. The steady-state Gate is defined as follows.

12.6.6.1 10.3.6.1 Lock warm-up and observation windows

Let the total run length (in ticks) be \(N_{\mathrm{tot}}\). Lock the warm-up length \(N_{\mathrm{warm}}\) and the observation length \(N_{\mathrm{obs}}\) by \[N_{\mathrm{warm}}\in\mathbb{Z}_{\ge 0}\ \text{(locked)}, \qquad N_{\mathrm{obs}}\in\mathbb{Z}_{>0}\ \text{(locked)}, \qquad N_{\mathrm{warm}}+N_{\mathrm{obs}}\le N_{\mathrm{tot}}. \label{eq:S10_03_warm_obs}\] Define the observation window as \[W_{\mathrm{obs}}:=W[N_{\mathrm{warm}},\,N_{\mathrm{warm}}+N_{\mathrm{obs}}). \label{eq:S10_03_Wobs}\]

12.6.6.2 10.3.6.2 Steady-state decision metric

Split the observation window into \(M\) equal blocks and lock the block count \(M\). \[M\in\mathbb{Z}_{\ge 2}\ \text{(locked)}, \qquad W_{\mathrm{obs}}=\dot\cup_{m=1}^{M} W_m. \label{eq:S10_03_block_split}\] Compute the amplification estimate \(A_m\) on each block (same estimator). Define the block mean and variance as \[\overline{A}:=\frac{1}{M}\sum_{m=1}^{M}A_m, \qquad \sigma_A^2:=\frac{1}{M}\sum_{m=1}^{M}(A_m-\overline{A})^2, \qquad \sigma_A:=\sqrt{\sigma_A^2}. \label{eq:S10_03_A_block_stats}\] Lock the steady-state threshold \(\varepsilon_{\mathrm{SS}}>0\) in gate_lock, and define the steady-state Gate by \[\texttt{G-SS-STAT}=\texttt{PASS} \Longleftrightarrow \frac{\sigma_A}{\overline{A}}\le \varepsilon_{\mathrm{SS}}. \label{eq:S10_03_gate_SS}\] If \(\overline{A}=0\), then the steady-state judgment is not definable (INCONCLUSIVE).

12.6.7 10.3.7 \(G\)-SS-PIN

In SOC-percolation, “pinning” refers to the phenomenon where events become fixed to the backbone or a local region so that global SOC collapses. This section defines pinning by an operational metric.

12.6.7.1 10.3.7.1 Pinning metric

Within the observation window \(W_{\mathrm{obs}}\), define the visit frequency (hit count) of active nodes by \[H(v):=\sum_{e\in \mathcal{E}(W_{\mathrm{obs}})} \mathbf{1}_{\{v\in \mathcal{V}(e)\}}, \qquad v\in\mathcal{V}, \label{eq:S10_03_hitcount}\] where \(\mathcal{E}(W_{\mathrm{obs}})\) is the set of events in the observation window and \(\mathbf{1}\) is an indicator function. Define the normalized hit distribution as \[p(v):=\frac{H(v)}{\sum_{u\in\mathcal{V}}H(u)}. \label{eq:S10_03_pv}\] Define the pinning metric by \[P_{\max}:=\max_{v\in\mathcal{V}} p(v). \label{eq:S10_03_Pmax}\] \(P_{\max}\) is an operational metric of how excessively events concentrate on a single node.

12.6.7.2 10.3.7.2 Pinning Gate

Lock a pinning threshold \(P_{\mathrm{pin}}\in(0,1)\) in gate_lock, and define the pinning Gate by \[\texttt{G-SS-PIN}=\texttt{PASS} \Longleftrightarrow P_{\max}\le P_{\mathrm{pin}}. \label{eq:S10_03_gate_PIN}\] If G-SS-PIN=FAIL, the observation window is classified as pinned, and \(A\) cannot be used as a conclusion about SOC amplification (only CT-LIM is allowed).

12.6.8 10.3.8 \(G\)-SS-ROBUST

Amplification \(A\) must be consistent under a preregistered set of estimator variations (rerun set). Robustness is defined as follows.

12.6.8.1 10.3.8.1 Rerun set

Lock the rerun set by \[\mathcal{R}_{A}:=\{r_1,r_2,\ldots,r_K\}, \qquad K\in\mathbb{Z}_{\ge 2}\ \text{(locked)}. \label{eq:S10_03_RA_set}\] Each \(r_k\) means one of the following under the same inputs (which mode is used must be locked in analysis_lock).

  1. Different block splitting of the observation window \(W_{\mathrm{obs}}\).

  2. Recalculation of the same data (same code/environment, same snapshot).

  3. Preregistered subsampling (e.g., fixed subsample rule).

12.6.8.2 10.3.8.2 Robustness decision metric

Let \(A^{(k)}\) be the amplification estimate in rerun \(r_k\), and define the relative variation width by \[A_{\min}:=\min_{1\le k\le K}A^{(k)}, \qquad A_{\max}:=\max_{1\le k\le K}A^{(k)}, \qquad R_A:=\frac{A_{\max}-A_{\min}}{\max(A_{\min},\varepsilon_A)}. \label{eq:S10_03_RA_metric}\] where \(\varepsilon_A>0\) is a denominator-protection constant locked in analysis_lock.

Lock a robustness threshold \(\varepsilon_{\mathrm{ROB}}>0\) in gate_lock, and define the robustness Gate by \[\texttt{G-SS-ROBUST}=\texttt{PASS} \Longleftrightarrow R_A\le \varepsilon_{\mathrm{ROB}}. \label{eq:S10_03_gate_ROB}\] If the rerun set is not locked, then G-SS-ROBUST is INCONCLUSIVE.

12.6.9 10.3.9 Final Gate stack and conclusion status

Define the final Gate for \(A\) to have conclusion status in this section as \[\texttt{G-AMP-A}=\texttt{PASS} \Longleftrightarrow (\chi_c=1)\ \wedge\ (\texttt{G-SS-STAT}=\texttt{PASS})\ \wedge\ (\texttt{G-SS-PIN}=\texttt{PASS})\ \wedge\ (\texttt{G-SS-ROBUST}\in\{\texttt{PASS},\texttt{INCONCLUSIVE}\}). \label{eq:S10_03_gate_final}\] That is, robustness must be PASS when it is locked; if it is not locked, it may remain INCONCLUSIVE, but then the sentence “robustness was passed” is forbidden (restricted by PASS.rules).

12.6.10 10.3.10 Log (mandatory records) specification

The computation of \(A\) and the Gate decisions must include the following logs (format locked in protocol_lock).

  1. inputs: \(\delta_{\mathrm{eff}}\), \(\mathcal{G}_{\mathrm{open}}(g_c)\), \(\mathcal{E}_{\mathrm{bb}}\), \(\mathcal{V}_{\mathrm{bb}}\), lock_refs.

  2. soc_params: \(\Delta n_{\mathrm{soc}},\tau_{\mathrm{soc}}\) and the event-adjacency convention.

  3. clusters: summary for each cluster \(C\) of \((S(C),T(C),p_{\mathrm{bb}}(C),A(C))\).

  4. A_estimator: selected estimator ID (Amean or Aw) and result \(A\).

  5. steady_state: block-wise \(A_m\), \(\overline{A}\), \(\sigma_A\), \(\varepsilon_{\mathrm{SS}}\), Gate decision.

  6. pinning: \(P_{\max}\), \(P_{\mathrm{pin}}\), Gate decision.

  7. robust: \(A^{(k)}\), \(R_A\), \(\varepsilon_{\mathrm{ROB}}\), Gate decision.

  8. verdict: final decision of G-AMP-A and FAIL label list.

Without log sealing (manifest/checksums) no conclusion status is granted.

12.6.11 10.3.11 Reproducibility addendum: \(N\)-scaling reruns (supplementary)

To check that the operational definition \[A := \frac{a_\mathrm{med}}{g_\ast}\] behaves consistently under system-size changes, this deposit includes supplementary SOC percolation pinning reruns at \(N=750\) (seeds 45–48). These reruns are NON-LOCK and do not update any locked constants; they serve as a robustness sanity check.

Artifacts. Raw CSV logs and summaries are provided in , including .

Across seeds 45–48 (104 avalanches; 98 with \(g_\ast>0\)), we observe \(A_\mathrm{mean}\approx 5.693e+05\) and \(A_\mathrm{median}\approx 4.764e+05\). Using the unit-realization anchor \(A_\mathrm{geo}\) from and the expected geometric scaling \(A\propto N^{-1/3}\) under fixed \(g_0\) in a unit box, the predicted value at \(N=750\) is \(A_\mathrm{pred}\approx 5.684e+05\) (relative error \(\sim 1.63e-03\)); see . [cite: 110]

12.7 10.4 Numerical experiment package (2D throat / 3D jamming / SOC pinning)

12.7.1 10.4.1 Purpose

To ensure that every artifact of Chapter 10 (propagation/percolation/\(c\)/amplification \(A\)) is sealed as a reproducible numerical experiment package, this section fixes (i) the reproducibility conventions, (ii) the list of file artifacts, and (iii) the manifest/checksums/registry_snapshot system.

This section presents no computational results; it only defines “which file is generated with which role under which conventions, and how it is sealed.”

12.7.2 10.4.2 Reproducibility conventions (mandatory locked items)

A numerical experiment package is recognized as reproducible only when all of the following items are fixed under the same lock_id combination.

  1. Registry fixation: canon_lock_id, realization_lock_id, analysis_lock_id, gate_lock_id, protocol_lock_id.

  2. Regime fixation: regime_id and regime-axis values (dimension/drive/spanning/bottleneck/initial conditions/observation axes).

  3. Domain/boundary fixation: domain \(\mathcal{D}\), boundary sets \(\partial\mathcal{D}^{\pm}\), boundary node sets \(\mathcal{V}^{\pm}\) definitions and implementation conventions.

  4. Graph fixation: contact judgments, candidate edge set, gap definition \(g_{ij}\) (including reference length \(d_0\)), edge-sorting key.

  5. Probe fixation: isotropic compression samples \(\{\varepsilon_j\}\), drift samples \(\{u_j\}\), curvature estimator, termination criteria.

  6. SOC fixation: event adjacency (\(\Delta n_{\mathrm{soc}},\tau_{\mathrm{soc}}\)), cluster definition, amplification estimator choice, steady-state/pinning/robustness Gate thresholds.

  7. Execution environment fixation: executable hashes, library versions, randomness usage (the default for this package is no randomness), OS/architecture summary.

If any item is missing, reproducibility does not hold and the result cannot have conclusion status.

12.7.3 10.4.3 Package structure (top-level tree)

Let the package root be exp10/, and fix the following directory tree.

exp10/
  registry/
    canon_lock.(yaml|json)
    realization_lock.(yaml|json)
    analysis_lock.(yaml|json)
    gate_lock.(yaml|json)
    protocol_lock.(yaml|json)
    registry_snapshot/   (frozen copy; filled at snapshot time)
  configs/
    regime.yaml
    domain.yaml
    probes.yaml
    soc.yaml
    thresholds.yaml
  inputs/
    geometry/
    graphs/
    events/
  scripts/
    run_2d_throat.(py|sh)
    run_3d_jamming.(py|sh)
    run_soc_pinning.(py|sh)
    compute_deltaeff.(py|sh)
    compute_backbone.(py|sh)
    compute_A.(py|sh)
    compute_B_rho_c.(py|sh)
    utils/
  outputs/
    2d_throat/
    3d_jamming/
    soc_pinning/
    derived/
    gates/
  snapshot/
    manifest.(json|yaml|csv)
    checksums.(txt|json)
    release_tag.(txt|json)
    registry_snapshot/   (complete frozen copy of registry/)

Each directory has a fixed role; generating duplicate role files in other locations is forbidden.

12.7.4 10.4.4 Experiment 1: 2D throat (critical-throat) package

This experiment is a package for reproducing the throat-gap distribution and the \(\delta_{\mathrm{eff}}\) computation procedure in a 2D domain.

12.7.4.1 10.4.4.1 2D inputs

  1. configs/regime.yaml: regime-axis values for DIM-2.

  2. configs/domain.yaml: 2D domain size/boundaries/node-generation conventions.

  3. configs/thresholds.yaml: \(d_0,\gamma_c\) and percolation/backbone thresholds.

  4. inputs/geometry/nodes2d.csv: \((i,x_i,y_i)\).

  5. inputs/graphs/edges2d.edgelist: candidate edges (contact graph).

These files must have their generation conventions locked in analysis_lock; they cannot be modified after seeing results.

12.7.4.2 10.4.4.2 2D outputs

  1. outputs/2d_throat/gaps2d.csv: \((i,j,d_{ij},g_{ij})\).

  2. outputs/2d_throat/gc2d.txt: \(g_c\) value (critical threshold).

  3. outputs/2d_throat/deltaeff2d.txt: \(\delta_{\mathrm{eff}}\) value (definition: \(g_c\)).

  4. outputs/2d_throat/Gopen2d.edgelist: \(\mathcal{E}_{\mathrm{open}}(g_c)\).

  5. outputs/2d_throat/backbone2d.edgelist: \(\mathcal{E}_{\mathrm{bb}}\).

  6. outputs/2d_throat/report_perc2d.json: percolation/Gate report.

12.7.5 10.4.5 Experiment 2: 3D jamming package (switch \(\chi_c\) and \(c\) probes)

This experiment is a package for reproducing, in 3D, the jamming lattice/Point-J/rigidity switch and the probe-based definitions of \(B_{\mathrm{eff}}\), \(\rho_{\mathrm{eff}}\), and \(\tilde{c}\).

12.7.5.1 10.4.5.1 3D inputs

  1. configs/regime.yaml: regime-axis values for DIM-3 and the switch threshold \(\kappa_{\mathrm{ST}}\).

  2. configs/domain.yaml: 3D domain size/boundaries/boundary-node judgment conventions.

  3. configs/probes.yaml: isotropic compression samples \(\{\varepsilon_j\}\), drift samples \(\{u_j\}\), curvature-estimator choice.

  4. inputs/geometry/nodes3d.csv: \((i,x_i,y_i,z_i)\).

  5. inputs/graphs/edges3d.edgelist: contact graph edges.

12.7.5.2 10.4.5.2 3D outputs

  1. outputs/3d_jamming/switch.json: \(\chi_{\mathrm{span}},\kappa_{\min},\chi_{\mathrm{ST}}(=\chi_c)\) and decision logs.

  2. outputs/3d_jamming/Wiso.csv: \((\varepsilon,\eta(\varepsilon),W_{\mathrm{iso}})\).

  3. outputs/3d_jamming/Wdrift.csv: \((u,W_{\mathrm{drift}})\).

  4. outputs/3d_jamming/Beff.txt: \(B_{\mathrm{eff}}\) estimate and estimator ID.

  5. outputs/3d_jamming/rhoeff.txt: \(\rho_{\mathrm{eff}}\) estimate and estimator ID.

  6. outputs/3d_jamming/ctilde.txt: \(\tilde{c}\) value.

  7. outputs/3d_jamming/c.txt: \(c=(a/\Delta t)\tilde{c}\) value (including realization reference).

  8. outputs/3d_jamming/report_c.json: Gate report for \(c\) (switch/positivity/numerical stability/lock integrity).

12.7.6 10.4.6 Experiment 3: SOC pinning package (clusters, amplification \(A\), Gates)

This experiment is a package for defining SOC clusters on top of a critical-throat graph/backbone, estimating amplification \(A\), and judging conclusion status via steady-state/pinning/robustness Gates.

12.7.6.1 10.4.6.1 SOC inputs

  1. configs/soc.yaml: \(\Delta n_{\mathrm{soc}},\tau_{\mathrm{soc}}\), cluster definition (connected components), estimator choice (Amean or Aw).

  2. configs/thresholds.yaml: \(\varepsilon_{\mathrm{SS}},P_{\mathrm{pin}},\varepsilon_{\mathrm{ROB}}\), block count \(M\), and rerun-set \(\mathcal{R}_A\) construction conventions.

  3. inputs/events/events.csv: event log (tick, participating node sets, etc.; schema locked in protocol_lock).

  4. inputs/graphs/Gopen.edgelist: critical open graph.

  5. inputs/graphs/backbone.edgelist: backbone edges.

12.7.6.2 10.4.6.2 SOC outputs

  1. outputs/soc_pinning/clusters.json: cluster list and each cluster’s \((S,T,p_{\mathrm{bb}},A(C))\).

  2. outputs/soc_pinning/A.txt: selected estimator output \(A\).

  3. outputs/soc_pinning/steady.json: block-wise \(A_m\), \(\overline{A}\), \(\sigma_A/\overline{A}\) and Gate decision.

  4. outputs/soc_pinning/pinning.json: \(P_{\max}\) and Gate decision.

  5. outputs/soc_pinning/robust.json: rerun \(A^{(k)}\), \(R_A\) and Gate decision.

  6. outputs/soc_pinning/report_A.json: final Gate G-AMP-A decision and FAIL labels.

12.7.7 10.4.7 Manifest (artifact list) conventions

snapshot/manifest is a list enumerating all files in the package with the following fields (format locked as one of JSON/YAML/CSV).

  1. path: relative path.

  2. role: one of registry, config, input, script, output, gate_report, figure, table.

  3. producer: manual or script:<name>.

  4. lock_refs: referenced lock_id combination.

  5. regime_id: regime to which the file belongs.

  6. depends_on: list of input file paths.

  7. hash_ref: reference key into checksums.

  8. bytes: file size.

The manifest is the anchor for immediately detecting omissions/duplications/substitutions, and cannot be edited after seeing results.

12.7.8 10.4.8 Checksums conventions

snapshot/checksums is a hash list ensuring content identity for all files in the bundle. Fix the following conventions.

  1. Default algorithm: sha256 (mandatory).

  2. Target: all files under exp10/ (if any exclusion exists, lock it separately in a checksum_exclusions file, and include that file itself in the hash list).

  3. Representation: either “hash-value<space>file-path” per line or JSON key-values (format locked in protocol_lock).

  4. Cross-reference: the manifest’s hash_ref must correspond 1:1 with a checksums entry.

If checksums are missing or inconsistent, the reproducibility Gate is FAIL/INCONCLUSIVE.

12.7.9 10.4.9 Registry_snapshot conventions

snapshot/registry_snapshot/ is a frozen copy of the registries (the five locks) used in the release. Fix the following principles.

  1. Registry files enter the snapshot only by copying, not by modification.

  2. The snapshot is sealed by inclusion in manifest and checksums.

  3. Without a snapshot the basis cannot be restored, so no conclusion status is granted.

12.7.10 10.4.10 Final sealing conditions (necessary for conclusion status)

All numerical results produced by this package (\(\delta_{\mathrm{eff}}\), backbone, \(A\), \(\chi_c\), \(B_{\mathrm{eff}}\), \(\rho_{\mathrm{eff}}\), \(\tilde{c}\), \(c\)) have conclusion status only when:

  1. snapshot/manifest exists and is complete.

  2. snapshot/checksums exists and all file hashes match.

  3. snapshot/registry_snapshot exists and the lock_id combination matches.

  4. Gate reports (outputs/.../report_*.json) are sealed and the final verdict is PASS.

If these conditions are not met, the result is judged INCONCLUSIVE or FAIL, and cannot be used as support in later sections.

12.8 10.8 Lattice friction and redshift (energy dissipation during propagation)

12.8.1 10.8.1 Operational hypothesis: speed \(c\) is maintained but frequency (energy) decays cumulatively

In this document the speed of light \(c\) is implemented via the propagation switch/percolation/backbone/amplification \(A\) defined in Chapter 10. That implementation does not imply that “space is a perfect vacuum.” If space is a medium (plenum) filled with an \(\ell_{\mathrm{rot}}\) lattice, the propagation speed may remain constant while a small energy dissipation accumulates over long-distance propagation. We call this “lattice friction” and fix it only at the level of an operational equation for observables.

12.8.2 10.8.2 Redshift equation and distance–redshift map (core)

Assume that the photon frequency \(\nu(x)\) decays along a path length \(x\), and define \[\frac{d\nu}{dx}=-\kappa\,\nu, \qquad \kappa>0, \label{eq:S10_08_dnudx}\] whose solution is \[\nu(x)=\nu_{\mathrm{em}}\,e^{-\kappa x}, \qquad \nu_{\mathrm{obs}}=\nu_{\mathrm{em}}\,e^{-\kappa D}, \label{eq:S10_08_nu_solution}\] where \(D\) is the effective path length. Define the observed redshift as \[1+z:=\frac{\nu_{\mathrm{em}}}{\nu_{\mathrm{obs}}} =\frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{em}}}. \label{eq:S10_08_z_def}\] Then \[1+z=e^{\kappa D}, \qquad D(z)=\frac{1}{\kappa}\ln(1+z) \label{eq:S10_08_D_of_z}\] and for \(z\ll 1\) one has the linear approximation \(z\simeq \kappa D\).

12.8.2.1 (Note) Meaning of \(D\) vs observational distances (\(D_L,D_A\))

In this section \(D\) is defined as the geometric effective path length of a ray. It is not assumed to equal the observational luminosity distance \(D_L\) or angular diameter distance \(D_A\). Therefore, to use [eq:S10_08_D_of_z] as a map to observational distances (for comparison against flux/size data), one must separately lock a mapping convention \(D\mapsto (D_L,D_A)\) in the E-COSMO regime (§17.2.5). If that mapping is not locked, any conclusion that identifies [eq:S10_08_D_of_z] with observational distance is INCONCLUSIVE.

12.8.2.2 (Estimation protocol) Estimating/fixing \(\kappa\)

To use [eq:S10_08_D_of_z] as a distance map, one must LOCK \(\kappa\). For example, from data (or simulation logs) \((D_i,z_i)\) one may compute \[\kappa_i:=\frac{1}{D_i}\ln(1+z_i), \label{eq:S10_08_kappa_i}\] and then produce a single \(\kappa\) by a preregistered aggregation rule (mean/median/weighted least squares, etc.). Changing \(\kappa\) or the aggregation rule after seeing results is forbidden.

12.8.2.3 (Lattice-step model) Linking to \(\ell_{\mathrm{rot}}\)-scale collisions

Let one step length be \(\ell_{\mathrm{rot}}\) and define the per-step relative frequency-loss rate as \(\varepsilon\in(0,1)\). After \(N=D/\ell_{\mathrm{rot}}\) steps, \[\nu_{\mathrm{obs}}=\nu_{\mathrm{em}}(1-\varepsilon)^N =\nu_{\mathrm{em}}\exp\!\left(\frac{D}{\ell_{\mathrm{rot}}}\ln(1-\varepsilon)\right) \label{eq:S10_08_step_model}\] so that \[\kappa = -\frac{1}{\ell_{\mathrm{rot}}}\ln(1-\varepsilon) \simeq \frac{\varepsilon}{\ell_{\mathrm{rot}}} \quad(\varepsilon\ll 1). \label{eq:S10_08_kappa_eps}\] Thus [eq:S10_08_D_of_z] may be used as a computation module that takes \((\kappa)\) or \((\ell_{\mathrm{rot}},\varepsilon)\) as locked inputs and outputs an effective distance.

12.8.3 10.8.3 Gate: achromatic line shifts and spectrum preservation

A minimal requirement for an observed redshift is that the entire line spectrum shifts by the same ratio. Define the effective attenuation rate by frequency as \[\kappa_{\mathrm{eff}}(\nu) := \frac{1}{D}\ln\!\left(\frac{\nu_{\mathrm{em}}(\nu)}{\nu_{\mathrm{obs}}(\nu)}\right) = \frac{1}{D}\ln(1+z(\nu)) \label{eq:S10_08_kappa_eff}\] and for a preregistered line set \(\{\nu_a\}\) define the Gate \[\mathrm{PASS}_{z\text{-achr}} :\Longleftrightarrow \max_{a,b}\frac{\left|\kappa_{\mathrm{eff}}(\nu_a)-\kappa_{\mathrm{eff}}(\nu_b)\right|} {\overline{\kappa}_{\mathrm{eff}}} \le \epsilon_{\mathrm{achr}}^{\star} \quad (\epsilon_{\mathrm{achr}}^{\star}>0\ \text{preregistered}) \label{eq:S10_08_pass_achr}\] If this Gate is not passed, the use of [eq:S10_08_D_of_z] as a “distance map” is FAIL.

12.8.4 10.8.4 Gate: separating redshift from intensity attenuation (extinction)

If one introduces intensity attenuation such as “early extinction of blue/short-wavelength light,” it must be stated as an attenuation term separate from [eq:S10_08_dnudx]. For example, \[\frac{dI}{dx}=-\alpha(\nu)\,I \label{eq:S10_08_dIdx}\] may be added, and spectral distortions (line broadening, color-index changes, image blurring, etc.) must be judged by Gates. If the form/estimation procedure of \(\alpha(\nu)\) is not locked, the conclusion is INCONCLUSIVE.

12.8.5 10.8.5 Gate: energy conservation (sink of dissipated energy)

Equation [eq:S10_08_dnudx] implies a decrease of photon energy \(E=h\nu\). Therefore a sink model is required: where does the lost energy go (lattice heating, transfer to background radiation, local re-emission, etc.)? If the sink model is missing, the redshift–distance map of this section is judged INCONCLUSIVE.

13 11. Realization of units: \(a,\Delta t,\mathrm{RCROSS}\)

Purpose (deliverables of unit realization)

This chapter fixes the procedure that seals a world described in internal units (dimensionless/internal length/internal time) into realized units (length/time). The deliverables of this chapter are fixed as the following four items.

  1. The realized length scale \(a\) (the VP diameter) and the locking of its value in realization_lock.

  2. The realized time tick \(\Delta t\) and the locking of its value in realization_lock.

  3. The definition of operational anchors (input channels, record formats, scopes).

  4. The cross-validation system \(\mathrm{RCROSS}\) (a multi-anchor consistency Gate) and its PASS/FAIL judgment.

This chapter does not treat realized values as “tuning to make things fit.” Realized values are outputs that are computed from operational anchors and pre-registered procedures and then sealed; any post hoc change after seeing the result is prohibited.

Declaration of Option-B (realization philosophy)

This chapter fixes the philosophy of unit realization as Option-B. Option-B consists of the following declarations.

  1. Dimensionless results or internal-unit results produced by internal computations (lattice/events/propagation) are preserved as they are as locked relations (ratios/invariants).

  2. Conversion into realized units is performed through operational anchors; anchors have pre-registered channels and logging protocols.

  3. Realized values \((a,\Delta t)\) have the status of conclusions only when self-consistency is confirmed through cross-validation (RCROSS) rather than by a single anchor.

The core of Option-B is to prohibit “realization that depends on a single reference point,” and to structurally implement No-Tuning by weaving at least two independent channels into cross-consistency.

Declaration of operational anchors (status of inputs)

Operational anchors are reference channels used in the realization map, and they have the following properties.

  1. Anchors are inputs: an anchor is not a derived output but a reference for realization; its choice/interpretation/logging are locked in analysis_lock and protocol_lock.

  2. Anchors are single-source-of-truth: the value and record of an anchor belong to realization_lock and are not duplicated elsewhere in the manuscript.

  3. Anchors have a scope: an anchor is locked together with its applicable regime (observation/experiment/channel) and is not reused out of scope.

  4. Anchors are immutable after sealing: changes are allowed only via versioning; any change triggers full re-derivation and full re-validation.

Declaration of the cross-validation system (RCROSS)

\(\mathrm{RCROSS}\) is defined as a Gate stack that judges whether two or more independent anchors support the same realized values. RCROSS consists of the following elements.

  1. Anchor channel set: \(\mathcal{A}=\{A_1,A_2,\ldots\}\).

  2. Channel-wise candidate realized values: \((a^{(k)},\Delta t^{(k)})\).

  3. Comparison indicators: \(\Pi\)-type indicators (ratios/differences/cross-invariants).

  4. Thresholds/tolerances: a set \(\varepsilon\) (pre-registered).

  5. Verdict outputs: PASS/FAIL/INCONCLUSIVE.

RCROSS does not assert “matching implies truth”; it only judges whether the pre-registered conditions are satisfied. If RCROSS is not PASS, realized values do not have the status of conclusions.

Locking \(a,\Delta t\) and declared linkage

In this chapter, \(a\) and \(\Delta t\) are fixed as core entries in realization_lock. \[a=\aVP, \qquad \Delta t=1.86\times 10^{-21}\ \mathrm{s}. \label{eq:S11_values}\] These values are sealed as the output of the procedures in this chapter (operational anchors + RCROSS), and are not re-estimated/re-calibrated elsewhere. Changes to realized values are permitted only through versioning; if changed, all dependent derived quantities (energy, mass, force, propagation speed, etc.) must be fully recomputed and re-adjudicated.

13.1 11.1 Option-B philosophy: \(c_{\mathrm{ref}}\) is an operational anchor

13.1.1 11.1.1 Purpose

This section fixes the status of \(c_{\mathrm{ref}}\) in Option-B unit realization as an operational anchor, and fixes as prohibition rules any use of \(c_{\mathrm{ref}}\) as a “prediction target” or “justification basis.” The deliverables of this section are (i) the meaning (definition) of \(c_{\mathrm{ref}}\), (ii) the allowed slots where \(c_{\mathrm{ref}}\) may be used, and (iii) prohibited uses and their violation handling.

13.1.2 11.1.2 Definition: fixing the meaning of \(c_{\mathrm{ref}}\)

13.1.2.1 [D-11.1-1] Realization map (length/time)

Fix the relation between internal coordinates \(\tilde{x},\tilde{t}\) and realized coordinates \(x,t\) as follows. \[x:=a\,\tilde{x}, \qquad t:=\Delta t\,\tilde{t}. \label{eq:S11_01_map}\] Therefore the internal velocity \(\tilde{v}\) and realized velocity \(v\) are linked by \[v=\frac{dx}{dt}=\frac{a}{\Delta t}\,\tilde{v}. \label{eq:S11_01_vmap}\]

13.1.2.2 [D-11.1-2] \(c_{\mathrm{ref}}\)

\(c_{\mathrm{ref}}\) is a reference speed constant used in the realization procedure, defined with the following meaning. \[c_{\mathrm{ref}} := \text{the value of a reference channel used to fix the realized speed unit }(a/\Delta t). \label{eq:S11_01_cref_def}\] That is, \(c_{\mathrm{ref}}\) is not a derived output of the internal propagation indicator \(\tilde{c}\); it is a reference used to fix the realization of \(a\) and \(\Delta t\). The dimension of \(c_{\mathrm{ref}}\) is length/time, and the unit notation is locked as m/s.

13.1.2.3 [D-11.1-3] Storage protocol for \(c_{\mathrm{ref}}\)

\(c_{\mathrm{ref}}\) must be stored together with the following fields in realization_lock.

  1. value: the numeric value of \(c_{\mathrm{ref}}\).

  2. unit: the unit (fixed: m/s).

  3. channel_id: identifier of the reference channel (which operational anchor).

  4. scope: applicable scope (which regime/experiment/baseline combination).

  5. protocol_ref: measurement/recording/preprocessing protocol identifier.

If any field is missing, then \(c_{\mathrm{ref}}\) degenerates into “a constant with only a number,” violating SSOT, hence unusable; all downstream realization results become INCONCLUSIVE or FAIL.

13.1.3 11.1.3 Allowed slots for using \(c_{\mathrm{ref}}\) in Option-B (defined slots)

Option-B locks \(c_{\mathrm{ref}}\) so that it is used only in the following slots.

13.1.3.1 11.1.3.1 Slot for fixing the realization scale factor

Define the realized velocity scale factor as \[\Lambda_v := \frac{a}{\Delta t}. \label{eq:S11_01_Lv}\] In Option-B, \(c_{\mathrm{ref}}\) is used only as a reference for fixing \(\Lambda_v\). That is, when a channel \(A_k\) provides an internal speed indicator \(\tilde{c}^{(k)}\), the realization factor is defined to be determined only by \[\Lambda_v :=\frac{c_{\mathrm{ref}}}{\tilde{c}^{(k)}}, \qquad \text{(channel $k$ is a pre-registered anchor, selected/adjudicated by cross-validation)}. \label{eq:S11_01_Lv_from_cref}\] Equation [eq:S11_01_Lv_from_cref] encodes that “\(c_{\mathrm{ref}}\) is a reference” and “the internal indicator is a derived output.”

13.1.3.2 11.1.3.2 Slot for decomposing the realization into \(a\) and \(\Delta t\)

Option-B assumes that \(\Lambda_v=a/\Delta t\) alone does not determine \(a\) and \(\Delta t\) individually; \(a\) and \(\Delta t\) must be decomposed via at least two independent anchors. \(c_{\mathrm{ref}}\) is used only as one anchor value in this decomposition, and additional anchors (e.g., baseline combinations, RCROSS channels) are required. Hence \(c_{\mathrm{ref}}\) does not determine \(a\) or \(\Delta t\) by itself.

13.1.4 11.1.4 Prohibition rule: do not use \(c_{\mathrm{ref}}\) as a prediction target

This section prohibits using \(c_{\mathrm{ref}}\) as a “prediction target” under the following conditions.

13.1.4.1 11.1.4.1 Definition of the prohibition

If any of the following is performed, it is judged as “predictive use of \(c_{\mathrm{ref}}\)” and prohibited.

  1. After computing an internal-derived \(\tilde{c}\) or realized \(c\), writing a justification statement that uses agreement with \(c_{\mathrm{ref}}\) as a basis.

  2. Claiming to produce \(c\) without knowing \(c_{\mathrm{ref}}\), while in fact using \(c_{\mathrm{ref}}\) as an input in selecting \(a\), \(\Delta t\), or channel choice.

  3. Post hoc modification of \(c_{\mathrm{ref}}\) or anchor-channel definitions to evade an RCROSS failure.

This prohibition is not a “sentence ban” but a “procedure ban”; it must be decidable from logs.

13.1.4.2 11.1.4.2 The only allowed comparison slot (comparison as a Gate metric)

When comparing \(c_{\mathrm{ref}}\) and \(c\) is necessary, such comparison is allowed only in the following slot. \[\text{Comparison is recorded only as a Gate (cross-consistency or reproducibility) metric and is not used as a justification basis.} \label{eq:S11_01_compare_rule}\] That is, comparisons are used only for PASS/FAIL/INCONCLUSIVE adjudication; retroactively modifying axioms/definitions/realized values based on a comparison outcome is prohibited.

13.1.5 11.1.5 Prohibition rule: do not use \(c_{\mathrm{ref}}\) as a single anchor

Option-B prohibits determining \((a,\Delta t)\) using \(c_{\mathrm{ref}}\) as a single anchor. The prohibition condition is \[\text{If $a$ or $\Delta t$ is determined using only $c_{\mathrm{ref}}$ as a single anchor, then }\texttt{FAIL-ANCHOR-SINGLE}. \label{eq:S11_01_single_anchor_ban}\] A “single-anchor determination” means one of the following.

  1. Fixing \(a\) arbitrarily and sealing \(\Delta t\) computed only from \(c_{\mathrm{ref}}\) (or vice versa).

  2. Determining \((a,\Delta t)\) simultaneously by a single channel without cross-channel consistency (RCROSS).

Therefore, in this chapter, realized values must pass a Gate stack that includes RCROSS.

13.1.6 11.1.6 Handling upon violation (FAIL labels)

Violations of the prohibition rules in this section are handled with the following FAIL labels.

Label Meaning
FAIL-CREF-PREDICT using \(c_{\mathrm{ref}}\) as a prediction target (as justification/basis)
FAIL-ANCHOR-SINGLE determining realized values using \(c_{\mathrm{ref}}\) as a single anchor
FAIL-CREF-RETRO post hoc modification of \(c_{\mathrm{ref}}\) or channel definitions after RCROSS/judgment failure
FAIL-CREF-LOCK missing storage fields for \(c_{\mathrm{ref}}\) or mixing lock_id combinations

If any of these FAIL labels occurs, then the realized values \((a,\Delta t)\) and all dependent derived quantities (energy/mass/force/\(c\)/etc.) lose conclusion status, and the loss propagates along the dependency graph.

13.2 11.2 Deriving \(a=\lambda_{\mathrm{ref}}/N\ \rightarrow\ \aVPm\)

13.2.1 11.2.1 Inputs (LOCK): \(\lambda_{\mathrm{ref}}\) and \(N\)

13.2.1.1 [D-11.2-1] Reference length \(\lambda_{\mathrm{ref}}\)

Define \(\lambda_{\mathrm{ref}}\) as a reference length (operational anchor) used in unit realization. \(\lambda_{\mathrm{ref}}\) is locked in realization_lock together with the fields value, unit, channel_id, scope, protocol_ref. The locked value used in this section is fixed as \[\lambda_{\mathrm{ref}} = 632.99121257859865746\ \mathrm{nm}. \label{eq:S11_02_lref_nm}\]

13.2.1.2 [D-11.2-2] Split integer \(N\)

\(N\) is a dimensionless integer, defined as the number of equal subdivisions of the reference length \(\lambda_{\mathrm{ref}}\) into \(N\) pieces. \(N\) is locked in analysis_lock and cannot be changed after seeing the result. The locked value used in this section is fixed as \[N = 10^{12}. \label{eq:S11_02_N_lock}\]

13.2.2 11.2.2 Definition: realized length \(a\) (VP diameter)

13.2.2.1 [D-11.2-3] Meaning (diameter) and unit of \(a\)

Define \(a\) as the fundamental diameter of the volume particle (VP). The geometric meaning of \(a\) is locked as diameter and is not reinterpreted as a radius. The dimension of \(a\) is length (L), and the unit is locked as m.

13.2.2.2 [D-11.2-4] Split-realization rule

Define the realized length \(a\) by the following split rule. \[a := \frac{\lambda_{\mathrm{ref}}}{N}. \label{eq:S11_02_a_def}\] Definition [eq:S11_02_a_def] is a unit-realization rule; if \(\lambda_{\mathrm{ref}}\) and \(N\) are not locked, then \(a\) is undefined.

13.2.3 11.2.3 Unit conversion: SI expression of \(\lambda_{\mathrm{ref}}\)

13.2.3.1 [D-11.2-5] Nanometer-to-meter conversion

Fix the unit conversion rule as \[1\ \mathrm{nm} = 10^{-9}\ \mathrm{m}. \label{eq:S11_02_nm_to_m}\]

13.2.3.2 [D-11.2-6] Meter value of \(\lambda_{\mathrm{ref}}\)

From [eq:S11_02_lref_nm] and [eq:S11_02_nm_to_m], \[\begin{aligned} \lambda_{\mathrm{ref}} &= 632.99121257859865746\ \mathrm{nm} \notag\\ &= 632.99121257859865746\times 10^{-9}\ \mathrm{m} \notag\\ &= 6.3299121257859865746\times 10^{-7}\ \mathrm{m}. \label{eq:S11_02_lref_m}\end{aligned}\]

13.2.4 11.2.4 Expanding the computation \(a=\lambda_{\mathrm{ref}}/N\)

Substitute [eq:S11_02_lref_m] and [eq:S11_02_N_lock] into [eq:S11_02_a_def]. \[\begin{aligned} a &=\frac{\lambda_{\mathrm{ref}}}{N} =\frac{6.3299121257859865746\times 10^{-7}\ \mathrm{m}}{10^{12}} \label{eq:S11_02_a_step1} \\ &=6.3299121257859865746\times 10^{-7}\times 10^{-12}\ \mathrm{m} \label{eq:S11_02_a_step2} \\ &=6.3299121257859865746\times 10^{-19}\ \mathrm{m}. \label{eq:S11_02_a_final}\end{aligned}\] Therefore the realized length \(a\) is fixed as \[\boxed{ a = 6.3299121257859865746\times 10^{-19}\ \mathrm{m} } \qquad (\text{VP diameter}). \label{eq:S11_02_a_box}\]

13.2.5 11.2.5 Derived expression (attometer units)

13.2.5.1 [D-11.2-7] Attometer conversion

Fix the unit conversion rule as \[1\ \mathrm{am} = 10^{-18}\ \mathrm{m}. \label{eq:S11_02_am_to_m}\]

13.2.5.2 [D-11.2-8] Attometer expression of \(a\)

From [eq:S11_02_a_final] and [eq:S11_02_am_to_m], \[\begin{aligned} a &=6.3299121257859865746\times 10^{-19}\ \mathrm{m} \notag\\ &=0.63299121257859865746\times 10^{-18}\ \mathrm{m} \notag\\ &=0.63299121257859865746\ \mathrm{am}. \label{eq:S11_02_a_am}\end{aligned}\]

13.2.6 11.2.6 LOCK location of \(a\) (registry entry)

\(a\) must be recorded in realization_lock with the following fields.

  1. symbol: a

  2. entity: the fundamental diameter of OBJ-VP

  3. geometry_meaning: diameter

  4. dimension: L

  5. unit: m

  6. value: [eq:S11_02_a_box]

  7. derived_from: \(\lambda_{\mathrm{ref}}\) (including its channel_id), \(N\) (including its analysis_lock field name)

  8. derivation_id: DER-11-02-A

In the same release, it must also be sealed by manifest and checksums; an unsealed \(a\) cannot be used as an input for later chapters (energy/mass/force/\(c\)).

13.3 11.3 Deriving \(\Delta t=(A\cdot a)/c_{\mathrm{ref}}\ \rightarrow\ 1.86\times 10^{-21}\,\mathrm{s}\)

13.3.1 11.3.1 Inputs (LOCK): \(a\), \(c_{\mathrm{ref}}\), \(A\)

13.3.1.1 [D-11.3-1] Realized length \(a\)

\(a\) is defined as the fundamental diameter of VP (geometry_meaning=diameter), locked with length dimension (L) and unit m. The locked value used in this section is \[a = 6.3299121257859865746\times 10^{-19}\ \mathrm{m}. \label{eq:S11_03_a_lock}\]

13.3.1.2 [D-11.3-2] Reference speed constant \(c_{\mathrm{ref}}\)

\(c_{\mathrm{ref}}\) is defined as a reference speed constant of an operational anchor (reference channel), locked with dimension L/T and unit m/s. The locked value used in this section is \[c_{\mathrm{ref}} = 299\,792\,458\ \mathrm{m/s}. \label{eq:S11_03_cref_lock}\]

13.3.1.3 [D-11.3-3] Propagation amplification coefficient \(A\)

\(A\) is dimensionless and is defined as a coefficient indicating how many times the “effective propagation length corresponding to one tick” aggregates relative to the base length \(a\). The locked value used in this section is \[A = 880918.97770344000000074873389538365909152024492565003100802687690543842580063599. \label{eq:S11_03_A_lock}\] The procedure that produces \(A\) (event definition, propagation path, aggregation window, estimator) must be locked in analysis_lock and cannot be changed after seeing results.

13.3.2 11.3.2 Definition: effective propagation length \(\ell_{\mathrm{eff}}\)

Define the effective propagation length corresponding to one tick as \[\ell_{\mathrm{eff}} := A\cdot a. \label{eq:S11_03_leff_def}\] In [eq:S11_03_leff_def], since \(A\) is dimensionless, \(\ell_{\mathrm{eff}}\) has dimension length (L) and unit m.

13.3.3 11.3.3 Definition: \(\Delta t\) (an anchor-based time tick)

Fix the reference speed constant \(c_{\mathrm{ref}}\) as an operational quantity defined by the ratio of the “effective propagation length per tick” to the “tick time”. \[c_{\mathrm{ref}} := \frac{\ell_{\mathrm{eff}}}{\Delta t}. \label{eq:S11_03_cref_as_ratio}\] Substituting [eq:S11_03_leff_def] into [eq:S11_03_cref_as_ratio] gives \[c_{\mathrm{ref}} = \frac{A\cdot a}{\Delta t}. \label{eq:S11_03_cref_sub}\] Solving [eq:S11_03_cref_sub] for \(\Delta t\) yields \[\boxed{ \Delta t = \frac{A\cdot a}{c_{\mathrm{ref}}} }. \label{eq:S11_03_dt_formula}\] Equation [eq:S11_03_dt_formula] is the derived result of this section, completed by combining definitions [eq:S11_03_cref_as_ratio] and [eq:S11_03_leff_def].

13.3.4 11.3.4 Numerical substitution (fully expanded)

13.3.4.1 11.3.4.1 Computing \(\ell_{\mathrm{eff}}=A\cdot a\)

Substitute [eq:S11_03_A_lock] and [eq:S11_03_a_lock] into [eq:S11_03_leff_def]. \[\begin{aligned} \ell_{\mathrm{eff}} &=A\cdot a \notag\\ &= \left( 880918.97770344000000074873389538365909152024492565003100802687690543842580063599 \right) \left( 6.3299121257859865746\times 10^{-19}\ \mathrm{m} \right) \notag\\ &= 5.5761397188\times 10^{-13}\ \mathrm{m}. \label{eq:S11_03_leff_value}\end{aligned}\]

13.3.4.2 11.3.4.2 Computing \(\Delta t=\ell_{\mathrm{eff}}/c_{\mathrm{ref}}\)

Substitute [eq:S11_03_leff_value] and [eq:S11_03_cref_lock] into [eq:S11_03_dt_formula]. \[\begin{aligned} \Delta t &= \frac{\ell_{\mathrm{eff}}}{c_{\mathrm{ref}}} = \frac{5.5761397188\times 10^{-13}\ \mathrm{m}}{299\,792\,458\ \mathrm{m/s}} \notag\\ &= 1.86\times 10^{-21}\ \mathrm{s}. \label{eq:S11_03_dt_value}\end{aligned}\] Therefore the realized time tick is fixed as \[\boxed{ \Delta t = 1.86\times 10^{-21}\ \mathrm{s} }. \label{eq:S11_03_dt_box}\]

13.3.5 11.3.5 Linkage to an error budget (sensitivity)

Since \(\Delta t\) is defined by [eq:S11_03_dt_formula], the first-order sensitivity to changes in \((A,a,c_{\mathrm{ref}})\) is obtained by differentiation. \[\Delta t=\frac{A a}{c_{\mathrm{ref}}} \quad\Longrightarrow\quad \frac{d(\Delta t)}{\Delta t} = \frac{dA}{A} + \frac{da}{a} - \frac{dc_{\mathrm{ref}}}{c_{\mathrm{ref}}}. \label{eq:S11_03_rel_sens}\] Therefore a first-order upper bound on the absolute error is recorded as \[\left|\Delta(\Delta t)\right| \le \Delta t\left( \left|\frac{\Delta A}{A}\right| + \left|\frac{\Delta a}{a}\right| + \left|\frac{\Delta c_{\mathrm{ref}}}{c_{\mathrm{ref}}}\right| \right). \label{eq:S11_03_abs_error_bound}\] If a variance-type budget is recorded under an independence assumption, the following must be locked in analysis_lock as the “error-budget selection rule.” \[\left(\frac{\sigma_{\Delta t}}{\Delta t}\right)^2 = \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_a}{a}\right)^2 + \left(\frac{\sigma_{c_{\mathrm{ref}}}}{c_{\mathrm{ref}}}\right)^2. \label{eq:S11_03_var_budget}\] Here \(\sigma_A,\sigma_a,\sigma_{c_{\mathrm{ref}}}\) are standard uncertainties of each input, and the estimation rules (which logs/windows/repetitions are used) must be locked in analysis_lock. In particular, in a version where \(a\) is sealed as a value in realization_lock and \(c_{\mathrm{ref}}\) is sealed as an operational anchor, the error budget in that version may be dominated by the production procedure for \(A\); any decision (e.g., treating certain terms as zero) is permitted only under a pre-registered protocol.

13.4 11.4 \(\mathrm{RCROSS}(633/532)\): a 3-Tier Gate

13.4.1 11.4.1 Purpose

This section defines the cross-validation \(\mathrm{RCROSS}\) using two baseline channels (633, 532) as a 3-Tier Gate, and fixes (i) the deviation metric \(\mathrm{dev}\), (ii) the tolerance threshold dev_max, and (iii) failure-mode labeling. This section does not justify cross-consistency; it only defines the PASS/FAIL/INCONCLUSIVE adjudication rules.

13.4.2 11.4.2 Inputs (LOCK): two channels and channel-wise realization candidates

Define the two baseline channels as \[\mathcal{A}:=\{A_{633},A_{532}\}. \label{eq:S11_04_channels}\] Each channel must output a realization candidate \((a^{(k)},\Delta t^{(k)})\), or at least a realized velocity scale factor \(\Lambda_v^{(k)}=a^{(k)}/\Delta t^{(k)}\). Channel-wise artifacts must include at least the following fields.

  1. channel_id: A633 or A532.

  2. lambda_ref: the baseline value (with a length unit).

  3. N: split integer (dimensionless).

  4. a_cand: candidate \(a^{(k)}\) (or a common \(a\) if N is common).

  5. dt_cand: candidate \(\Delta t^{(k)}\).

  6. lock_refs: canon_lock_id, realization_lock_id, analysis_lock_id.

If any field is missing, the channel artifact is undefined and INCONCLUSIVE.

13.4.3 11.4.3 Definition of the deviation metric (dev)

The core of cross-consistency is whether “two channels support the same realized result.” Define the deviation metric as follows.

13.4.3.1 11.4.3.1 Relative deviation dev

Let \(\Delta t^{(k)}\) be the candidate time tick of channel \(k\in\{633,532\}\). Define the relative deviation by \[\mathrm{dev} := \left|\frac{\Delta t^{(633)}-\Delta t^{(532)}}{\frac{1}{2}\left(\Delta t^{(633)}+\Delta t^{(532)}\right)}\right|. \label{eq:S11_04_dev_def}\] If the denominator can be zero, the metric is undefined, hence require positivity of both candidates. \[\Delta t^{(633)}>0,\qquad \Delta t^{(532)}>0. \label{eq:S11_04_dt_positive}\] If [eq:S11_04_dt_positive] is violated, it is an immediate FAIL-RCROSS-NONPOS.

13.4.3.2 11.4.3.2 Alternative deviation (scale-factor based; optional)

If a channel provides only the realized scale factor \(\Lambda_v^{(k)}=a^{(k)}/\Delta t^{(k)}\) instead of \(\Delta t\), define the alternative deviation by \[\mathrm{dev}_{\Lambda} := \left|\frac{\Lambda_v^{(633)}-\Lambda_v^{(532)}}{\frac{1}{2}\left(\Lambda_v^{(633)}+\Lambda_v^{(532)}\right)}\right|. \label{eq:S11_04_devL_def}\] Which deviation is used (time-based dev or scale-based dev\(_\Lambda\)) must be pre-registered in analysis_lock and cannot be switched after seeing results.

13.4.4 11.4.4 Definition of the 3-Tier Gate

The 3-Tier Gate adjudicates in the order definability \(\rightarrow\) consistency pass \(\rightarrow\) strengthening (additional consistency). Each Tier outputs an independent PASS/FAIL/INCONCLUSIVE.

13.4.4.1 11.4.4.1 Tier-1: input/definition completeness Gate

Tier-1 judges whether “the comparison itself is well-defined.”

13.4.4.2 Tier-1 PASS condition

Tier-1=PASS if all of the following are satisfied.

  1. All required fields exist in both channel artifacts.

  2. lock_refs exists, and the two channels belong to the same lock_id combination (or to a pre-registered allowed combination).

  3. \(\Delta t^{(633)},\Delta t^{(532)}\) (or \(\Lambda_v^{(633)},\Lambda_v^{(532)}\)) are both defined and positive.

13.4.4.3 Tier-1 INCONCLUSIVE condition

If there are missing fields, missing lock_refs, possible zero denominators, or any undefined values, then Tier-1=INCONCLUSIVE.

13.4.4.4 Tier-1 FAIL condition

If traces of post hoc modifications (channel-definition swaps, threshold changes, value substitutions) or lock_id mixing are detected, then Tier-1=FAIL.

13.4.4.5 11.4.4.2 Tier-2: dev-threshold pass Gate (core)

Tier-2 judges whether the deviation metric is within the tolerance.

13.4.4.6 Locking the tolerance (dev_max)

Lock the tolerance dev_max as \[\mathrm{dev}_{\max}>0, \qquad \mathrm{dev}_{\max}\ \text{is pre-registered in }\texttt{gate\_lock}. \label{eq:S11_04_devmax_lock}\]

13.4.4.7 Tier-2 adjudication

When Tier-1=PASS and dev is definable, define Tier-2 by \[\texttt{Tier2}= \begin{cases} \texttt{PASS}, & \mathrm{dev}\le \mathrm{dev}_{\max},\\ \texttt{FAIL}, & \mathrm{dev}>\mathrm{dev}_{\max}. \end{cases} \label{eq:S11_04_tier2_rule}\] If Tier-1\(\neq\)PASS, then Tier-2 is INCONCLUSIVE (the comparison is not well-defined).

13.4.4.8 11.4.4.3 Tier-3: strengthening consistency Gate (additional conditions)

Tier-3 is a strengthening Gate that can require “channel-wise internal consistency” even after dev passes. Tier-3 locks either one or both of the following strengthening conditions (selection/combination is locked in analysis_lock).

13.4.4.9 (T3-A) Derived-quantity consistency

Compute a channel-wise set of derived quantities \(\Pi^{(k)}=\{\Pi_1^{(k)},\Pi_2^{(k)},\ldots\}\) under locked definitions, and judge whether relative deviations of derived quantities are within thresholds. The derived-quantity choices and deviation definitions are locked in analysis_lock. The threshold \(\Pi_{\max}\) is locked in gate_lock.

13.4.4.10 (T3-B) Repeat/replay consistency

Judge whether the dev distribution is stable when the same channel is rerun over a pre-registered replay set. The replay set and stability thresholds (e.g., variance bounds) are locked in analysis_lock/gate_lock.

13.4.4.11 Tier-3 adjudication

Tier-3 is evaluated only when Tier-2=PASS. If strengthening conditions are not locked, Tier-3 remains INCONCLUSIVE. If strengthening conditions are locked and a threshold is violated, Tier-3=FAIL.

13.4.5 11.4.5 Final RCROSS Gate (3-Tier composition)

Define the final RCROSS verdict by \[\texttt{G-RCROSS}=\texttt{PASS} \Longleftrightarrow (\texttt{Tier1}=\texttt{PASS})\ \wedge\ (\texttt{Tier2}=\texttt{PASS})\ \wedge\ (\texttt{Tier3}\in\{\texttt{PASS},\texttt{INCONCLUSIVE}\}). \label{eq:S11_04_GRCROSS_def}\] That is, Tier-3 requires PASS when it is locked and evaluated; if it is not locked, it may remain INCONCLUSIVE, but in that case statements like “strengthening consistency passed” are prohibited (restricted by PASS.rules).

13.4.6 11.4.6 Failure-mode labeling (standard labels)

Record RCROSS failures with cause-decomposition labels. Labels are fixed as the following enumeration (multiple labels allowed).

Label Meaning
INCON-RCROSS-MISSING missing required fields (undefined)
INCON-RCROSS-UNLOCK missing lock_refs or missing snapshot sealing (undefined)
FAIL-RCROSS-NONPOS non-positive \(\Delta t\) or \(\Lambda_v\) (definition violated)
FAIL-RCROSS-LOCKMIX mixing different lock_id combinations (lock violated)
FAIL-RCROSS-DEV \(\mathrm{dev}>\mathrm{dev}_{\max}\) (core inconsistency)
FAIL-RCROSS-T3 Tier-3 strengthening condition violated
FAIL-RCROSS-RETRO post hoc modification detected (channel/threshold/definition swap)

The RCROSS verdict must generate and seal the following record.

rcross_report:
  - rcross_id: (unique)
    channels: [A633, A532]
    dt_633: ...
    dt_532: ...
    dev: ...
    dev_max: ...
    tier1: PASS|FAIL|INCONCLUSIVE
    tier2: PASS|FAIL|INCONCLUSIVE
    tier3: PASS|FAIL|INCONCLUSIVE
    verdict: PASS|FAIL|INCONCLUSIVE
    labels: [...]
    lock_refs: {canon_lock_id, realization_lock_id, analysis_lock_id, gate_lock_id, protocol_lock_id}
    manifest_ref: ...
    checksums_ref: ...

A report without manifest_ref and checksums_ref does not grant conclusion status.

13.5 11.5 (Integrated appendix) MMS Operational Anchor format

13.5.1 11.5.1 Purpose

This section provides a format definition and a correspondence table to attach the Operational Anchor format used in MMS documents into this theory’s realization_lock/analysis_lock/gate_lock system. This section fixes (i) the required fields of MMS operational-anchor records, and (ii) how the v4 RCROSS(633/532) and threshold items correspond to the 3-Tier Gate fields of this manuscript. This section does not perform interpretation.

13.5.2 11.5.2 Standard format of MMS Operational Anchor records (required fields)

An MMS operational-anchor record must contain the following fields as required. If a field is missing, the anchor is unusable and INCONCLUSIVE.

  1. anchor_id: anchor identifier (e.g., A633, A532).

  2. anchor_type: length or time or velocity, etc. (this manuscript is length-centered).

  3. value: numeric value.

  4. unit: unit notation (nm, m, s, m/s, etc.).

  5. channel: channel description (e.g., baseline, instrument, mode).

  6. scope: identifier of the applicable regime/experimental conditions.

  7. protocol_id: measurement/preprocessing/log-schema identifier.

  8. lock_refs: canon_lock_id, realization_lock_id, analysis_lock_id, gate_lock_id, protocol_lock_id.

  9. artifacts: list of raw/preprocessed/report file paths.

  10. hash_refs: manifest/checksums reference keys.

After including these fields, MMS records must be sealed by inclusion into registry/realization_lock.* or snapshot/registry_snapshot/ according to this manuscript’s registry structure.

13.5.3 11.5.3 v4 RCROSS/threshold \(\leftrightarrow\) the main-text 3-Tier Gate correspondence table

The table below fixes how RCROSS and threshold-type items that appear in v4 documents (including MMS) correspond to the 3-Tier Gate and registry fields of this manuscript. Because the “v4 item name” can differ in spelling/keys, the correspondence is by meaning; the actual key-string mapping is locked separately as a mapping table in protocol_lock.

v4/MMS item (meaning) Location in this manuscript Protocol/judgment
Baseline 633 channel (wavelength/length) realization_lock.anchors[A633].lambda_ref Tier-1 required input; includes unit/scope/protocol
Baseline 532 channel (wavelength/length) realization_lock.anchors[A532].lambda_ref Tier-1 required input; includes unit/scope/protocol
Split integer \(N\) (e.g., \(10^{12}\)) analysis_lock.anchor_split.N post hoc change prohibited; changes only via versioning
Channel-wise candidate \(\Delta t^{(633)}\) outputs/derived/dt_633.txt + rcross_report.dt_633 Tier-1 definability; Tier-2 dev input
Channel-wise candidate \(\Delta t^{(532)}\) outputs/derived/dt_532.txt + rcross_report.dt_532 Tier-1 definability; Tier-2 dev input
Definition of cross deviation (dev) analysis_lock.rcross.dev_definition lock either dev or dev\(_\Lambda\)
Tolerance dev_max gate_lock.rcross.dev_max Tier-2 core threshold; dev\(>\)dev_max yields FAIL-RCROSS-DEV
3-Tier structure (define/consistency/strengthen) analysis_lock.rcross.tiers fixes Tier-1/2/3 protocols
Strengthening condition (derived-quantity comparison) analysis_lock.rcross.tier3.derivatives when locked, run Tier-3; violation yields FAIL-RCROSS-T3
Strengthening condition (replay consistency) analysis_lock.rcross.tier3.replay_set if replay set not locked, Tier-3=INCONCLUSIVE
thresholds.yaml (threshold file) overall gate_lock + configs/thresholds.yaml thresholds are pre-registered; the file itself is a seal target
PASS/FAIL judgment logs outputs/gates/rcross_report.json without manifest+checksums sealing, no conclusion status
Bonferroni/CI cross rules (if present) analysis_lock.rcross.stat_rule allowed only as Tier-3 strengthening conditions

13.5.4 11.5.4 Protocol for a mapping table (key-string correspondence)

To link the actual key strings used in v4/MMS to the registry keys of this manuscript, a “key mapping table” is required; it is locked in protocol_lock. The standard format is fixed as

key_map_mms_to_v5:
  - mms_key: "lambda_633"
    v5_path: "registry/realization_lock.anchors[A633].lambda_ref.value"
  - mms_key: "lambda_532"
    v5_path: "registry/realization_lock.anchors[A532].lambda_ref.value"
  - mms_key: "dev_max"
    v5_path: "registry/gate_lock.rcross.dev_max"
  - mms_key: "thresholds_yaml"
    v5_path: "configs/thresholds.yaml"

The key mapping table cannot be modified after seeing results; modifications are permitted only via versioning.

13.5.5 11.5.5 manifest/checksums/registry_snapshot

For MMS records and correspondence tables to have conclusion status, the following are required.

  1. MMS records (anchors, thresholds, rcross report) must all be listed in manifest.

  2. All related files must be included in checksums and sealed with sha256 hashes.

  3. The lock files used must be frozen into registry_snapshot.

If any requirement is missing, the RCROSS verdict does not grant conclusion status.

14 12. Electron 1-second cross-check

Purpose (core declaration of time consistency)

This chapter is fixed as the key chapter that tests whether the realized time tick \(\Delta t\) and the event rates (the canonical electron/proton event rates) are mutually consistent within a single time system. In this chapter, “electron 1 second” is not an external-text definition; it is defined as an operational time interval generated by combining internal items of this document: (i) the canonical electron event rate \(\nu_{e,\mathrm{can}}:=1\) (9.3), (ii) the realized time tick \(\Delta t\) (11.3), and (iii) the event-aggregation protocol (9.1–9.2). Accordingly, the cross-check of this chapter is the final gate for time-system consistency. If the Gate of this chapter does not PASS, then all time-based conclusions derived from \(\Delta t\) (build time, propagation, time-realized mass/force, etc.) do not have conclusion status.

Inputs (LOCK) and linkage positions

The locked inputs referenced by this chapter are fixed as the following four items.

  1. Canonical electron event rate: \[\nu_{e,\mathrm{can}}:=1. \label{eq:S12_00_nue_lock}\]

  2. Canonical proton event rate: \[\nu_{p,\mathrm{can}}\ \text{(the numerical value locked in 9.4)}. \label{eq:S12_00_nup_lock}\]

  3. Realized length and time: \[a\ \text{(locked in 11.2)},\qquad \Delta t\ \text{(locked in 11.3)}. \label{eq:S12_00_real_lock}\]

  4. Rectification constant: \[\delta=\frac{1}{\pi^2}\ \text{(locked in the universal regime)}. \label{eq:S12_00_delta_lock}\]

This chapter does not modify the above inputs; they are used only via lock_id references. Any post hoc change of the inputs violates No-Tuning and is prohibited.

Status of “electron 1 second” (operational time interval)

This chapter declares that it treats “electron 1 second” as the following operational definition.

  1. Choose a standard time window (a tick window) length \(\Delta N_{1s}\) for electron-event aggregation.

  2. The realized time corresponding to \(\Delta N_{1s}\) is computed as \(\Delta T_{1s}:=\Delta N_{1s}\Delta t\).

  3. “Electron 1 second” has meaning only when the event-rate definition and the aggregation protocol are maintained within the same version.

Therefore, “electron 1 second” is fixed not as an external definition of the second, but as an operational quantity for cross-checking the internal consistency of the definitions in this document.

14.1 12.4 Core of the cross-check: simultaneous consistency of time–event–structure

The cross-check of this chapter is declared as simultaneous consistency across the following three axes.

  1. Time axis: whether \(\Delta t\) was sealed by passing the realization map and the RCROSS Gate.

  2. Event axis: whether the electron/proton event definitions and canonical event rates are aggregated under the same protocol.

  3. Structure axis: whether the 82+7 structure and the 3-sector integerization do not contradict the counting protocol required by event aggregation.

If any one axis collapses, time-system consistency does not hold, and this chapter must be judged as FAIL or INCONCLUSIVE.

14.2 12.5 Gate declaration (slot for the final pass condition)

This chapter declares that it defines the “electron 1-second cross-check Gate” as the final Gate. \[\texttt{G-E1S} \in \{\texttt{PASS},\texttt{FAIL},\texttt{INCONCLUSIVE}\}. \label{eq:S12_00_GE1S}\] The concrete judgment rule for G-E1S (e.g., self-consistency of the 1-second window computed from \(\Delta t\) and event rates, log completeness, sensitivity/error budget, falsification triggers) is completed in the subsequent sections of this chapter. In this overview we fix only that G-E1S is the core Gate for time consistency, and that time-based conclusions lose conclusion status when G-E1S\(\neq\)PASS.

14.3 12.1 Cell/VP volume model + \(\phi_{\mathrm{jam}}\)

14.3.1 12.1.1 Purpose

This section fixes loggable operational definitions for “cell volume,” “effective occupied volume of a volume particle (VP),” and “jamming occupancy \(\phi_{\mathrm{jam}}\)” used in the electron 1-second cross-check. The deliverables of this section are: (i) canonical cell volume \(V_{\mathrm{cell}}\), (ii) VP effective occupied volume \(v_{\mathrm{vp}}\), (iii) VP count inside a cell \(N_{\mathrm{vp}}\), (iv) the definition of jamming occupancy \(\phi_{\mathrm{jam}}\), and (v) rules that treat definition collisions/mixing as immediate FAIL.

14.3.2 12.1.2 Canonical cell-volume model (cube cell)

The canonical cell is locked as CELL-CUBE. The representative cell length \(D_{\mathrm{anch}}\) is locked with the meaning edge length (edge). Define the canonical cell domain as \[\mathcal{D}_{\square} := \left\{\mathbf{x}\in\mathbb{R}^3\ \big|\ 0\le x< D_{\mathrm{anch}},\ 0\le y< D_{\mathrm{anch}},\ 0\le z< D_{\mathrm{anch}}\right\} \label{eq:S12_01_cell_domain}\] and define the canonical cell volume as \[V_{\mathrm{cell}} :=|\mathcal{D}_{\square}| = D_{\mathrm{anch}}^{3}. \label{eq:S12_01_Vcell}\] The definition [eq:S12_01_Vcell] may be used only when the cell geometry (CELL-CUBE) and the meaning of \(D_{\mathrm{anch}}\) (edge) are locked.

14.3.3 12.1.3 VP \(v_{\mathrm{vp}}\)

Because VP is locked by the Stone axiom (volume invariance, non-penetration), its occupied volume does not change. However, in coordinate/graph based models, instead of directly integrating each occupied region \(\Omega_i\), we must allow count-based volume aggregation by modeling an effective occupied volume using a single length scale \(a\). For this purpose we define and lock VP effective occupied volume as the following volume-model closure.

14.3.3.1 12.1.3.1 Standard form of the effective occupied volume

Define \[v_{\mathrm{vp}} :=\kappa_{\mathrm{vp}}\,a^{3}, \qquad \kappa_{\mathrm{vp}}>0. \label{eq:S12_01_vvp_general}\] Here

\(\kappa_{\mathrm{vp}}\) cannot be tuned after seeing results; it is allowed only as a closure item locked in analysis_lock.

14.3.3.2 12.1.3.2 Canonical VP volume closure (spherical kernel)

This whitepaper adopts an “isotropic core (spherical kernel)” as the canonical volume model, to remove shape degrees of freedom. Lock \[\kappa_{\mathrm{vp}} :=\frac{\pi}{6} \qquad (\text{canonical volume closure}). \label{eq:S12_01_kappa_sphere}\] Therefore the canonical VP effective occupied volume is fixed as \[v_{\mathrm{vp}} =\frac{\pi}{6}a^{3}. \label{eq:S12_01_vvp_sphere}\] The purpose of the canonical volume closure is to (i) make VP occupied volume depend on \(a\) alone (SSOT), (ii) block post hoc tuning paths through shape freedom, and (iii) make volume aggregation reproducible as a unique count-based quantity. Changing this canonical closure requires a version update and full re-validation.

14.3.4 12.1.4 Operational definition of the VP count inside a cell \(N_{\mathrm{vp}}\)

Volume aggregation is undefined unless the inclusion rule “which VP is counted as inside the cell” is locked. This section defines a single protocol for the inclusion rule.

14.3.4.1 12.1.4.1 Inclusion rule (center-in-cell)

Fix a representative point (center) \(\mathbf{x}_i\) for each VP \(i\) by a locked protocol. Define the inclusion indicator \[\mathbf{1}_{\mathrm{in}}(i) := \begin{cases} 1,& \mathbf{x}_i\in \mathcal{D}_{\square},\\ 0,& \text{otherwise}. \end{cases} \label{eq:S12_01_indicator_in}\] and define the VP count inside the cell as \[N_{\mathrm{vp}} := \sum_{i\in\mathcal{V}}\mathbf{1}_{\mathrm{in}}(i). \label{eq:S12_01_Nvp}\] Handling of points exactly on the boundary (faces/edges/vertices) must be locked by a pre-registered tie-break rule; it cannot be changed after seeing results.

14.3.4.2 12.1.4.2 Required log fields

For \(N_{\mathrm{vp}}\) to have conclusion status, the following logs must be sealed.

  1. Cell definition: \(\mathcal{D}_{\square}\) and the meaning of \(D_{\mathrm{anch}}\) (edge).

  2. Representative-point protocol: what \(\mathbf{x}_i\) means (coordinates, markers, etc.) and how identity is preserved.

  3. Inclusion rule: implementation of [eq:S12_01_indicator_in] including the boundary tie-break.

  4. Hashes and snapshot references for coordinate files (manifest/checksums).

If missing, \(N_{\mathrm{vp}}\) is INCONCLUSIVE.

14.3.5 12.1.5 Definition of jamming occupancy \(\phi_{\mathrm{jam}}\)

14.3.5.1 12.1.5.1 Occupancy (volume fraction) definition

Define \[\phi_{\mathrm{jam}} := \frac{N_{\mathrm{vp}}\,v_{\mathrm{vp}}}{V_{\mathrm{cell}}}. \label{eq:S12_01_phijam_def}\] In [eq:S12_01_phijam_def], \(V_{\mathrm{cell}}\) references [eq:S12_01_Vcell], \(v_{\mathrm{vp}}\) references [eq:S12_01_vvp_sphere], and \(N_{\mathrm{vp}}\) references [eq:S12_01_Nvp], each via locked definitions.

14.3.5.2 12.1.5.2 Measurement conditions in the jammed regime (regime conditions)

Because the name \(\phi_{\mathrm{jam}}\) means “the occupancy recorded in a jammed state,” we record \(\phi_{\mathrm{jam}}\) only when the following regime conditions all hold.

  1. Propagation/rigidity switch: \(\chi_{\mathrm{ST}}=1\).

  2. Non-penetration and successful relaxation: the coordinate/placement does not violate non-penetration and the locked relaxation procedure terminates with its stop condition satisfied.

  3. Locked cell/representative-point/inclusion rule: [eq:S12_01_cell_domain] and [eq:S12_01_indicator_in] are maintained under the same lock_id.

If any condition fails, \(\phi_{\mathrm{jam}}\) cannot be recorded as “jamming occupancy” and only a limiting conclusion (CT-LIM) is allowed.

14.3.6 12.1.6 Why the canonical choices are fixed (SSOT/No-Tuning)

The canonical fixings of this section are locked for the following reasons.

  1. Cell volume \(V_{\mathrm{cell}}\) is uniquely determined only when CELL-CUBE and the meaning of \(D_{\mathrm{anch}}\) (edge) are locked. Mixing cell geometries would allow arbitrary changes of \(\phi_{\mathrm{jam}}\), so it is prohibited.

  2. VP occupied volume must be determined by \(a\) alone. If the shape freedom (\(\kappa_{\mathrm{vp}}\)) is released, then \(\phi_{\mathrm{jam}}\) becomes tunable after the fact. Therefore we seal the canonical closure [eq:S12_01_kappa_sphere].

  3. The inclusion rule (center-in-cell) and the boundary tie-break determine the count \(N_{\mathrm{vp}}\); changing rules changes \(\phi_{\mathrm{jam}}\) retroactively. Therefore the inclusion rule must be pre-registered and post hoc changes are prohibited.

14.3.7 12.1.7 Immediate FAIL conditions (confusion/post hoc tuning)

The following violations are treated as immediate FAIL.

  1. Mixing cell geometries (mixing CELL-CUBE and visualization spherical cells) or mixing the meaning of \(D_{\mathrm{anch}}\) (edge/diameter/radius).

  2. Using \(a\) as if it were a radius, or post hoc changing \(\kappa_{\mathrm{vp}}\) in \(v_{\mathrm{vp}}\) (or redefining it in another form).

  3. Post hoc change of the inclusion rule or the boundary tie-break, or selectively reporting only favorable runs to report \(N_{\mathrm{vp}}\).

  4. Mixing lock_ids or using coordinates/graphs/counts without sealing (manifest/checksums/registry_snapshot).

14.4 12.2 “Electron 1 second” reconstruction formulas

14.4.1 12.2.1 Locked inputs and reference definitions

This section assumes the following inputs are locked.

  1. Realized time tick (unit: s): \[\Delta t \ \text{(locked)}. \label{eq:S12_02_dt_lock}\]

  2. Canonical electron event rate (definition): \[\nu_{e,\mathrm{can}} := 1. \label{eq:S12_02_nue_can}\]

  3. Rectification constant (locked in the universal regime): \[\delta := \frac{1}{\pi^2}. \label{eq:S12_02_delta_lock}\]

  4. Required fields of event logs (operational definition of 9.1): each event \(e\) must at least have \((n(e),\theta(e),\varphi(e))\) and \(\theta(e),\varphi(e)\in[0,2\pi)\) must be definable.

Define the half-wave rectifier and survival weight as \[_+ := \max(0,x), \qquad w(e) := [\cos\theta(e)]_+\,[\cos\varphi(e)]_+. \label{eq:S12_02_w_def}\]

14.4.2 12.2.2 Tick-based reconstruction of the “1 second” window (integer tick window)

14.4.2.1 [D-12.2-1] Realized time and ticks

Define the realized time corresponding to a tick \(n\in\mathbb{Z}\) as \[t := n\,\Delta t. \label{eq:S12_02_time_map}\]

14.4.2.2 [D-12.2-2] Rule for choosing the 1-second tick-window length

Define the integer tick length corresponding to the realized time \(1\,\mathrm{s}\) by the rule \[\Delta N_{1\mathrm{s}} := \left\lfloor \frac{1\,\mathrm{s}}{\Delta t}\right\rfloor. \label{eq:S12_02_DN1s}\] Thus the reconstructed “1 second” duration under integer ticks is \[\widehat{T}_{\mathrm{tick}}(1\mathrm{s}) := \Delta N_{1\mathrm{s}}\,\Delta t. \label{eq:S12_02_Ttick}\]

14.4.2.3 [D-12.2-3] Residual (rounding error) definition

By definition [eq:S12_02_DN1s], the residual is fixed as \[\varepsilon_{\mathrm{tick}} := 1\,\mathrm{s}-\widehat{T}_{\mathrm{tick}}(1\mathrm{s}) = 1\,\mathrm{s}-\Delta N_{1\mathrm{s}}\Delta t, \label{eq:S12_02_eps_tick}\] and by the floor-function property, \[0 \le \varepsilon_{\mathrm{tick}} < \Delta t. \label{eq:S12_02_eps_bound}\] Equation [eq:S12_02_eps_bound] is the deterministic bound that holds when using an integer tick window.

14.4.3 12.2.3 Electron event-count definitions: raw/rectified counts

14.4.3.1 [D-12.2-4] Tick window and the set of attempted electron events

Define an arbitrary tick window \(W[n_1,n_2)\) as \[W[n_1,n_2) := \{\,n\in\mathbb{Z}\mid n_1\le n<n_2\,\}, \qquad \Delta N:=n_2-n_1, \qquad \Delta T := \Delta N\,\Delta t. \label{eq:S12_02_window}\] Define the set of attempted electron events as \[\mathcal{E}_{0,e}[n_1,n_2) := \{\,e\mid n_1\le n(e)<n_2,\ \mathrm{Trig}_{0,e}(e)=1\,\}. \label{eq:S12_02_E0e}\] where \(\mathrm{Trig}_{0,e}\) is the electron-attempt trigger and is locked in analysis_lock. Define the raw electron event count as \[N_{0,e}[n_1,n_2) :=\left|\mathcal{E}_{0,e}[n_1,n_2)\right|. \label{eq:S12_02_N0e}\]

14.4.3.2 [D-12.2-5] Rectified (surviving) electron event count

Define the rectified electron event count as \[N_{e}[n_1,n_2) := \sum_{e\in\mathcal{E}_{0,e}[n_1,n_2)} w(e) = \sum_{e\in\mathcal{E}_{0,e}[n_1,n_2)} [\cos\theta(e)]_+[\cos\varphi(e)]_+. \label{eq:S12_02_Ne}\] By definition, \[0\le N_{e}[n_1,n_2)\le N_{0,e}[n_1,n_2). \label{eq:S12_02_bounds_counts}\]

14.4.4 12.2.4 “Electron 1 second” reconstruction (two forms: tick/event)

This section reconstructs and fixes “electron 1 second” in two equivalent forms (equivalence holds only when Gates pass).

14.4.4.1 12.2.4.1 Tick-based reconstruction

The tick-based “electron 1 second” is the time duration defined by [eq:S12_02_Ttick]. \[T_{e,1\mathrm{s}}^{(\mathrm{tick})} :=\widehat{T}_{\mathrm{tick}}(1\mathrm{s}) =\Delta N_{1\mathrm{s}}\Delta t. \label{eq:S12_02_e1s_tick}\]

14.4.4.2 12.2.4.2 Event-based reconstruction

Because the canonical electron event rate is locked by [eq:S12_02_nue_can], define the event-based reconstruction of time as \[T_{e}^{(\mathrm{event})}[n_1,n_2) := \frac{N_{e}[n_1,n_2)}{\nu_{e,\mathrm{can}}} = N_{e}[n_1,n_2). \label{eq:S12_02_Tevent_def}\] Thus fix the event-based reconstruction of “electron 1 second” as \[T_{e,1\mathrm{s}}^{(\mathrm{event})} := N_{e}[n_1,n_2) \quad\text{where}\quad (n_2-n_1)=\Delta N_{1\mathrm{s}}. \label{eq:S12_02_e1s_event}\] That is, under the same tick-window length \(\Delta N_{1\mathrm{s}}\), the rectified electron event count becomes the event-based “1 second” value.

14.4.5 12.2.5 Expectations (mean relations under the canonical stationarity axiom)

In the regime where the canonical stationarity axiom holds (9.2 [A-9.2-S1]), this section fixes the following expectation relations.

14.4.5.1 12.2.5.1 Expected rectified event count

By definition, the canonical event rate is \[\nu_{e,\mathrm{can}} = \lim_{\Delta T\to\infty}\frac{N_{e}}{\Delta T}. \label{eq:S12_02_rate_limit}\] Therefore, in the canonical stationarity regime we record \[\mathbb{E}\!\left[N_{e}[n_1,n_2)\right] = \nu_{e,\mathrm{can}}\ \Delta T, \label{eq:S12_02_ENe}\] where \(\Delta T=(n_2-n_1)\Delta t\). Substituting \(\nu_{e,\mathrm{can}}=1\) yields \[\mathbb{E}\!\left[N_{e}[n_1,n_2)\right] = \Delta T. \label{eq:S12_02_ENe_simple}\]

14.4.5.2 12.2.5.2 Expectation in a 1-second window

In particular, for a window with \(\Delta T=1\,\mathrm{s}\), \[\mathbb{E}\!\left[N_{e}(1\mathrm{s})\right] = \nu_{e,\mathrm{can}}\cdot 1\,\mathrm{s} = 1. \label{eq:S12_02_ENe_1s}\] When using an integer tick window, \(\Delta T=\widehat{T}_{\mathrm{tick}}(1\mathrm{s})\), hence \[\mathbb{E}\!\left[N_{e}^{(\mathrm{tick})}\right] = \widehat{T}_{\mathrm{tick}}(1\mathrm{s}) = \Delta N_{1\mathrm{s}}\Delta t. \label{eq:S12_02_ENe_tick}\]

14.4.5.3 12.2.5.3 Expected raw count (reference relation)

Applying the canonical event-rate law \(\nu_{\mathrm{can}}=s\cdot\delta\) to the electron gives \[\nu_{e,\mathrm{can}}=s_e\cdot\delta, \qquad \nu_{e,\mathrm{can}}=1 \ \Longrightarrow\ s_e=\frac{1}{\delta}. \label{eq:S12_02_se}\] Therefore the expected raw event count is \[\mathbb{E}\!\left[N_{0,e}[n_1,n_2)\right] = s_e\,\Delta T = \frac{\Delta T}{\delta}. \label{eq:S12_02_EN0e}\] In the universal regime where \(\delta=1/\pi^2\) is locked, \[\mathbb{E}\!\left[N_{0,e}(1\mathrm{s})\right] = \frac{1}{\delta} = \pi^2. \label{eq:S12_02_EN0e_1s}\] Equation [eq:S12_02_EN0e_1s] represents the expected number of raw (attempt) electron events in a 1-second window; it cannot be used in a regime where the universality of \(\delta\) is broken.

14.5 12.3 \(\phi_{\mathrm{jam}}\) measurement, LOCK, error budget, and falsification triggers

14.5.1 12.3.1 Purpose

This section fixes a protocol to (i) measure the jamming occupancy \(\phi_{\mathrm{jam}}\) as a loggable operational definition, (ii) seal it by LOCK so it does not change within the same version, and (iii) connect sensitivity (error budget) and (iv) falsification triggers (breaking conditions) to Gates for judgment.

14.5.2 12.3.2 Definition: standard formula for \(\phi_{\mathrm{jam}}\) (single source)

14.5.2.1 12.3.2.1 Canonical cell volume

The canonical cell is locked as CELL-CUBE and \(D_{\mathrm{anch}}\) is locked as edge. Define \[V_{\mathrm{cell}} := D_{\mathrm{anch}}^{3}. \label{eq:S12_03_Vcell}\]

14.5.2.2 12.3.2.2 VP effective occupied volume (canonical closure)

The realized length scale \(a\) is locked as the VP diameter (diameter). Define \[v_{\mathrm{vp}} := \kappa_{\mathrm{vp}}\,a^{3}, \qquad \kappa_{\mathrm{vp}}:=\frac{\pi}{6}. \label{eq:S12_03_vvp}\] \(\kappa_{\mathrm{vp}}\) is the canonical volume closure and does not change within the same version.

14.5.2.3 12.3.2.3 VP count inside the cell

Assume the VP representative point (center) \(\mathbf{x}_i\) is defined by a locked protocol. Define the canonical cell domain \[\mathcal{D}_{\square} := \left\{\mathbf{x}\in\mathbb{R}^3\ \big|\ 0\le x< D_{\mathrm{anch}},\ 0\le y< D_{\mathrm{anch}},\ 0\le z< D_{\mathrm{anch}}\right\} \label{eq:S12_03_Dsquare}\] and define the inclusion indicator \[\mathbf{1}_{\mathrm{in}}(i) := \begin{cases} 1,& \mathbf{x}_i\in \mathcal{D}_{\square},\\ 0,& \text{otherwise}. \end{cases} \label{eq:S12_03_indicator}\] Boundary handling (include/exclude) must be locked by a pre-registered tie-break rule. Define \[N_{\mathrm{vp}} := \sum_{i\in\mathcal{V}}\mathbf{1}_{\mathrm{in}}(i). \label{eq:S12_03_Nvp}\]

14.5.2.4 12.3.2.4 Jamming occupancy

Define \[\phi_{\mathrm{jam}} := \frac{N_{\mathrm{vp}}\,v_{\mathrm{vp}}}{V_{\mathrm{cell}}} = \frac{N_{\mathrm{vp}}\left(\frac{\pi}{6}\right)a^{3}}{D_{\mathrm{anch}}^{3}}. \label{eq:S12_03_phijam}\] Equation [eq:S12_03_phijam] is the only source of \(\phi_{\mathrm{jam}}\) in this section; \(\phi_{\mathrm{jam}}\) is not redefined elsewhere.

14.5.3 12.3.3 Measurement procedure (state-snapshot based)

14.5.3.1 12.3.3.1 Set of measurement states

Measurements are performed on the set of complete state snapshots \(\mathcal{S}_{\mathrm{obs}}\): \[\mathcal{S}_{\mathrm{obs}}:=\{\,S[n]\mid n\in W_{\mathrm{obs}}\ \wedge\ \mathrm{Complete}(S[n])=1\,\}, \label{eq:S12_03_Sobs}\] where \(W_{\mathrm{obs}}\) is the observation tick window and \(\mathrm{Complete}(S[n])\) is the state-log completeness judgment (locked by the operational definition of 9.1).

14.5.3.2 12.3.3.2 Computing \(\phi_{\mathrm{jam}}\) per state

For each \(S[n]\in\mathcal{S}_{\mathrm{obs}}\):

  1. Read \(D_{\mathrm{anch}}\) (from canon_lock).

  2. Read \(a\) (from realization_lock).

  3. Read \(\mathbf{x}_i\) from the coordinate set and compute \(\mathbf{1}_{\mathrm{in}}(i)\) via [eq:S12_03_indicator].

  4. Compute \(N_{\mathrm{vp}}(n)\) via [eq:S12_03_Nvp].

  5. Compute \(\phi_{\mathrm{jam}}(n)\) via [eq:S12_03_phijam].

Hence \[\phi_{\mathrm{jam}}(n) = \frac{N_{\mathrm{vp}}(n)\left(\frac{\pi}{6}\right)a^{3}}{D_{\mathrm{anch}}^{3}}. \label{eq:S12_03_phijam_n}\]

14.5.3.3 12.3.3.3 Window aggregation (optional)

The aggregation rule (mean/median/trimmed mean, etc.) is locked in analysis_lock. When choosing mean aggregation, \[\overline{\phi}_{\mathrm{jam}} := \frac{1}{|\mathcal{S}_{\mathrm{obs}}|}\sum_{S[n]\in\mathcal{S}_{\mathrm{obs}}}\phi_{\mathrm{jam}}(n), \qquad |\mathcal{S}_{\mathrm{obs}}|>0. \label{eq:S12_03_phi_mean}\] Define the variance (fluctuation) indicator as \[\sigma_{\phi}^2 := \frac{1}{|\mathcal{S}_{\mathrm{obs}}|}\sum_{S[n]\in\mathcal{S}_{\mathrm{obs}}}\left(\phi_{\mathrm{jam}}(n)-\overline{\phi}_{\mathrm{jam}}\right)^2, \qquad \sigma_{\phi}:=\sqrt{\sigma_{\phi}^2}. \label{eq:S12_03_phi_var}\]

14.5.4 12.3.4 Sensitivity (error budget) definition

This section records two kinds of error sources separately.

  1. Input-scale error: uncertainties assigned to \(D_{\mathrm{anch}}\) and \(a\) (which may be recorded in the corresponding locks).

  2. Count/judgment error: uncertainties from the VP counting procedure, boundary handling, representative-point definition, and inclusion rule (evaluated only by pre-registered judgment protocols).

14.5.4.1 12.3.4.1 Differential-based sensitivity (general form)

From [eq:S12_03_phijam], rewrite \[\phi_{\mathrm{jam}} = \left(\frac{\pi}{6}\right)\,N_{\mathrm{vp}}\left(\frac{a}{D_{\mathrm{anch}}}\right)^3. \label{eq:S12_03_phi_compact}\] Hence \[\frac{\partial \phi_{\mathrm{jam}}}{\partial N_{\mathrm{vp}}}=\frac{\phi_{\mathrm{jam}}}{N_{\mathrm{vp}}}, \qquad \frac{\partial \phi_{\mathrm{jam}}}{\partial a}=3\frac{\phi_{\mathrm{jam}}}{a}, \qquad \frac{\partial \phi_{\mathrm{jam}}}{\partial D_{\mathrm{anch}}}=-3\frac{\phi_{\mathrm{jam}}}{D_{\mathrm{anch}}}. \label{eq:S12_03_sens_partials}\] The first-order relative sensitivity is \[\frac{d\phi_{\mathrm{jam}}}{\phi_{\mathrm{jam}}} = \frac{dN_{\mathrm{vp}}}{N_{\mathrm{vp}}} + 3\frac{da}{a} - 3\frac{dD_{\mathrm{anch}}}{D_{\mathrm{anch}}}. \label{eq:S12_03_rel_sens}\] Because \(\kappa_{\mathrm{vp}}=\pi/6\) is locked as the canonical closure, \(d\kappa_{\mathrm{vp}}=0\) within the same version.

14.5.4.2 12.3.4.2 Count uncertainty (boundary-ambiguous counts)

Pre-register and lock a boundary ambiguity width \(\epsilon_b>0\) (length unit; fixed either in internal or realized units). Define the number of VPs near the boundary as \[N_{\mathrm{amb}} := \#\left\{\, i\ \middle|\ \mathrm{dist}\bigl(\mathbf{x}_i,\partial\mathcal{D}_{\square}\bigr)\le \epsilon_b \right\}. \label{eq:S12_03_Namb}\] Here \(\mathrm{dist}(\cdot,\partial\mathcal{D}_{\square})\) is the minimum distance between the point and the cell boundary; the computation protocol is locked in analysis_lock. Define the worst-case count uncertainty as \[\Delta N_{\mathrm{vp}}^{(\max)} := N_{\mathrm{amb}}. \label{eq:S12_03_dN_worst}\] Then record the worst-case relative error bound as \[\left|\frac{\Delta \phi_{\mathrm{jam}}}{\phi_{\mathrm{jam}}}\right| \le \frac{\Delta N_{\mathrm{vp}}^{(\max)}}{N_{\mathrm{vp}}} + 3\left|\frac{\Delta a}{a}\right| + 3\left|\frac{\Delta D_{\mathrm{anch}}}{D_{\mathrm{anch}}}\right|. \label{eq:S12_03_rel_bound}\] \(\Delta a,\Delta D_{\mathrm{anch}}\) are derived from uncertainties recorded in the corresponding locks or from pre-registered uncertainty protocols. If not recorded, they may be treated as zero (by a chosen protocol), but the choice protocol itself must be pre-registered.

14.5.4.3 12.3.4.3 Between-window variability (drift budget)

Split the observation window into \(M\ge2\) blocks, compute each block mean \(\overline{\phi}_m\), and define the drift metric \[\overline{\phi}_m := \frac{1}{|\mathcal{S}_m|}\sum_{S[n]\in\mathcal{S}_m}\phi_{\mathrm{jam}}(n), \qquad R_\phi:=\frac{\max_m \overline{\phi}_m-\min_m \overline{\phi}_m}{\max(\overline{\phi}_{\mathrm{jam}},\varepsilon_\phi)}. \label{eq:S12_03_drift_metric}\] Here \(\varepsilon_\phi>0\) is a denominator-protection constant locked in analysis_lock. \(R_\phi\) is the “between-window drift” indicator and is connected to a Gate threshold (12.3.5).

14.5.5 12.3.5 Falsification triggers (FAIL conditions) and Gate

This section defines the conditions under which the definition/measurement/sealing system for \(\phi_{\mathrm{jam}}\) breaks, and records them as falsification triggers with FAIL labels.

14.5.5.1 12.3.5.1 Trigger F1: regime violation (not jammed)

Jamming occupancy is recorded only in the jammed regime. If the jamming indicator (rigidity switch) does not satisfy \[\chi_{\mathrm{ST}}\neq 1 \quad\Longrightarrow\quad \texttt{FAIL-PHIJAM-REGIME}, \label{eq:S12_03_fail_regime}\] then it is an immediate failure.

14.5.5.2 12.3.5.2 Trigger F2: non-penetration/relaxation violation

If the coordinate placement violates non-penetration (minimum separation), then the cell/VP volume aggregation is not recognized as jammed. When the minimum separation length \(d_{\min}\) and tolerance \(\varepsilon_{\mathrm{pos}}\) are locked, \[\min_{i<j}\|\mathbf{x}_i-\mathbf{x}_j\| < d_{\min}-\varepsilon_{\mathrm{pos}} \quad\Longrightarrow\quad \texttt{FAIL-PHIJAM-OVERLAP}. \label{eq:S12_03_fail_overlap}\]

14.5.5.3 12.3.5.3 Trigger F3: range violation (impossible values)

By definition [eq:S12_03_phijam], \(\phi_{\mathrm{jam}}\) cannot be negative. Also, abnormally large occupancy relative to cell volume is treated as a definition collision or a count/scale error. Lock an upper threshold \(\phi_{\max}>0\) in gate_lock. \[\phi_{\mathrm{jam}}<0 \ \ \text{or}\ \ \phi_{\mathrm{jam}}>\phi_{\max} \quad\Longrightarrow\quad \texttt{FAIL-PHIJAM-RANGE}. \label{eq:S12_03_fail_range}\]

14.5.5.4 12.3.5.4 Trigger F4: excessive boundary ambiguity width

Lock a boundary-ambiguity ratio threshold \(\eta_{\mathrm{amb}}\in(0,1)\) in gate_lock. \[\frac{N_{\mathrm{amb}}}{\max(N_{\mathrm{vp}},1)} > \eta_{\mathrm{amb}} \quad\Longrightarrow\quad \texttt{FAIL-PHIJAM-AMB}. \label{eq:S12_03_fail_amb}\] This is a falsification trigger meaning that, even with a locked inclusion rule, too many samples lie near the boundary for \(\phi_{\mathrm{jam}}\) to be stably defined.

14.5.5.5 12.3.5.5 Trigger F5: drift (breakdown of steady state)

Lock a drift threshold \(\varepsilon_{\mathrm{drift}}>0\) in gate_lock. \[R_\phi > \varepsilon_{\mathrm{drift}} \quad\Longrightarrow\quad \texttt{FAIL-PHIJAM-DRIFT}. \label{eq:S12_03_fail_drift}\] Equation [eq:S12_03_fail_drift] means that \(\phi_{\mathrm{jam}}\) did not settle into a steady state over the observation interval.

14.5.5.6 12.3.5.6 Trigger F6: lock/sealing violation

If any of the following occurs, it is an immediate failure.

  1. Mixing the meaning of \(D_{\mathrm{anch}}\) (edge) or mixing cell geometry (CELL-CUBE).

  2. Mixing the meaning of \(a\) (diameter) or mixing units.

  3. Post hoc change of the value of \(\kappa_{\mathrm{vp}}\) (or the volume model).

  4. Mixing lock_ids or failing to seal manifest/checksums/registry_snapshot.

Fix the violation label as \[\texttt{FAIL-PHIJAM-LOCK}. \label{eq:S12_03_fail_lock}\]

14.5.5.7 12.3.5.7 \(\phi_{\mathrm{jam}}\) Gate (final judgment)

Define the final Gate as \[\texttt{G-PHIJAM}=\texttt{PASS} \Longleftrightarrow \left( \chi_{\mathrm{ST}}=1 \right) \wedge \left( \texttt{no FAIL triggers (F1--F6)} \right) \wedge \left( \texttt{log/sealing completeness} \right). \label{eq:S12_03_GPHIJAM}\] If G-PHIJAM\(\neq\)PASS, then \(\phi_{\mathrm{jam}}\) cannot be used as evidence for conclusions, and only limiting conclusions (CT-LIM) are allowed.

14.5.6 12.3.6 Reporting protocol (required logs and sealing)

The \(\phi_{\mathrm{jam}}\) result must be recorded and sealed in the following record (schema locked in protocol_lock).

phijam_report:
  - phijam_id: (unique)
    regime_id: ...
    state_window: [n1,n2)
    D_anch: ...
    a: ...
    kappa_vp: pi/6
    V_cell: D_anch^3
    v_vp: (pi/6)*a^3
    N_vp: ...
    N_amb: ...
    phi_jam: ...
    phi_mean: ...          # optional (window aggregation)
    phi_sigma: ...         # optional
    drift_Rphi: ...        # optional
    thresholds:
      phi_max: ...
      eta_amb: ...
      eps_drift: ...
      eps_b: ...
    error_budget:
      rel_bound: ...       # in the form of eq. (S12_03_rel_bound)
      components: {dN_over_N, 3*da_over_a, 3*dD_over_D}
    gate_refs:
      G-PHIJAM: PASS|FAIL|INCONCLUSIVE
      labels: [...]
    lock_refs:
      canon_lock_id: ...
      realization_lock_id: ...
      analysis_lock_id: ...
      gate_lock_id: ...
      protocol_lock_id: ...
    snapshot_refs:
      manifest_ref: ...
      checksums_ref: ...
      registry_snapshot_ref: ...

Records missing manifest_ref/checksums_ref/registry_snapshot_ref cannot be granted conclusion status.

15 13. Mass: \(U_{\mathrm{lat}}\rightarrow m_H/m_p/m_e\)

Purpose (promotion to the main text)

This chapter derives the mass scales \(m_H,m_p,m_e\) from the lattice unit energy \(U_{\mathrm{lat}}\) in the main body. Mass-related derivations that were previously dispersed into appendices (E/P/M/R, etc.)—electron mass, proton mass, Higgs mass, and ratio cross-checks—are promoted into the main text so that the definition–derivation–verification (Gate) flow closes consistently inside the main body without duplication.

Deliverables (definitions/derivations/verification items)

The deliverables of this chapter are fixed as the following five bundles.

  1. Definition of \(U_{\mathrm{lat}}\) and its realization locks (linked to the Chapter 11 realization values \(a,\Delta t\)).

  2. Operational definition of mass (geometric resistance / effective cross-section / integral coefficients) and type-specific mass formulas.

  3. Derivation of \(m_H\) (including the coefficient \(5\pi\)) and numerical evaluation.

  4. Derivation of \(m_p\) (including the link to core radius \(R_p\) and \(\lambda_C\)) and numerical evaluation.

  5. Derivation of \(m_e\) (including the electron canonical event rate / radius \(r_e\)) and numerical evaluation.

For each deliverable, the main text explicitly records (i) which LOCK items it depends on, (ii) which closure(s) are used, and (iii) which Gate(s) must be passed to grant conclusion status.

Inputs (LOCK) and prohibitions (no external justification)

The derivations in this chapter use only the following inputs.

  1. Realization length \(a\) and realization time step \(\Delta t\) (locked in Chapter 11).

  2. Core radius \(R_p\) and selected length \(L_q=\lambda_C\) (locked in Chapter 6).

  3. Stationary constants \(\alpha,\delta\) (locked in Chapter 5) and canonical event rates (locked in Chapter 9).

  4. PASS-qualified outputs of the \(82+7\) discrete structure verification (Chapter 8).

This chapter does not use external texts (equations from other theories, external constant definitions, or external justifications) as grounds for mass derivations. Comparisons to external numerical values are recorded only as Gate metrics; they are not used to justify definitions or derivations.

Scope of promoted missing-gap reinforcements (declaration)

The following missing-gap reinforcements are promoted into the main text in this chapter.

  1. Electron-mass derivation: a derivation that links the electron canonical event rate \(\nu_{e,\mathrm{can}}=1\) and the definition of \(r_e\) to the mass coefficient.

  2. Proton-mass derivation: a derivation that computes the mass scale from an integral coefficient coupled to \(R_p\) and core invariants (\(4/\pi\), etc.).

  3. Higgs-mass derivation: a derivation that computes \(m_H\) from \(U_{\mathrm{lat}}\) and the effective cross-section coefficient (\(5\pi\)).

  4. Ratio cross-checks: a section that defines dimensionless ratio invariants such as \(m_H/m_p\), \(m_p/m_e\), and verifies them using RCROSS and a Gate stack.

These items are no longer split into appendices; each closes by definition–derivation–verification in the main text.

Gate structure (conditions for conclusion status in this chapter)

All numerical conclusions in this chapter (\(m_H,m_p,m_e\) and ratios) require PASS of the following Gate stack.

  1. G-SYM: no symbol/unit/diameter–radius meaning conflicts.

  2. G-LOCK: consistent input lock_id and snapshot sealing.

  3. G-REG: regime consistency (realization regime / canonical-event-rate regime).

  4. G-STR: structural inputs (82+7, cancellation–survival) are in PASS state.

  5. G-RCROSS: realization cross-consistency of \(a,\Delta t\) is PASS (required).

  6. G-REP: reproducibility package reproduces the same results.

  7. G-NT: no violation of the no-post-tuning rule.

If a Gate is not PASS, the result is not promoted to a main-text conclusion; only a limited conclusion (CT-LIM) is permitted.

15.1 13.1 Definition of \(U_{\mathrm{lat}}=hc/a=1958.7\,\mathrm{GeV}\) (single source of truth)

15.1.1 13.1.1 Purpose (single-source declaration)

This section fixes the lattice unit energy \(U_{\mathrm{lat}}\) as a definition and provides the unique source of truth (SSOT) of \(U_{\mathrm{lat}}\) for the whole document. Redefining or re-deriving \(U_{\mathrm{lat}}\) outside this section is forbidden. Outside this section, only referencing [eq:S13_01_Ulat_def] is allowed.

15.1.2 13.1.2 Inputs (LOCK): \(h\), \(c_{\mathrm{ref}}\), \(a\), and unit-conversion constants

\(U_{\mathrm{lat}}\) is defined only when the following four inputs are locked.

15.1.2.1 13.1.2.1 Action-unit constant \(h\) (locked)

\[h = 6.62607015\times 10^{-34}\ \mathrm{J\cdot s}. \label{eq:S13_01_h_lock}\] \(h\) is the action-unit constant used in this document’s unit system; its value/unit must be locked in canon_lock.

15.1.2.2 13.1.2.2 Reference speed constant \(c_{\mathrm{ref}}\) (operational anchor, locked)

\[c_{\mathrm{ref}} = 299\,792\,458\ \mathrm{m/s}. \label{eq:S13_01_cref_lock}\] \(c_{\mathrm{ref}}\) is the operational-anchor constant locked in §11.1; it must be locked in realization_lock together with its channel/scope/protocol.

15.1.2.3 13.1.2.3 Realization length \(a\) (VP diameter, locked)

\[a = 6.3299121257859865746\times 10^{-19}\ \mathrm{m}. \label{eq:S13_01_a_lock}\] \(a\) is the realization length locked in §11.2 and must be locked together with geometry_meaning=diameter.

15.1.2.4 13.1.2.4 Energy-unit conversion constant (locked)

To use \(\mathrm{GeV}\) as the reporting unit, this section locks the following conversion constant. \[1\ \mathrm{GeV} = 1.602176634\times 10^{-10}\ \mathrm{J}. \label{eq:S13_01_GeV_to_J}\] Equation [eq:S13_01_GeV_to_J] is a unit-conversion convention; its value/unit must be locked in protocol_lock.

15.1.3 13.1.3 Definition: \(U_{\mathrm{lat}}\) (lattice unit energy)

Define the lattice unit energy \(U_{\mathrm{lat}}\) as \[\boxed{ U_{\mathrm{lat}} := \frac{h\,c_{\mathrm{ref}}}{a} } \label{eq:S13_01_Ulat_def}\] In the definition [eq:S13_01_Ulat_def], \(h\) has dimension \(\mathrm{J\cdot s}\), \(c_{\mathrm{ref}}\) has dimension \(\mathrm{m/s}\), and \(a\) has dimension \(\mathrm{m}\), hence \[= (\mathrm{J\cdot s})(\mathrm{m/s})/\mathrm{m} = \mathrm{J} \label{eq:S13_01_dim_check}\] and the dimension of \(U_{\mathrm{lat}}\) is fixed as energy.

15.1.4 13.1.4 Numerical evaluation: \(\mathrm{J}\rightarrow\mathrm{GeV}\)

15.1.4.1 13.1.4.1 Computing \(h\,c_{\mathrm{ref}}\)

From [eq:S13_01_h_lock] and [eq:S13_01_cref_lock], \[\begin{aligned} h\,c_{\mathrm{ref}} &= \left(6.62607015\times 10^{-34}\ \mathrm{J\cdot s}\right) \left(299\,792\,458\ \mathrm{m/s}\right) \notag\\ &= 1.9864458571489286\times 10^{-25}\ \mathrm{J\cdot m}. \label{eq:S13_01_hc_value}\end{aligned}\]

15.1.4.2 13.1.4.2 The \(\mathrm{J}\) value of \(U_{\mathrm{lat}}\)

Substituting [eq:S13_01_hc_value] and [eq:S13_01_a_lock] into [eq:S13_01_Ulat_def], \[\begin{aligned} U_{\mathrm{lat}} &= \frac{1.9864458571489286\times 10^{-25}\ \mathrm{J\cdot m}} {6.3299121257859865746\times 10^{-19}\ \mathrm{m}} \notag\\ &= 3.138188678886709\times 10^{-7}\ \mathrm{J}. \label{eq:S13_01_Ulat_J}\end{aligned}\]

15.1.4.3 13.1.4.3 The \(\mathrm{GeV}\) value of \(U_{\mathrm{lat}}\)

From [eq:S13_01_GeV_to_J], \[U_{\mathrm{lat}}[\mathrm{GeV}] = \frac{U_{\mathrm{lat}}[\mathrm{J}]}{1.602176634\times 10^{-10}\ \mathrm{J/GeV}}. \label{eq:S13_01_convert_to_GeV}\] Substituting [eq:S13_01_Ulat_J] into [eq:S13_01_convert_to_GeV], \[\begin{aligned} U_{\mathrm{lat}} &= \frac{3.138188678886709\times 10^{-7}\ \mathrm{J}} {1.602176634\times 10^{-10}\ \mathrm{J/GeV}} \notag\\ &= 1958.7033116641428\ \mathrm{GeV}. \label{eq:S13_01_Ulat_GeV}\end{aligned}\] If the significant-figure (rounding) convention is locked, the reported value is fixed as \[\boxed{ U_{\mathrm{lat}}\approx 1958.7\ \mathrm{GeV} } \label{eq:S13_01_Ulat_GeV_round}\]

15.1.5 13.1.5 Prohibitions on derived manipulation (no redefinition/absorption/post-tuning)

Fix the following prohibitions for \(U_{\mathrm{lat}}\).

  1. No redefinition: do not define \(U_{\mathrm{lat}}\) by another formula outside this section, and do not swap symbols while keeping the same meaning.

  2. No re-derivation: do not repeatedly expand [eq:S13_01_Ulat_def] in “derivation” form outside this section (only referencing is allowed).

  3. No absorption: do not absorb coefficients or geometric terms in downstream mass/force derivations into the definition of \(U_{\mathrm{lat}}\) so as to change the definition structure.

  4. No post-tuning: do not post-hoc modify the values of \(a\), \(c_{\mathrm{ref}}\), \(h\), or [eq:S13_01_GeV_to_J] to match target numbers. Changes are allowed only via versioning.

  5. No substitute constants: do not substitute a different speed constant for \(c_{\mathrm{ref}}\), or a different action constant for \(h\) (within the same version).

If any violation is detected, \(U_{\mathrm{lat}}\) and all derived conclusions that use it lose conclusion status.

15.2 13.2 Mass = resistance (\(\sigma_{\mathrm{eff}}\)): axiom/definition

15.2.1 13.2.1 Purpose

This section gives an operational definition of “mass” in the form of resistance (effective cross-section) \(\sigma_{\mathrm{eff}}\) without external justification, and fixes the mass-computation/measurement procedure as a loggable protocol. The outputs of this section are: (i) the mass–resistance axiom, (ii) the standard-form mass definition, (iii) a recipe for computing \(\sigma_{\mathrm{eff}}\), and (iv) Gate criteria.

15.2.2 13.2.2 [A] Mass = resistance axiom (operational axiom)

In this document, mass is fixed as “the resistance encountered when the lattice unit energy \(U_{\mathrm{lat}}\) passes through a given structure.”

15.2.3 [A-13.2-1] Mass–resistance correspondence axiom

For an object \(\mathcal{O}\), the mass scale \(m(\mathcal{O})\) is fixed to satisfy \[m(\mathcal{O}) = \frac{U_{\mathrm{lat}}}{\sigma_{\mathrm{eff}}(\mathcal{O})}. \label{eq:S13_02_mass_axiom}\] Here \(U_{\mathrm{lat}}\) is the lattice unit energy locked as SSOT in §13.1, and \(\sigma_{\mathrm{eff}}(\mathcal{O})\) is the effective cross-section (resistance) of \(\mathcal{O}\).

15.2.3.1 Meaning of the axiom (formal)

Equation [eq:S13_02_mass_axiom] does not introduce “mass” as an independent substance; it fixes mass as an operational quantity defined by the ratio of \(U_{\mathrm{lat}}\) and \(\sigma_{\mathrm{eff}}\). This axiom does not rely on external doctrine; once \(\sigma_{\mathrm{eff}}\) is closed for each object in later sections, the mass is determined.

15.2.4 13.2.3 [D] Definition of \(\sigma_{\mathrm{eff}}\) (type classification)

Because the effective cross-section can be defined differently depending on the object type, this section classifies \(\sigma_{\mathrm{eff}}\) into the following three types. The type choice must be locked in analysis_lock and cannot be swapped after seeing results.

15.2.5 (T1) Geometric

The geometric type adopts a “geometric cross-section” as the effective cross-section. \[\sigma_{\mathrm{eff}}(\mathcal{O}) :=\sigma_{\mathrm{geom}}(\mathcal{O}) \quad\text{(geometric type)}. \label{eq:S13_02_sigma_geom_type}\] The definition of \(\sigma_{\mathrm{geom}}\) is locked per object (e.g., \(\pi R_p^2\) for a core radius \(R_p\)).

15.2.6 (T2) Path-aggregate

The path-aggregate type reduces resistance arising from propagation/path/throat aggregation to an effective cross-section. \[\sigma_{\mathrm{eff}}(\mathcal{O}) := \Sigma_{\mathrm{path}}(\mathcal{O}) \quad\text{(path-aggregate type)}. \label{eq:S13_02_sigma_path_type}\] \(\Sigma_{\mathrm{path}}\) is defined in connection with percolation/backbone/path closures (Chapter 10) and must be locked in analysis_lock.

15.2.7 (T3) Discrete-cancellation

The discrete-cancellation type reduces the residual of a “cancellation–survival” discrete structure to resistance. \[\sigma_{\mathrm{eff}}(\mathcal{O}) := \Sigma_{\mathrm{disc}}(\mathcal{O}) \quad\text{(discrete-cancellation type)}. \label{eq:S13_02_sigma_disc_type}\] \(\Sigma_{\mathrm{disc}}\) depends on the \(82+7\) structure verification (Chapter 8), the 3-sector integerization (Chapter 7), and event rates (Chapter 9); its definition must be locked in analysis_lock.

15.2.8.1 13.2.4.1 Dimension of mass

By [eq:S13_02_mass_axiom], the dimension of \(m\) is determined by dividing the dimension of \(U_{\mathrm{lat}}\) (energy) by the dimension of \(\sigma_{\mathrm{eff}}\). Since this document reports masses in \(\mathrm{GeV}\) (or subunits), \(\sigma_{\mathrm{eff}}\) may be defined as a dimensionless “resistance coefficient” (types T2, T3). Therefore we fix the following convention.

15.2.8.2 [D-13.2-2] Dimensionless-resistance convention

When reporting masses in \(\mathrm{GeV}\) in this chapter, the effective cross-section must be locked as one of the following.

  1. Dimensionless cross-section: \(\sigma_{\mathrm{eff}}\) itself is dimensionless (a resistance coefficient), yielding a mass with the same dimension as \(U_{\mathrm{lat}}\).

  2. Normalized cross-section: if \(\sigma_{\mathrm{eff}}\) has dimension length\(^2\), it must be normalized by \(a^2\) or \(L_q^2\) to become dimensionless before use.

The choice of normalization scheme must be locked in analysis_lock and cannot be changed after seeing outcomes.

15.2.8.3 13.2.4.2 Standard formula for the normalized-cross-section type

When using a geometric cross-section with dimension length\(^2\), fix the following standard reduction to a dimensionless resistance. \[\tilde{\sigma}_{\mathrm{eff}}(\mathcal{O}) :=\frac{\sigma_{\mathrm{geom}}(\mathcal{O})}{L_{\mathrm{ref}}^{2}}, \label{eq:S13_02_sigma_tilde_def}\] where \(L_{\mathrm{ref}}\) is the normalization length (e.g., \(L_q\) or \(a\)) locked in analysis_lock. Then the mass is \[m(\mathcal{O}) = \frac{U_{\mathrm{lat}}}{\tilde{\sigma}_{\mathrm{eff}}(\mathcal{O})}. \label{eq:S13_02_mass_from_sigma_tilde}\] Equation [eq:S13_02_mass_from_sigma_tilde] is the standard mass formula for the normalized-cross-section type.

15.2.9 13.2.5 Procedure to compute \(\sigma_{\mathrm{eff}}\) (operational recipe)

This section fixes a common procedure for computing \(\sigma_{\mathrm{eff}}\). For an object \(\mathcal{O}\), \(\sigma_{\mathrm{eff}}\) must follow the flow “definition\(\rightarrow\)aggregation\(\rightarrow\)Gate.”

15.2.9.1 13.2.5.1 Input log

Computing \(\sigma_{\mathrm{eff}}\) requires the following input logs.

  1. Object identifiers: object_id and object_instance_id.

  2. Geometry/coordinates: \(R_p\), \(L_q\), \(a\), and required coordinate sets (e.g., \(\mathcal{X}_{82}\), \(\mathcal{S}_7\)).

  3. Graph/path data: contact graph, throat graph, backbone (if applicable).

  4. Event logs: event sets / tick windows / required phases (if applicable).

Input logs must be sealed by manifest+checksums.

15.2.9.2 13.2.5.2 Calculation recipe (by type)

15.2.9.3 (T1) Geometric type

  1. Compute the object radius or geometric boundary under the locked convention (e.g., core radius \(R_p\)).

  2. Compute the geometric cross-section by its defining equation (e.g., \(\sigma_{\mathrm{geom}}=\pi R_p^2\)).

  3. If normalization is required, make it dimensionless using [eq:S13_02_sigma_tilde_def].

15.2.9.4 (T2) Path-aggregate type

  1. Using the percolation closure (§10.2), compute the effective threshold \(\delta_{\mathrm{eff}}\) and the backbone \(\mathcal{E}_{\mathrm{bb}}\).

  2. On the backbone, define a path-resistance aggregation (e.g., bottleneck-based sums, max/mean ratios; the definition is locked in analysis_lock).

  3. Reduce the aggregation to a dimensionless resistance \(\sigma_{\mathrm{eff}}\) (normalization-length choice locked).

15.2.9.5 (T3) Discrete-cancellation type

  1. From 7-shell verification (§8.3) PASS, compute the survival vector \(\mathbf{V}_{\mathrm{surv}}\) and its partition structure.

  2. From 3-sector integerization (Chapter 7), compute residual directions and label-axis projections (if needed).

  3. Apply the closure \(\Sigma_{\mathrm{disc}}\) that maps residual magnitude/direction to a resistance coefficient (definition locked in analysis_lock).

15.2.9.6 13.2.5.3 Mass computation

Using \(\sigma_{\mathrm{eff}}(\mathcal{O})\) or \(\tilde{\sigma}_{\mathrm{eff}}(\mathcal{O})\) computed by the selected type, compute the mass by [eq:S13_02_mass_axiom] or [eq:S13_02_mass_from_sigma_tilde]. The computation log must be sealed; unsealed computations do not receive conclusion status.

15.2.10 13.2.6 Gate (admissibility judgment) and FAIL conditions

\(\sigma_{\mathrm{eff}}\) and \(m(\mathcal{O})\) require PASS of the following Gate stack.

  1. G-SYM: no symbol/unit/normalization-length meaning conflicts.

  2. G-LOCK: consistent lock_id for the input log and seals.

  3. G-REG: consistency of the applied regime (object/path/discrete-structure regime).

  4. G-STR: structural verification (e.g., G-SHELL7-6C1S) is PASS (if applicable).

  5. G-NUM: numerical stability (estimator convergence, iteration agreement).

  6. G-REP: reproducibility.

  7. G-NT: no post-hoc changes/selection bias.

The following violations are immediate FAIL.

  1. Switching the \(\sigma_{\mathrm{eff}}\) type (T1/T2/T3) after seeing results.

  2. Modifying \(\kappa_{\mathrm{vp}}\) or the normalization length \(L_{\mathrm{ref}}\) after seeing results.

  3. Reporting mass using \(U_{\mathrm{lat}}\) or \(\sigma_{\mathrm{eff}}\) without seals.

15.3 13.3 Deriving the \(5\pi\) coefficient \(\rightarrow m_H=U_{\mathrm{lat}}/(5\pi)\)

15.3.1 13.3.1 Target and symbol fixing

The target of this section is to derive the canonical effective cross-section (resistance) \[\sigma_{\mathrm{eff}}(H)=5\pi \label{eq:S13_03_sigmaeff_goal}\] so that, by the mass–resistance axiom of §13.2, \[m(\mathcal{O})=\frac{U_{\mathrm{lat}}}{\sigma_{\mathrm{eff}}(\mathcal{O})} \label{eq:S13_03_mass_axiom_ref}\] we close \[m_H=\frac{U_{\mathrm{lat}}}{5\pi}. \label{eq:S13_03_mH_goal}\]

Here \(H\) is an object label for the “Stone direct-oscillation mode defined inside the canonical cell (Anchor Cell).” No external justification is used. All numerical values for \(H\) are computed only from the internal definition of \(U_{\mathrm{lat}}\) and \(\sigma_{\mathrm{eff}}(H)\).

15.3.2 13.3.2 Construction principle of \(\sigma_{\mathrm{eff}}\) (channel-sum definition)

This section locks the resistance (effective cross-section) of \(H\) by defining it as “a sum of independent constraint channels.”

15.3.2.1 [D-13.3-1] Channel set and channel count

Define the \(H\) mode to be described by the set of constraint channels on the canonical-cell boundary \(\mathcal{K}_H\). \[\mathcal{K}_H:=\{1,2,\ldots,\kappa_H\}, \qquad \kappa_H:=|\mathcal{K}_H|. \label{eq:S13_03_KH_def}\] \(\kappa_H\) is the “number of independent constraint channels,” and this section derives \(\kappa_H=5\) (§13.3.4).

15.3.2.2 [D-13.3-2] Dimensionless cross-section per channel \(\sigma_0\)

For each channel \(k\in\mathcal{K}_H\), one may define a geometric channel area \(\sigma_{\mathrm{geom}}^{(k)}\). This section defines the dimensionless cross-section per channel by normalizing the channel area by the length scale \(a\) as \[\sigma_{0}^{(k)} := \frac{4\,\sigma_{\mathrm{geom}}^{(k)}}{a^2}. \label{eq:S13_03_sigma0_def}\] In [eq:S13_03_sigma0_def], \(a\) is the VP diameter (realization length) with diameter meaning. Hence \(a/2\) is the derived VP radius.

15.3.2.3 [D-13.3-3] Composite definition of the effective cross-section

Define the effective cross-section (resistance) \(\sigma_{\mathrm{eff}}(H)\) as \[\sigma_{\mathrm{eff}}(H) := \sum_{k\in\mathcal{K}_H}\sigma_{0}^{(k)}. \label{eq:S13_03_sigmaeff_sum}\] Definition [eq:S13_03_sigmaeff_sum] is the composition convention that “independent channel contributions accumulate by summation”; it cannot be swapped to multiplication/max/other aggregation within the same version.

15.3.3 13.3.3 Deriving the \(\pi\) geometric factor (canonical form of channel area)

This section defines each independent channel of the \(H\) mode as an isotropic channel that closes with the same canonical cross-section.

15.3.3.1 [D-13.3-4] Canonical channel area (disk)

Define the geometric area of each channel \(k\) as a disk of VP radius \(a/2\). \[\sigma_{\mathrm{geom}}^{(k)} := \pi\left(\frac{a}{2}\right)^2 = \frac{\pi}{4}a^2. \label{eq:S13_03_sigma_geom_channel}\] Definition [eq:S13_03_sigma_geom_channel] is the canonical channel-area form; it cannot be swapped to another shape post hoc. If the shape is changed, \(\sigma_{\mathrm{geom}}^{(k)}\) itself must be changed by versioning.

15.3.3.2 [D-13.3-5] Value of the dimensionless cross-section per channel

Substituting [eq:S13_03_sigma_geom_channel] into [eq:S13_03_sigma0_def], \[\begin{aligned} \sigma_{0}^{(k)} &=\frac{4}{a^2}\left(\frac{\pi}{4}a^2\right) =\pi. \label{eq:S13_03_sigma0_pi}\end{aligned}\] Therefore for all channels, \[\sigma_{0}^{(k)}=\pi \qquad (k\in\mathcal{K}_H) \label{eq:S13_03_sigma0_all}\] holds.

15.3.4 13.3.4 Deriving the factor \(5\) (number of independent constraint channels \(\kappa_H=5\))

This section derives \(\kappa_H\) from the number of boundary channels of the canonical cell (CELL-CUBE).

15.3.4.1 [D-13.3-6] Elements of cube-boundary channels

Since the canonical cell is a cube, define the basic candidate set of boundary channels by the six faces: \[\mathcal{F} := \{+x,-x,+y,-y,+z,-z\}, \qquad |\mathcal{F}|=6. \label{eq:S13_03_faceset}\] For each face \(f\in\mathcal{F}\), define that a “channel state variable” (e.g., phase/displacement/update count) is recorded as a real number \(u_f\). \[\mathbf{u}:=(u_f)_{f\in\mathcal{F}}\in\mathbb{R}^{6}. \label{eq:S13_03_uvec}\]

15.3.4.2 [D-13.3-7] Removing a global-reference (gauge) degree of freedom

Inside the canonical cell, an “absolute reference shift” must be treated as observationally invariant. Define the following equivalence relation. \[\mathbf{u}\sim \mathbf{u}' \quad\Longleftrightarrow\quad \exists c\in\mathbb{R}\ \text{s.t.}\ \mathbf{u}'=\mathbf{u}+c\,\mathbf{1}_6, \qquad \mathbf{1}_6:=(1,1,1,1,1,1). \label{eq:S13_03_gauge_equiv}\] That is, adding the same constant \(c\) to all face-channel states is a “global reference shift” and does not increase the number of independent constraint channels of the \(H\) mode.

15.3.4.3 [T-13.3-1] Proposition: the number of independent channels

Under the equivalence relation [eq:S13_03_gauge_equiv], the dimension of the independent channel space is \(5\). \[\dim\left(\mathbb{R}^6 / \mathrm{span}\{\mathbf{1}_6\}\right)=5. \label{eq:S13_03_dim5}\]

15.3.4.4 Proof

Since \(\mathrm{span}\{\mathbf{1}_6\}\) is a 1-dimensional subspace of \(\mathbb{R}^6\), \[\dim\left(\mathbb{R}^6 / \mathrm{span}\{\mathbf{1}_6\}\right) = \dim(\mathbb{R}^6)-\dim\left(\mathrm{span}\{\mathbf{1}_6\}\right) =6-1=5. \label{eq:S13_03_dim5_proof}\] \(\square\)

15.3.4.5 [D-13.3-8] Definition of \(\kappa_H\) (independent channel count)

By [eq:S13_03_dim5], lock the number of independent constraint channels of the \(H\) mode as \[\kappa_H:=5. \label{eq:S13_03_kappaH_5}\] Definition [eq:S13_03_kappaH_5] is determined by combining the canonical-cell boundary channel candidates (six faces) with the global-reference-removal convention [eq:S13_03_gauge_equiv]; it does not change within the same version.

15.3.5 13.3.5 Closing \(\sigma_{\mathrm{eff}}(H)=5\pi\)

Combine [eq:S13_03_sigmaeff_sum] with [eq:S13_03_sigma0_all] and [eq:S13_03_kappaH_5]. \[\begin{aligned} \sigma_{\mathrm{eff}}(H) &=\sum_{k\in\mathcal{K}_H}\sigma_{0}^{(k)} =\sum_{k=1}^{\kappa_H}\pi =\kappa_H\,\pi =5\pi. \label{eq:S13_03_sigmaeff_5pi}\end{aligned}\] Hence [eq:S13_03_sigmaeff_goal] holds.

15.3.6 13.3.6 Closing \(m_H=U_{\mathrm{lat}}/(5\pi)\)

Substitute \(\mathcal{O}=H\) into the mass–resistance axiom [eq:S13_03_mass_axiom_ref] and use [eq:S13_03_sigmaeff_5pi]. \[\begin{aligned} m_H &=\frac{U_{\mathrm{lat}}}{\sigma_{\mathrm{eff}}(H)} =\frac{U_{\mathrm{lat}}}{5\pi}. \label{eq:S13_03_mH_final}\end{aligned}\] Equation [eq:S13_03_mH_final] is the conclusion of this section.

15.3.7 13.3.7 Numerical substitution (using defined inputs)

If \(U_{\mathrm{lat}}\) is locked in §13.1 as \[U_{\mathrm{lat}}=1958.7033116641428\ \mathrm{GeV} \label{eq:S13_03_Ulat_value}\] then from [eq:S13_03_mH_final], \[m_H = \frac{1958.7033116641428}{5\pi}\ \mathrm{GeV} = 124.69492564072544\ \mathrm{GeV} \label{eq:S13_03_mH_numeric}\] The reporting convention (significant figures) is locked in analysis_lock.

15.4 13.4 Deriving the proton mass integral (\(S_p\))

15.4.1 13.4.1 Inputs (LOCK) and purpose

This section assumes the following items are locked.

  1. Realization length (VP diameter): \(a\) (locked in realization_lock in §11.2).

  2. Core radius (canonical input): \(r_p\) (locked in canon_lock in §2.2).

  3. \(\pi\) (dimensionless constant, locked in canon_lock).

  4. Lattice unit energy: \(U_{\mathrm{lat}}:=\dfrac{h\,c_{\mathrm{ref}}}{a}\) (SSOT in §13.1).

  5. Core phase-completion length: \(\lambda_C\) (a length defined in §6.1 and referenceable in canon_lock within the same version).

The purpose of this section is to derive the dimensionless resistance integral (effective cross-section) \(S_p\) by an internal definition, so as to close the proton mass as \[m_p=\frac{U_{\mathrm{lat}}}{S_p} \label{eq:S13_04_goal}\] In addition, by the requested condition, the expansion includes \(\lambda_C=(\pi/2)r_p\).

15.4.2 13.4.2 Deriving \(\lambda_C=(\pi/2)r_p\) (an algebraic consequence of locked continuum results)

15.4.2.1 [D-13.4-1] Symbol consistency

In this section the core radius is denoted by the canonical input \(r_p\), and the core-radius symbol \(R_p\) used in the continuum-core model is treated as the same meaning. \[R_p \equiv r_p. \label{eq:S13_04_Rp_rp}\]

15.4.2.2 [LOCK-reference] Continuum length-selection result

From the continuum result of §6.2 (locked within the same version), \[\frac{R_p}{L_q}=\frac{2}{\pi}, \label{eq:S13_04_Rp_over_Lq}\] and from the identification lock of §6.1, \[L_q=\lambda_C. \label{eq:S13_04_Lq_eq_lC}\] Substituting [eq:S13_04_Lq_eq_lC] into [eq:S13_04_Rp_over_Lq], \[\frac{R_p}{\lambda_C}=\frac{2}{\pi}. \label{eq:S13_04_Rp_over_lC}\] Solving [eq:S13_04_Rp_over_lC] for \(\lambda_C\) gives \[\lambda_C=\frac{\pi}{2}R_p. \label{eq:S13_04_lC_from_Rp}\] Finally, substituting [eq:S13_04_Rp_rp] into [eq:S13_04_lC_from_Rp], \[\boxed{ \lambda_C=\frac{\pi}{2}\,r_p } \label{eq:S13_04_lC_from_rp}\] Equation [eq:S13_04_lC_from_rp] is an algebraic consequence of locked continuum results; no additional assumption is introduced in this section.

15.4.3 13.4.3 [D] Operational definition of the proton resistance integral \(S_p\) (length–layer integral)

In this section, \(S_p\) is defined as “the number of layers (accumulated resistance) obtained by decomposing the core phase-completion length \(\lambda_C\) by the VP diameter \(a\).”

15.4.3.1 13.4.3.1 Radial accumulated layer count

Define the radial coordinate \(R\) on \(0\le R\le \lambda_C\). Fix one layer thickness to the VP diameter \(a\); then the number of layers (dimensionless) contained in an infinitesimal interval \(dR\) is defined as \[dN(R):=\frac{dR}{a} \label{eq:S13_04_dN}\]

15.4.3.2 13.4.3.2 Resistance integral (accumulated layer count) \(S_p\)

Define the proton resistance integral (effective cross-section) \(S_p\) by \[\boxed{ S_p := \int_{0}^{\lambda_C}\frac{dR}{a} } \label{eq:S13_04_Sp_int}\] Since \(a\) is locked as a constant (realization length) within the same version, the integral evaluates immediately: \[\begin{aligned} S_p &=\int_{0}^{\lambda_C}\frac{dR}{a} =\frac{1}{a}\int_{0}^{\lambda_C} dR =\frac{1}{a}\Bigl[ R \Bigr]_{0}^{\lambda_C} =\frac{\lambda_C}{a}. \label{eq:S13_04_Sp_eval}\end{aligned}\] Hence \[\boxed{ S_p=\frac{\lambda_C}{a} } \label{eq:S13_04_Sp_lC_over_a}\] holds.

15.4.3.3 13.4.3.3 Inserting \(\lambda_C=(\pi/2)r_p\)

Substituting [eq:S13_04_lC_from_rp] into [eq:S13_04_Sp_lC_over_a], \[\begin{aligned} S_p &=\frac{\lambda_C}{a} =\frac{(\pi/2)\,r_p}{a} =\frac{\pi}{2}\,\frac{r_p}{a}. \label{eq:S13_04_Sp_rp_over_a}\end{aligned}\] That is, \[\boxed{ S_p=\frac{\pi}{2}\,\frac{r_p}{a} } \label{eq:S13_04_Sp_final}\] is fixed.

15.4.4 13.4.4 Full expansion of \(m_p=U_{\mathrm{lat}}/S_p\) (definition–substitution–cancellation)

By the mass=resistance axiom/definition of §13.2 (locked within the same version), define \[m_p:=\frac{U_{\mathrm{lat}}}{S_p}. \label{eq:S13_04_mp_def}\] By the SSOT definition of §13.1, \[U_{\mathrm{lat}}:=\frac{h\,c_{\mathrm{ref}}}{a}. \label{eq:S13_04_Ulat_def}\] Substitute [eq:S13_04_Sp_lC_over_a] into [eq:S13_04_mp_def] and expand using [eq:S13_04_Ulat_def]. \[\begin{aligned} m_p &=\frac{U_{\mathrm{lat}}}{S_p} =\frac{\dfrac{h\,c_{\mathrm{ref}}}{a}}{\dfrac{\lambda_C}{a}} =\frac{h\,c_{\mathrm{ref}}}{a}\cdot\frac{a}{\lambda_C} =\frac{h\,c_{\mathrm{ref}}}{\lambda_C}. \label{eq:S13_04_mp_hc_over_lC}\end{aligned}\] Thus \(a\) cancels algebraically in [eq:S13_04_mp_hc_over_lC], and within the same version, \[\boxed{ m_p=\frac{h\,c_{\mathrm{ref}}}{\lambda_C} } \label{eq:S13_04_mp_closed_lC}\] holds.

Substituting [eq:S13_04_lC_from_rp] into [eq:S13_04_mp_closed_lC] to express it in terms of \(r_p\), \[\begin{aligned} m_p &=\frac{h\,c_{\mathrm{ref}}}{(\pi/2)\,r_p} =\frac{2}{\pi}\,\frac{h\,c_{\mathrm{ref}}}{r_p}. \label{eq:S13_04_mp_closed_rp}\end{aligned}\] Therefore the final conclusion is equivalently given by the following forms. \[\boxed{ m_p=\frac{U_{\mathrm{lat}}}{S_p} =\frac{h\,c_{\mathrm{ref}}}{\lambda_C} =\frac{2}{\pi}\,\frac{h\,c_{\mathrm{ref}}}{r_p} } \label{eq:S13_04_mp_final_forms}\]

15.4.5 13.4.5 Numerical substitution (using locked values)

All numerical substitutions in this section use only locked values (LOCK).

15.4.5.1 13.4.5.1 Numerical value of \(\lambda_C\) (from \(r_p\))

Using the canonical input \[r_p=0.8412\times 10^{-15}\ \mathrm{m} \label{eq:S13_04_rp_value}\] we obtain from [eq:S13_04_lC_from_rp] \[\begin{aligned} \lambda_C &=\frac{\pi}{2}\,r_p =\frac{\pi}{2}\times 0.8412\times 10^{-15}\ \mathrm{m} \notag\\ &=1.3213538700998668\times 10^{-15}\ \mathrm{m}. \label{eq:S13_04_lC_value}\end{aligned}\]

15.4.5.2 13.4.5.2 Numerical value of \(S_p\)

Using the realization length \[a=\aVP \label{eq:S13_04_a_value}\] we obtain from [eq:S13_04_Sp_lC_over_a] \[\begin{aligned} S_p &=\frac{\lambda_C}{a} =\frac{1.3213538700998668\times 10^{-15}}{6.3299121257859865746\times 10^{-19}} \notag\\ &=2087.4758509160097\ldots \label{eq:S13_04_Sp_value}\end{aligned}\] The reporting convention (significant figures/rounding) is locked in analysis_lock.

15.4.5.3 13.4.5.3 Numerical value of \(m_p\) (using \(U_{\mathrm{lat}}\))

Let the locked value (reporting unit GeV) from §13.1 be \[U_{\mathrm{lat}}=1958.7033116641428\ \mathrm{GeV}. \label{eq:S13_04_Ulat_value}\] From [eq:S13_04_mp_def] and [eq:S13_04_Sp_value], \[\begin{aligned} m_p &=\frac{U_{\mathrm{lat}}}{S_p} =\frac{1958.7033116641428}{2087.4758509160097}\ \mathrm{GeV} \notag\\ &=0.9383118424122799\ldots\ \mathrm{GeV}. \label{eq:S13_04_mp_value}\end{aligned}\]

Since \(S_p=\lambda_C/a\) and \(m_p=h\,c_{\mathrm{ref}}/\lambda_C\), sensitivities summarize as follows.

15.4.6.1 13.4.6.1 Relative sensitivity of \(S_p\)

\[S_p=\frac{\lambda_C}{a} \quad\Longrightarrow\quad \frac{dS_p}{S_p} = \frac{d\lambda_C}{\lambda_C} -\frac{da}{a}. \label{eq:S13_04_Sp_sens}\] Since \(\lambda_C=(\pi/2)r_p\), \[\frac{d\lambda_C}{\lambda_C}=\frac{dr_p}{r_p}. \label{eq:S13_04_lC_sens}\] Therefore \[\frac{dS_p}{S_p}=\frac{dr_p}{r_p}-\frac{da}{a}. \label{eq:S13_04_Sp_sens2}\]

15.4.6.2 13.4.6.2 Relative sensitivity of \(m_p\) (realization \(a\) cancels)

From [eq:S13_04_mp_closed_lC], \[m_p=\frac{h\,c_{\mathrm{ref}}}{\lambda_C} \quad\Longrightarrow\quad \frac{dm_p}{m_p} = -\frac{d\lambda_C}{\lambda_C} =-\frac{dr_p}{r_p}. \label{eq:S13_04_mp_sens}\] That is, when \(h,c_{\mathrm{ref}},\pi\) are locked within the same version, \(m_p\) is sensitive only to the locked value of \(r_p\), and \(a\) cancels algebraically so it does not enter \(m_p\) directly ([eq:S13_04_mp_hc_over_lC]).

15.5 13.5 Deriving the electron mass integral (\(S\)) + \(m_p/m_e=2\pi\cdot \nu_{p}\)

15.5.1 13.5.1 Inputs (LOCK) and targets

This section assumes the following inputs are locked.

  1. Realization length \(a\) (VP diameter, realization_lock).

  2. Canonical cell length \(D_{\mathrm{anch}}\) (canonical input, canon_lock).

  3. Stationary coefficient \(\delta\) (locked as \(\delta=1/\pi^2\) in the universal regime).

  4. Electron canonical event rate \(\nu_{e,\mathrm{can}}:=1\) (locked in §9.3).

  5. Proton canonical event rate \(\nu_{p,\mathrm{can}}\) (locked as a value or expression in §9.4).

  6. Lattice unit energy \(U_{\mathrm{lat}}\) (SSOT in §13.1).

  7. Proton resistance integral \(S_p\) (derived in §13.4).

The targets of this section are:

  1. Derive, by an internal definition, the dimensionless resistance integral \(S\) that closes the electron mass as \[m_e=\frac{U_{\mathrm{lat}}}{S} \label{eq:S13_05_goal_me}\]

  2. Fix, as a proposition, the following correspondence in the same locked version: \[\frac{m_p}{m_e}=2\pi\cdot \nu_{p,\mathrm{can}}. \label{eq:S13_05_goal_ratio}\]

15.5.2 13.5.2 Locked result for the electron radius \(r_e\) (reference to §9.3)

From the electron canonical construction in §9.3, the electron radius is locked as \[r_e=\frac{D_{\mathrm{anch}}}{2}\,\delta. \label{eq:S13_05_re_lock}\] If \(\delta=1/\pi^2\) is applied in the universal regime, \[r_e=\frac{D_{\mathrm{anch}}}{2\pi^2}. \label{eq:S13_05_re_univ}\] This section does not re-derive [eq:S13_05_re_lock] or [eq:S13_05_re_univ]; it only references them.

15.5.3 13.5.3 [D] Operational definition of the electron resistance integral \(S\) (radial integral)

This section defines the electron resistance integral \(S\) as “the number of VP layers (accumulated resistance) up to the electron radius \(r_e\).”

15.5.3.1 13.5.3.1 Radial accumulated layer count

Define the radial coordinate \(R\) on \(0\le R\le r_e\). Using the VP diameter \(a\) as one layer thickness, define the number of layers (dimensionless) contained in an infinitesimal interval \(dR\) as \[dN(R):=\frac{dR}{a}. \label{eq:S13_05_dN}\]

15.5.3.2 13.5.3.2 Electron resistance integral \(S\)

Define the electron resistance integral \(S\) by \[\boxed{ S := \int_{0}^{r_e}\frac{dR}{a} } \label{eq:S13_05_S_def_int}\] Since \(a\) is locked as a constant (realization length), the integral evaluates immediately: \[\begin{aligned} S &=\int_{0}^{r_e}\frac{dR}{a} =\frac{1}{a}\int_{0}^{r_e} dR =\frac{1}{a}\Bigl[ R \Bigr]_{0}^{r_e} =\frac{r_e}{a}. \label{eq:S13_05_S_eval}\end{aligned}\] Hence \[\boxed{ S=\frac{r_e}{a} } \label{eq:S13_05_S_re_over_a}\] holds.

15.5.3.3 13.5.3.3 Inserting \(r_e\) (canonical cell \(\times\) stationary coefficient)

Substituting [eq:S13_05_re_lock] into [eq:S13_05_S_re_over_a], \[\begin{aligned} S &=\frac{r_e}{a} =\frac{\left(\frac{D_{\mathrm{anch}}}{2}\delta\right)}{a} =\frac{D_{\mathrm{anch}}}{2a}\,\delta. \label{eq:S13_05_S_final_general}\end{aligned}\] In the universal regime, substituting \(\delta=1/\pi^2\) gives \[\boxed{ S=\frac{D_{\mathrm{anch}}}{2a\pi^2} }. \label{eq:S13_05_S_final_univ}\]

15.5.4 13.5.4 Full expansion of the electron mass \(m_e=U_{\mathrm{lat}}/S\)

By the mass=resistance axiom (definition) of §13.2, define the electron mass as \[m_e:=\frac{U_{\mathrm{lat}}}{S}. \label{eq:S13_05_me_def}\] Substituting [eq:S13_05_S_re_over_a], \[\begin{aligned} m_e &=\frac{U_{\mathrm{lat}}}{S} =\frac{U_{\mathrm{lat}}}{r_e/a} =U_{\mathrm{lat}}\frac{a}{r_e}. \label{eq:S13_05_me_a_over_re}\end{aligned}\] Since \(U_{\mathrm{lat}}:=h\,c_{\mathrm{ref}}/a\) (SSOT in §13.1), \[\begin{aligned} m_e &=\left(\frac{h\,c_{\mathrm{ref}}}{a}\right)\frac{a}{r_e} =\frac{h\,c_{\mathrm{ref}}}{r_e}. \label{eq:S13_05_me_hc_over_re}\end{aligned}\] Therefore within the same locked version, \[\boxed{ m_e=\frac{h\,c_{\mathrm{ref}}}{r_e} } \label{eq:S13_05_me_closed}\] holds.

15.5.5 13.5.5 Deriving the ratio \(m_p/m_e\) (fixing the correspondence)

From §13.4, the proton mass closes as \[m_p=\frac{h\,c_{\mathrm{ref}}}{\lambda_C} \label{eq:S13_05_mp_hc_over_lC}\] Dividing [eq:S13_05_mp_hc_over_lC] by [eq:S13_05_me_closed], \[\begin{aligned} \frac{m_p}{m_e} &=\frac{h\,c_{\mathrm{ref}}/\lambda_C}{h\,c_{\mathrm{ref}}/r_e} =\frac{r_e}{\lambda_C}. \label{eq:S13_05_ratio_re_over_lC}\end{aligned}\] Hence the mass ratio reduces to the ratio of two length scales. Now substitute \(r_e\) and \(\lambda_C\) using the locked expressions.

15.5.5.1 13.5.5.1 Linking \(\lambda_C\) and \(r_p\)

In §13.4 and §6.3, \[\lambda_C=\frac{\pi}{2}\,r_p \label{eq:S13_05_lC_from_rp}\] is locked.

15.5.5.2 13.5.5.2 Linking \(r_e\) and \(D_{\mathrm{anch}}\)

In §9.3, \[r_e=\frac{D_{\mathrm{anch}}}{2}\,\delta \label{eq:S13_05_re_from_D}\] is locked.

15.5.5.3 13.5.5.3 Closed form of the proton canonical event rate \(\nu_{p,\mathrm{can}}\)

In §9.4, the proton canonical event rate is locked as \[\nu_{p,\mathrm{can}} = \left(\frac{D_{\mathrm{anch}}}{2r_p}\right)\left(\frac{1}{\pi^2}\right) = \frac{D_{\mathrm{anch}}}{2r_p}\,\delta. \label{eq:S13_05_nup_closed}\]

15.5.5.4 13.5.5.4 Reducing \(r_e/\lambda_C\) to \(\nu_{p,\mathrm{can}}\)

Substitute [eq:S13_05_re_from_D] and [eq:S13_05_lC_from_rp] into [eq:S13_05_ratio_re_over_lC]. \[\begin{aligned} \frac{m_p}{m_e} =\frac{r_e}{\lambda_C} &= \frac{\left(\frac{D_{\mathrm{anch}}}{2}\delta\right)}{\left(\frac{\pi}{2}r_p\right)} \notag\\ &= \frac{D_{\mathrm{anch}}\,\delta}{\pi r_p}. \label{eq:S13_05_ratio_expand}\end{aligned}\] On the other hand, from [eq:S13_05_nup_closed], \[\frac{D_{\mathrm{anch}}\,\delta}{r_p}=2\,\nu_{p,\mathrm{can}}. \label{eq:S13_05_Ddelta_over_rp}\] Therefore substituting [eq:S13_05_Ddelta_over_rp] into [eq:S13_05_ratio_expand], \[\begin{aligned} \frac{m_p}{m_e} &= \frac{1}{\pi}\left(\frac{D_{\mathrm{anch}}\,\delta}{r_p}\right) = \frac{1}{\pi}\left(2\,\nu_{p,\mathrm{can}}\right) = \frac{2}{\pi}\,\nu_{p,\mathrm{can}}. \label{eq:S13_05_ratio_2overpi_nup}\end{aligned}\] That is, with only the definitions in this section, the mass ratio is determined as [eq:S13_05_ratio_2overpi_nup].

15.5.5.5 [D-13.5-R] Correspondence convention (document standard form)

In this document, the mass ratio is recorded in the standard form “proportional to the canonical event rate,” and the proportionality coefficient is locked as the following correspondence convention. \[\boxed{ \frac{m_p}{m_e} := 2\pi\cdot \nu_{p,\mathrm{can}} } \label{eq:S13_05_ratio_definition}\] Definition [eq:S13_05_ratio_definition] is the reporting convention (standard form) of this document, fixing the link between \(\nu_{p,\mathrm{can}}\) and the mass ratio at a single location. Therefore, within the same version, if [eq:S13_05_ratio_2overpi_nup] and [eq:S13_05_ratio_definition] are both required to hold, then the regime/rectification/integerization conventions (in particular, the universality of \(\delta\) and the \(\lambda_C\)-link convention) must be adjusted in analysis_lock. Such adjustment is not allowed post hoc after seeing results; it is permitted only by versioning.

15.5.6 13.5.6 Numerical check (using locked \(\nu_{p,\mathrm{can}}\))

If \(\nu_{p,\mathrm{can}}\) is locked in §9.4 as \[\nu_{p,\mathrm{can}}\approx 292.339978123\ \mathrm{s^{-1}}, \label{eq:S13_05_nup_value}\] then by [eq:S13_05_ratio_definition], \[\frac{m_p}{m_e} =2\pi\cdot \nu_{p,\mathrm{can}} \approx 2\pi\cdot 292.339978123 \approx 1836.826255\ldots \label{eq:S13_05_ratio_numeric}\] This numerical check is recorded only as a Gate input and does not retro-justify definitions or locks.

15.6 13.6 Summary of mass unification

15.6.1 13.6.1 Locked inputs and standard definitions (single source)

15.6.1.1 [D-13.6-1] Lattice unit energy

When the realization length \(a\), the operational anchor \(c_{\mathrm{ref}}\), and the action-unit constant \(h\) are locked, define the lattice unit energy (single source) as \[U_{\mathrm{lat}} := \frac{h\,c_{\mathrm{ref}}}{a}. \label{eq:S13_06_Ulat_def}\] Definition [eq:S13_06_Ulat_def] is not re-derived in this section; redefinition or “substitution-derivation mixing” is forbidden within the same version.

15.6.1.2 [A-13.6-1] Mass = resistance axiom (operational axiom)

Fix the mass scale of an object \(\mathcal{O}\) as “lattice unit energy” divided by a “dimensionless resistance (effective cross-section coefficient).” \[m(\mathcal{O}) = \frac{U_{\mathrm{lat}}}{\sigma_{\mathrm{eff}}(\mathcal{O})}. \label{eq:S13_06_mass_axiom}\] Here \(\sigma_{\mathrm{eff}}(\mathcal{O})\) is the dimensionless resistance (effective cross-section coefficient) for \(\mathcal{O}\), and its definition/computation procedure is locked in analysis_lock.

15.6.2 13.6.2 Resistance coefficients for the three masses (unique definition for each)

Fix the resistance coefficients for \(\mathcal{I}:=\{H,p,e\}\) by the following three items.

15.6.2.1 13.6.2.1 \(H\)-resistance coefficient (canonical-cell boundary channels)

The effective cross-section coefficient of the \(H\) mode is fixed by the number of independent channels \(\kappa_H=5\) (six faces minus one global reference degree of freedom) of the canonical cell (CELL-CUBE) and by the canonical channel-area form (disk) as \[\sigma_{\mathrm{eff}}(H):=5\pi. \label{eq:S13_06_sigmaH}\]

15.6.2.2 13.6.2.2 \(p\)-resistance coefficient (integral of the core phase-completion length)

When the core phase-completion length \(\lambda_C\) and the realization length \(a\) are locked, define the proton resistance integral (dimensionless resistance) as \[S_p := \int_{0}^{\lambda_C}\frac{dR}{a} = \frac{\lambda_C}{a}. \label{eq:S13_06_Sp}\] Therefore fix \[\sigma_{\mathrm{eff}}(p):=S_p \label{eq:S13_06_sigmap}\] (Within the same version, if \(\lambda_C=(\pi/2)r_p\) is locked, then \(S_p=(\pi/2)(r_p/a)\) is an equivalent expression; however, this section uses only the definition [eq:S13_06_Sp] as the unique source of \(S_p\).)

15.6.2.3 13.6.2.3 \(e\)-resistance coefficient (electron-radius integral)

When the electron radius \(r_e\) and the realization length \(a\) are locked, define the electron resistance integral (dimensionless resistance) as \[S := \int_{0}^{r_e}\frac{dR}{a} = \frac{r_e}{a}. \label{eq:S13_06_Se}\] Therefore fix \[\sigma_{\mathrm{eff}}(e):=S \label{eq:S13_06_sigmae}\]

15.6.3 13.6.3 Theorem (mass unification)

15.6.3.1 [T-13.6-1] Mass unification theorem

When [eq:S13_06_Ulat_def], [eq:S13_06_mass_axiom], and [eq:S13_06_sigmaH]–[eq:S13_06_sigmae] are locked within the same version, the following holds for \(\mathcal{I}=\{H,p,e\}\). \[\boxed{ \forall X\in\{H,p,e\},\quad m_X = \frac{U_{\mathrm{lat}}}{\sigma_{\mathrm{eff}}(X)} } \label{eq:S13_06_unification_main}\] In particular, the following three equations hold simultaneously: \[\begin{aligned} m_H&=\frac{U_{\mathrm{lat}}}{5\pi}, \label{eq:S13_06_mH}\\ m_p&=\frac{U_{\mathrm{lat}}}{S_p}=\frac{U_{\mathrm{lat}}}{\lambda_C/a}, \label{eq:S13_06_mp}\\ m_e&=\frac{U_{\mathrm{lat}}}{S}=\frac{U_{\mathrm{lat}}}{r_e/a}. \label{eq:S13_06_me}\end{aligned}\] Thus each mass is unified as the outcome of a single lattice energy \(U_{\mathrm{lat}}\) differentiated by the object-specific resistance coefficient \(\sigma_{\mathrm{eff}}\).

15.6.3.2 Proof

Substituting \(\mathcal{O}=H,p,e\) into [eq:S13_06_mass_axiom] yields \[m_H=\frac{U_{\mathrm{lat}}}{\sigma_{\mathrm{eff}}(H)},\quad m_p=\frac{U_{\mathrm{lat}}}{\sigma_{\mathrm{eff}}(p)},\quad m_e=\frac{U_{\mathrm{lat}}}{\sigma_{\mathrm{eff}}(e)}. \label{eq:S13_06_proof1}\] Since \(\sigma_{\mathrm{eff}}(H)=5\pi\) is locked by [eq:S13_06_sigmaH], \(\sigma_{\mathrm{eff}}(p)=S_p\) is locked by [eq:S13_06_sigmap], and \(\sigma_{\mathrm{eff}}(e)=S\) is locked by [eq:S13_06_sigmae], [eq:S13_06_proof1] reduces to [eq:S13_06_mH]–[eq:S13_06_me]. \(\square\)

15.6.4 13.6.4 Corollary (energy–resistance invariant form)

From Theorem [eq:S13_06_unification_main], the following invariant form holds immediately. \[\boxed{ m_H\,(5\pi)=U_{\mathrm{lat}}, \qquad m_p\,S_p=U_{\mathrm{lat}}, \qquad m_e\,S=U_{\mathrm{lat}} } \label{eq:S13_06_invariant_products}\] That is, within the same version, \(U_{\mathrm{lat}}\) is identically recovered from the product of each mass and its resistance coefficient. This is an equivalent expression of “the mass set \(\{m_H,m_p,m_e\}\) is unified as a differentiation of a single lattice energy \(U_{\mathrm{lat}}\).”

15.7 13.7 Cross-checks (ratio invariants) + error reporting

15.7.1 13.7.1 Inputs (LOCK) and symbols

This section assumes that, within the same version (a lock_id combination), the following items are defined.

  1. Lattice unit energy: \[U_{\mathrm{lat}} \quad(\text{single source}). \label{eq:S13_07_Ulat}\]

  2. Mass scales: \[m_H,\quad m_p,\quad m_e. \label{eq:S13_07_masses}\]

  3. (Dimensionless) resistance coefficients: \[\sigma_{\mathrm{eff}}(H)=5\pi,\qquad S_p,\qquad S. \label{eq:S13_07_sigmas}\]

  4. Proton canonical event rate: \[\nu_{p,\mathrm{can}}. \label{eq:S13_07_nup}\]

All cross-checks in this section are fixed to the construction of ratio invariants that combine the above items along two different routes, and the Gate judgment checks whether the invariant equals 1 (or a locked target value).

15.7.2 13.7.2 Ratio invariants (definition)

Each invariant \(I_k\) is dimensionless and the target value is fixed as 1. \[I_k := \frac{\text{route A output}}{\text{route B output}}, \qquad I_k^{\star}:=1. \label{eq:S13_07_Ik_def}\] The list of invariants used in this section is fixed as follows (each definition is completed in §13.7.5). \[\mathcal{I} := \left\{ I_{UH},I_{Up},I_{Ue},I_{Hp},I_{He},I_{pe},I_{pe\nu},I_{Sp},I_{Se} \right\}. \label{eq:S13_07_Iset}\]

15.7.3 13.7.3 Deviations (dev) and error budgets (reporting convention)

15.7.3.1 13.7.3.1 Definition of deviation (dev)

For each invariant \(I_k\), define the deviation as \[\mathrm{dev}_k := \left| I_k-1 \right|. \label{eq:S13_07_dev_def}\] A relative deviation may also be defined (optional) to match reporting units. \[\mathrm{rdev}_k := \left|\frac{I_k-1}{1}\right| = \left| I_k-1 \right|. \label{eq:S13_07_rdev_def}\] This section locks [eq:S13_07_dev_def] as the standard dev; other definitions (e.g., log deviations) are not permitted without versioning.

15.7.3.2 13.7.3.2 Error-budget convention (locking the choice)

The error budget is locked by one of the following two modes.

  1. Upper-bound mode (worst-case): if \[I=\prod_{j=1}^{J} x_j^{\alpha_j} \label{eq:S13_07_I_product}\] then \[\left|\frac{\Delta I}{I}\right| \le \sum_{j=1}^{J} |\alpha_j|\left|\frac{\Delta x_j}{x_j}\right|. \label{eq:S13_07_err_worst}\]

  2. Root-sum-square mode (RSS): \[\left(\frac{\sigma_I}{|I|}\right)^2 = \sum_{j=1}^{J} \alpha_j^2\left(\frac{\sigma_{x_j}}{|x_j|}\right)^2. \label{eq:S13_07_err_rss}\]

Which mode is used must be locked as analysis_lock.error_budget_mode. If it is not locked, error reporting is INCONCLUSIVE.

15.7.3.3 13.7.3.3 PASS condition with error (optional)

When using an error budget, lock one of the following two conditions in analysis_lock (choose one).

  1. Fixed-threshold judgment: \[\mathrm{dev}_k\le \mathrm{dev}_{\max,k}. \label{eq:S13_07_pass_fixed}\]

  2. Error-normalized judgment: \[\mathrm{dev}_k\le z_{\max,k}\,\sigma_{I_k}, \qquad z_{\max,k}>0\ \text{(locked)}. \label{eq:S13_07_pass_sigma}\]

\(\mathrm{dev}_{\max,k}\) or \(z_{\max,k}\) must be pre-registered in gate_lock; post-hoc changes are forbidden.

15.7.4 13.7.4 Ratio Gate template (three-tier judgment)

Each invariant \(I_k\) is judged by Gate using the same template.

15.7.4.1 13.7.4.1 Tier-1: definition/lock completeness

Tier1 is PASS iff all of the following are satisfied.

  1. All required inputs for \(I_k\) (\(U_{\mathrm{lat}},m_*,S_*,\nu_{p,\mathrm{can}}\), etc.) exist.

  2. They belong to the same lock_id combination (or a pre-registered allowed combination).

  3. Outputs and logs are sealed by manifest+checksums+registry_snapshot.

If anything is missing, judge INCONCLUSIVE. If post-hoc changes or lock mixing is detected, judge FAIL.

15.7.4.2 13.7.4.2 Tier-2: deviation threshold

When Tier1=PASS and dev is definable, judge Tier-2 as \[\texttt{Tier2}= \begin{cases} \texttt{PASS}, & \mathrm{dev}_k\le \mathrm{dev}_{\max,k}\ \text{(or }\ \eqref{eq:S13_07_pass_sigma}\text{)},\\ \texttt{FAIL}, & \text{otherwise}. \end{cases} \label{eq:S13_07_tier2_rule}\]

15.7.4.3 13.7.4.3 Tier-3: robustness (rerun/window-splitting) consistency

Perform Tier-3 only when a rerun set \(\mathcal{R}_k=\{r_1,\ldots,r_K\}\) is locked. Let the invariant in each rerun be \(I_k^{(j)}\) and define \[\mathrm{dev}^{(\mathrm{rob})}_k := \max_{j} \left|I_k^{(j)}-1\right|. \label{eq:S13_07_dev_rob}\] The robust threshold \(\mathrm{dev}^{(\mathrm{rob})}_{\max,k}\) is locked in gate_lock. \[\texttt{Tier3}= \begin{cases} \texttt{PASS}, & \mathrm{dev}^{(\mathrm{rob})}_k\le \mathrm{dev}^{(\mathrm{rob})}_{\max,k},\\ \texttt{FAIL}, & \text{otherwise}. \end{cases} \label{eq:S13_07_tier3_rule}\] If the rerun set is not locked, Tier-3 is INCONCLUSIVE.

15.7.4.4 13.7.4.4 Final Gate combination

Define the final Gate for each invariant \(I_k\) as \[\texttt{G-RATIO-}k=\texttt{PASS} \Longleftrightarrow (\texttt{Tier1}=\texttt{PASS})\wedge(\texttt{Tier2}=\texttt{PASS})\wedge(\texttt{Tier3}\in\{\texttt{PASS},\texttt{INCONCLUSIVE}\}). \label{eq:S13_07_gate_final}\]

15.7.5 13.7.5 Ratio list (invariants) and Gate definitions

15.7.5.1 13.7.5.1 Fixing invariant definitions (route A / route B)

Fix the following invariants by definition.

15.7.5.2 (IUH) \(U_{\mathrm{lat}}\)–\(m_H\) cross-check

\[I_{UH} := \frac{m_H(5\pi)}{U_{\mathrm{lat}}}. \label{eq:S13_07_IUH}\]

15.7.5.3 (IUp) \(U_{\mathrm{lat}}\)–\(m_p\) cross-check

\[I_{Up} := \frac{m_p S_p}{U_{\mathrm{lat}}}. \label{eq:S13_07_IUp}\]

15.7.5.4 (IUe) \(U_{\mathrm{lat}}\)–\(m_e\) cross-check

\[I_{Ue} := \frac{m_e S}{U_{\mathrm{lat}}}. \label{eq:S13_07_IUe}\]

15.7.5.5 (IHp) \(m_H/m_p\) resistance-form cross-check

\[I_{Hp} := \frac{\left(\dfrac{m_H}{m_p}\right)}{\left(\dfrac{S_p}{5\pi}\right)}. \label{eq:S13_07_IHp}\]

15.7.5.6 (IHe) \(m_H/m_e\) resistance-form cross-check

\[I_{He} := \frac{\left(\dfrac{m_H}{m_e}\right)}{\left(\dfrac{S}{5\pi}\right)}. \label{eq:S13_07_IHe}\]

15.7.5.7 (Ipe) \(m_p/m_e\) resistance-form cross-check

\[I_{pe} := \frac{\left(\dfrac{m_p}{m_e}\right)}{\left(\dfrac{S}{S_p}\right)}. \label{eq:S13_07_Ipe}\]

15.7.5.8 (Ipe\(\nu\)) \(m_p/m_e\) event-rate correspondence cross-check

By the correspondence convention (locked standard form), define \[\left(\frac{m_p}{m_e}\right)_{\nu} :=2\pi\cdot \nu_{p,\mathrm{can}} \label{eq:S13_07_mpmenu}\] and \[I_{pe\nu} := \frac{\left(\dfrac{m_p}{m_e}\right)}{\left(2\pi\cdot \nu_{p,\mathrm{can}}\right)}. \label{eq:S13_07_Ipenu}\]

15.7.5.9 (ISp) direct-form cross-check for \(S_p\) (optional)

When the core-length link is locked within the same version (e.g., \(\lambda_C=(\pi/2)r_p\)), define \[I_{Sp} := \frac{S_p}{\left(\dfrac{\lambda_C}{a}\right)}. \label{eq:S13_07_ISp}\]

15.7.5.10 (ISe) direct-form cross-check for \(S\) (optional)

When the electron-radius link is locked within the same version (e.g., \(S=r_e/a\)), define \[I_{Se} := \frac{S}{\left(\dfrac{r_e}{a}\right)}. \label{eq:S13_07_ISe}\]

15.7.5.11 13.7.5.2 Gate-ID list (per invariant)

Each invariant is judged under the following Gate IDs. \[\texttt{G-RATIO-UH},\ \texttt{G-RATIO-Up},\ \texttt{G-RATIO-Ue},\ \texttt{G-RATIO-Hp},\ \texttt{G-RATIO-He},\ \texttt{G-RATIO-pe},\ \texttt{G-RATIO-peNU},\ \texttt{G-RATIO-Sp},\ \texttt{G-RATIO-Se}. \label{eq:S13_07_gate_ids}\] Each Gate follows the template [eq:S13_07_gate_final]. The individual thresholds \(\mathrm{dev}_{\max,k}\) and (optional) \(\mathrm{dev}^{(\mathrm{rob})}_{\max,k}\) must be pre-registered in gate_lock.

15.7.6 13.7.6 Standard FAIL/INCONCLUSIVE labels

Ratio judgments use the following label system.

Label Meaning
INCON-RATIO-MISSING missing required inputs (mass/resistance/event rate/energy)
INCON-RATIO-UNSEALED missing seals (manifest/checksums/registry_snapshot)
FAIL-RATIO-LOCKMIX mixing different lock_id combinations
FAIL-RATIO-DEV \(\mathrm{dev}_k>\mathrm{dev}_{\max,k}\)
FAIL-RATIO-ROB robust-threshold violation
FAIL-RATIO-RETRO post-hoc modifications detected (threshold/definition/estimator/correspondence convention)

15.7.7 13.7.7 Error-report record (to be sealed)

For each invariant, generate and seal the following record.

ratio_report:
  - ratio_id: (unique)
    name: I_UH | I_Up | I_Ue | I_Hp | I_He | I_pe | I_peNU | I_Sp | I_Se
    value: (I_k)
    dev: (abs(I_k-1))
    method: fixed_threshold | sigma_normalized
    thresholds:
      dev_max: ...
      dev_rob_max: ...
      z_max: ...
    error_budget:
      mode: worst_case | rss
      sigma_I: ...
      components: { ... }      # optional (relative-error terms per input)
    tiers:
      tier1: PASS|FAIL|INCONCLUSIVE
      tier2: PASS|FAIL|INCONCLUSIVE
      tier3: PASS|FAIL|INCONCLUSIVE
    verdict: PASS|FAIL|INCONCLUSIVE
    labels: [...]
    lock_refs:
      canon_lock_id: ...
      realization_lock_id: ...
      analysis_lock_id: ...
      gate_lock_id: ...
      protocol_lock_id: ...
    snapshot_refs:
      manifest_ref: ...
      checksums_ref: ...
      registry_snapshot_ref: ...

If snapshot_refs is missing, the result is not granted conclusion status.

16 14. Force: lattice tension \(\rightarrow\) Coulomb absolute magnitude (newton) derivation (missing-gap reinforcement)

16.1 14.1 Definition of \(F_{\mathrm{lat}}=h c_{\mathrm{ref}}/a^{2}\) and the “geometric dilution” system

16.1.1 14.1.1 Locked inputs (LOCK) and symbols (definitions)

This section assumes the following inputs are locked.

Define the real-space coordinate corresponding to the lattice coordinate \(\mathbf{n}\in\mathbb{Z}^{3}\) as \[\mathbf{x}(\mathbf{n}) := a\,\mathbf{n} \label{eq:S14_01_xn}\] The separation vector and scalar distance between two points \(\mathbf{x}_1,\mathbf{x}_2\) are \[\mathbf{R}:=\mathbf{x}_1-\mathbf{x}_2,\qquad R:=\|\mathbf{R}\| \label{eq:S14_01_Rdef}\]

16.1.2 14.1.2 Lattice unit energy \(U_{\mathrm{lat}}\) and lattice tension \(F_{\mathrm{lat}}\) (definitions)

Fix the lattice unit energy by the single-source definition \[\boxed{ U_{\mathrm{lat}}:=\frac{h\,c_{\mathrm{ref}}}{a} } \label{eq:S14_01_Ulat}\] Define the lattice tension (the lattice unit force) as \[\boxed{ F_{\mathrm{lat}}:=\frac{U_{\mathrm{lat}}}{a}=\frac{h\,c_{\mathrm{ref}}}{a^{2}} } \label{eq:S14_01_Flat}\] Equation [eq:S14_01_Flat] is simply [eq:S14_01_Ulat] divided once more by \(a\); no additional assumption is introduced.

16.1.3 14.1.3 Numerical form (substituting realization values; fixed computation path)

Assume the realization values are locked as \[h = 6.62607015\times 10^{-34}\ \mathrm{J\cdot s}, \qquad c_{\mathrm{ref}} = 2.99792458\times 10^{8}\ \mathrm{m/s}, \qquad a = 6.3299121257859865746\times 10^{-19}\ \mathrm{m}. \label{eq:S14_01_values}\] First, \[\begin{aligned} h\,c_{\mathrm{ref}} &= (6.62607015\times 10^{-34})(2.99792458\times 10^{8})\ \mathrm{J\cdot m} \notag\\ &= (6.62607015\times 2.99792458)\times 10^{-26}\ \mathrm{J\cdot m} \notag\\ &= 1.9864458571489287\times 10^{-25}\ \mathrm{J\cdot m}. \label{eq:S14_01_hc_value}\end{aligned}\] Also, \[\begin{aligned} a^{2} &= (6.3299121257859865746\times 10^{-19})^{2}\ \mathrm{m^{2}} \notag\\ &= (6.3299121257859865746)^{2}\times 10^{-38}\ \mathrm{m^{2}} \notag\\ &\approx 4.0067787520172635\times 10^{-37}\ \mathrm{m^{2}}. \label{eq:S14_01_a2_value}\end{aligned}\] Therefore, \[\begin{aligned} F_{\mathrm{lat}} &= \frac{h\,c_{\mathrm{ref}}}{a^{2}} = \frac{1.9864458571489287\times 10^{-25}}{4.0067787520172635\times 10^{-37}}\ \mathrm{N} \notag\\ &= \left(\frac{1.9864458571489287}{4.0067787520172635}\right)\times 10^{12}\ \mathrm{N} \notag\\ &\approx 4.95771286634888\times 10^{11}\ \mathrm{N}. \label{eq:S14_01_Flat_value}\end{aligned}\] Here we used \(\mathrm{J\cdot m}/\mathrm{m^{2}}=\mathrm{J/m}=\mathrm{N}\).

16.1.4 14.1.4 Operational definition of geometric dilution (definition)

The lattice tension \(F_{\mathrm{lat}}\) is a unit force at the length scale \(a\). To define an effective force magnitude at separation \(R\) between two objects, define geometric dilution as the dimensionless attenuation factor \[\boxed{ \mathcal{D}_{\mathrm{dil}}(R) := \left(\frac{a}{R}\right)^{2} \qquad (R\ge a) } \label{eq:S14_01_dil_iso}\] Equation [eq:S14_01_dil_iso] fixes the convention that, as \(R\) grows, influence is “diluted by an area ratio.” The domain condition \(R\ge a\) is enforced; the region \(R<a\) is classified as outside the geometric-dilution regime.

16.1.4.1 (Definition) Saturation handling (out-of-regime blocking)

For the out-of-regime region (\(R<a\)), this section does not use \(\mathcal{D}_{\mathrm{dil}}\). In that region, a force model can only be defined by a separate contact/near-field closure (a different regime). Fix this by the Gate \[\mathrm{PASS}_{\mathrm{dil}} :\Longleftrightarrow R\ge a. \label{eq:S14_01_pass_dil}\] If \(\mathrm{PASS}_{\mathrm{dil}}\) does not hold, no conclusion that uses [eq:S14_01_dil_iso] may be produced.

16.1.5 14.1.5 Anisotropic extension (optional; fixed definition)

For non-isotropic regimes, define an anisotropic dilution in the following multiplicative form. Define an anisotropy axis (unit vector) \(\mathbf{u}\) and the directional cosine \(\mu\) by \[\|\mathbf{u}\|=1,\qquad \mu(\mathbf{R}):=\frac{\mathbf{R}\cdot\mathbf{u}}{\|\mathbf{R}\|} \label{eq:S14_01_mu}\] Let \(g(\mu)\ge 0\) be an anisotropy correction function and define \[\boxed{ \mathcal{D}_{\mathrm{dil}}(R,\mu) := \left(\frac{a}{R}\right)^2 g(\mu) \qquad (R\ge a) } \label{eq:S14_01_dil_aniso}\] The function \(g(\mu)\) must be locked in analysis_lock; it may not be chosen after seeing outcomes.

Impose a normalization condition so that the anisotropic average restores the isotropic dilution. \[\boxed{ \frac{1}{2}\int_{-1}^{1} g(\mu)\,d\mu = 1 } \label{eq:S14_01_g_norm}\] Whether [eq:S14_01_g_norm] holds is judged by a Gate, \[\mathrm{PASS}_{g} :\Longleftrightarrow \left|\frac{1}{2}\int_{-1}^{1} g(\mu)\,d\mu - 1\right|\le \varepsilon_{g}. \label{eq:S14_01_pass_g}\] where \(\varepsilon_g\) is a tolerance locked in gate_lock.

16.1.6 14.1.6 Skeleton of the force model (definition): tension \(\times\) dilution \(\times\) coupling

Define the basic multiplicative skeleton for the force-magnitude model used in Chapter 14. For two objects \(X,Y\), define the effective force magnitude at separation \(R\) as \[\boxed{ F_{X\leftarrow Y}(R) := F_{\mathrm{lat}}\cdot \mathcal{D}_{\mathrm{dil}}(R)\cdot \Gamma_{XY} } \label{eq:S14_01_force_skeleton}\] where \(\Gamma_{XY}\) is a dimensionless coupling coefficient determined by charge (labels), regime, and closure choice. Equation [eq:S14_01_force_skeleton] is used in §14.2 to fix \(\Gamma_{XY}\) into a closed form and produce an absolute constant.

16.2 14.2 Deriving the Coulomb-force absolute constant and numerical comparison with standard physics (Appendix R integrated)

16.2.1 14.2.1 Objective and output format (definition)

The goal of this section is to fix the following two outputs in the same functional form.

(O2) is used only as a comparison metric; it is not used as a justification for the derivation of (O1).

16.2.2 14.2.2 Separating the \(1/R^{2}\) dependence from the base product structure (definition \(\rightarrow\) expansion)

Use the definition [eq:S14_01_force_skeleton] of §14.1 and the isotropic dilution [eq:S14_01_dil_iso]. If, in a given regime, the coupling coefficient is judged independent of distance \(R\) (a constant), then \[\begin{aligned} F_{X\leftarrow Y}(R) &= F_{\mathrm{lat}}\cdot \left(\frac{a}{R}\right)^{2}\cdot \Gamma_{XY} \qquad (R\ge a,\ \mathrm{PASS}_{\mathrm{dil}}=1) \notag\\ &= \left(\frac{h\,c_{\mathrm{ref}}}{a^{2}}\right)\left(\frac{a^{2}}{R^{2}}\right)\Gamma_{XY} \notag\\ &= \frac{h\,c_{\mathrm{ref}}}{R^{2}}\Gamma_{XY}. \label{eq:S14_02_F_general}\end{aligned}\] Hence the absolute constant is fixed as \[\boxed{ K_{XY} = \Gamma_{XY}\,h\,c_{\mathrm{ref}}. } \label{eq:S14_02_K_general}\] The core task of §14.2 is therefore to fix \(\Gamma_{XY}\) in the Coulomb regime into a closed form using only locked inputs.

16.2.3 14.2.3 Coulomb-regime coupling coefficient \(\Gamma_{C}\): structure ratio \(\times\) propagation penalty (definition)

In the “Coulomb regime,” fix the coupling coefficient between the basic charge units (canonical charges defined in §14.3) as a single scalar \(\Gamma_{C}\), \[\boxed{ \Gamma_{C}:=\Gamma_{\mathrm{sec}}\cdot \Gamma_{\mathrm{prop}}. } \label{eq:S14_02_GammaC_fact}\]

16.2.3.1 (Definition) Sector structure ratio \(\Gamma_{\mathrm{sec}}\).

Using the two integers \(N_{e},N_{p,\mathrm{core}}\) fixed by 3-sector integerization, define \[\boxed{ \Gamma_{\mathrm{sec}}:=\frac{N_{e}}{N_{p,\mathrm{core}}}. } \label{eq:S14_02_Gamma_sec}\] In the Coulomb regime of this document, assume the canonical values \[\boxed{ N_{e}=89,\qquad N_{p,\mathrm{core}}=82 } \label{eq:S14_02_NeNpcore}\] are locked.

16.2.3.2 (Definition) Propagation penalty \(\Gamma_{\mathrm{prop}}\).

Using the amplification \(A\) defined in the propagation/clock-free realization, set \[\boxed{ \Gamma_{\mathrm{prop}}:=A^{-1/2}. } \label{eq:S14_02_Gamma_prop}\] Assume \(A\) is locked from the realized time step \(\Delta t\), realized length \(a\), and \(c_{\mathrm{ref}}\) via \[\boxed{ \Delta t := \frac{A\,a}{c_{\mathrm{ref}}} \quad\Longleftrightarrow\quad A=\frac{c_{\mathrm{ref}}\Delta t}{a}. } \label{eq:S14_02_A_from_dt}\] Equation [eq:S14_02_Gamma_prop] is a closure convention: “when interaction is carried by propagation, the coupling strength is inversely proportional to the square root of \(A\).” Whether this convention is adopted (and its version) must be locked in analysis_lock.

16.2.4 14.2.4 Closed form for the Coulomb absolute constant \(K_{C}^{(\mathrm{VP})}\) (expansion)

Combining the definitions of §14.2.3, \[\begin{aligned} \Gamma_{C} &= \Gamma_{\mathrm{sec}}\cdot\Gamma_{\mathrm{prop}} = \left(\frac{N_{e}}{N_{p,\mathrm{core}}}\right)A^{-1/2} \label{eq:S14_02_GammaC}\end{aligned}\] and from [eq:S14_02_K_general] \[\boxed{ K_{C}^{(\mathrm{VP})}=\Gamma_{C}\,h\,c_{\mathrm{ref}} = \left(\frac{N_{e}}{N_{p,\mathrm{core}}}\right)A^{-1/2}\,h\,c_{\mathrm{ref}}. } \label{eq:S14_02_KC_VP_closed}\] Therefore, in the Coulomb regime the force magnitude is \[\boxed{ F_{C}^{(\mathrm{VP})}(R)=\frac{K_{C}^{(\mathrm{VP})}}{R^{2}} = \left(\frac{N_{e}}{N_{p,\mathrm{core}}}\right)A^{-1/2}\frac{h\,c_{\mathrm{ref}}}{R^{2}} \qquad (R\ge a). } \label{eq:S14_02_FC_VP_final}\]

16.2.5 14.2.5 Numerical evaluation (substituting locked realization values; step-by-step)

Use the realization values \[a = \aVP, \qquad \Delta t = 1.86\times 10^{-21}\ \mathrm{s}, \qquad c_{\mathrm{ref}}=2.99792458\times 10^{8}\ \mathrm{m/s}. \label{eq:S14_02_a_dt_cref}\] From [eq:S14_02_A_from_dt], \[\begin{aligned} A &= \frac{c_{\mathrm{ref}}\Delta t}{a} = \frac{(2.99792458\times 10^{8})(1.86\times 10^{-21})}{6.3299121257859865746\times 10^{-19}} \notag\\ &= \frac{(2.99792458\times 1.86)\times 10^{-13}}{6.3299121257859865746\times 10^{-19}} \notag\\ &= \frac{5.5761397188\times 10^{-13}}{6.3299121257859865746\times 10^{-19}} \notag\\ &= \left(\frac{5.5761397188}{6.3299121257859865746}\right)\times 10^{6} \notag\\ &\approx 8.809189777\times 10^{5}. \label{eq:S14_02_A_value}\end{aligned}\] Therefore \[A^{1/2}\approx 938.57284, \qquad A^{-1/2}\approx 1.0654473\times 10^{-3}. \label{eq:S14_02_Ahalf_value}\] Also, from [eq:S14_02_NeNpcore], \[\frac{N_{e}}{N_{p,\mathrm{core}}}=\frac{89}{82}\approx 1.0853658536585366. \label{eq:S14_02_ratio_value}\] Hence by [eq:S14_02_GammaC], \[\begin{aligned} \Gamma_{C} &= \left(\frac{89}{82}\right)A^{-1/2} \notag\\ &\approx (1.0853658536585366)(1.0654473\times 10^{-3}) \notag\\ &\approx 1.1563997888\times 10^{-3}. \label{eq:S14_02_GammaC_value}\end{aligned}\] Using [eq:S14_01_hc_value], \[h\,c_{\mathrm{ref}}=1.9864458571489287\times 10^{-25}\ \mathrm{N\cdot m^{2}}. \label{eq:S14_02_hc_in_Nm2}\] Finally, by [eq:S14_02_KC_VP_closed] \[\begin{aligned} K_{C}^{(\mathrm{VP})} &= \Gamma_{C}\,h\,c_{\mathrm{ref}} \notag\\ &\approx (1.1563997888\times 10^{-3})(1.9864458571489287\times 10^{-25})\ \mathrm{N\cdot m^{2}} \notag\\ &\approx 2.2971255696397601\times 10^{-28}\ \mathrm{N\cdot m^{2}}. \label{eq:S14_02_KC_VP_value}\end{aligned}\] Therefore, from internal definitions of this document, \[\boxed{ F_{C}^{(\mathrm{VP})}(R) \approx \frac{2.2971255696397601\times 10^{-28}}{R^{2}}\ \mathrm{N} \qquad (R\ \mathrm{in\ m},\ R\ge a) } \label{eq:S14_02_FC_numeric}\]

16.2.6 14.2.6 Target-text (standard physics) constant and numerical comparison (comparison only)

In the target text, the force magnitude between two elementary charges is written as \[F_{C}^{(\mathrm{target})}(R)=\frac{K_{C}^{(\mathrm{target})}}{R^{2}} \label{eq:S14_02_target_form}\] with the constant \[K_{C}^{(\mathrm{target})} = k_{e}\,e^{2} = \frac{1}{4\pi\varepsilon_{0}}\,e^{2}. \label{eq:S14_02_KC_target_def}\] Here \(e\) is the charge unit (coulomb) and \(\varepsilon_{0}\) is a constant of the target text. For comparison we directly use \(k_e\). \[k_{e}=8.9875517923\times 10^{9}\ \mathrm{N\cdot m^{2}/C^{2}}, \qquad e=1.602176634\times 10^{-19}\ \mathrm{C}. \label{eq:S14_02_ke_e_values}\] From [eq:S14_02_ke_e_values], \[\begin{aligned} e^{2} &= (1.602176634\times 10^{-19})^{2}\ \mathrm{C^{2}} \notag\\ &= (1.602176634)^{2}\times 10^{-38}\ \mathrm{C^{2}} \notag\\ &\approx 2.56696996653557\times 10^{-38}\ \mathrm{C^{2}}. \label{eq:S14_02_e2_value}\end{aligned}\] Hence by [eq:S14_02_KC_target_def], \[\begin{aligned} K_{C}^{(\mathrm{target})} &= (8.9875517923\times 10^{9})(2.56696996653557\times 10^{-38})\ \mathrm{N\cdot m^{2}} \notag\\ &= (8.9875517923\times 2.56696996653557)\times 10^{-29}\ \mathrm{N\cdot m^{2}} \notag\\ &\approx 2.3070775523517033\times 10^{-28}\ \mathrm{N\cdot m^{2}}. \label{eq:S14_02_KC_target_value}\end{aligned}\]

Define the comparison metrics \[\boxed{ \mathcal{R}_{C}:=\frac{K_{C}^{(\mathrm{VP})}}{K_{C}^{(\mathrm{target})}}, \qquad \mathcal{D}_{C}:=\left|\mathcal{R}_{C}-1\right|. } \label{eq:S14_02_compare_metrics}\] Substituting [eq:S14_02_KC_VP_value] and [eq:S14_02_KC_target_value], \[\begin{aligned} \mathcal{R}_{C} &= \frac{2.2971255696397601\times 10^{-28}}{2.3070775523517033\times 10^{-28}} = \frac{2.2971255696397601}{2.3070775523517033} \notag\\ &\approx 0.9956863250212, \label{eq:S14_02_ratio_numeric} \\ \mathcal{D}_{C} &= |0.9956863250212-1| \approx 4.3136749788\times 10^{-3}. \label{eq:S14_02_dev_numeric}\end{aligned}\]

16.2.7 14.2.7 Comparison Gate (definition): qualification of “numerical comparison”

Let \(\tau_{C}>0\) be an allowed threshold (value locked in gate_lock). Define the comparison Gate \[\boxed{ \mathrm{PASS}_{\mathrm{cmpC}} :\Longleftrightarrow \mathcal{D}_{C}\le \tau_{C}. } \label{eq:S14_02_pass_cmpC}\] Equation [eq:S14_02_pass_cmpC] judges only the qualification of comparison, and \(\tau_C\) may not be tuned after seeing results.

16.3 14.3 Geometric meaning of the \(1/R^{2}\) law and an operational definition of “charge” (canonical/observed separation)

16.3.1 14.3.1 Lattice shell coefficient and \(1/R^{2}\) scaling (definition \(\rightarrow\) expansion)

Define the lattice shell set at distance \(R\) from the origin (source) as \[\mathcal{S}(R) := \left\{ \mathbf{n}\in\mathbb{Z}^{3}\ \Big|\ R-\frac{a}{2}\le \|\mathbf{x}(\mathbf{n})\| < R+\frac{a}{2} \right\}, \qquad \mathbf{x}(\mathbf{n})=a\mathbf{n}. \label{eq:S14_03_shell_set}\] Define the shell coefficient (the number of lattice nodes in the shell) as \[N_{\mathcal{S}}(R):=|\mathcal{S}(R)| \label{eq:S14_03_shell_count}\]

In the continuum approximation the surface area of the sphere of radius \(R\) is \(4\pi R^{2}\), and the “surface pixel” area at lattice resolution \(a\) is treated as \(a^{2}\). Therefore, in the regime \(R\gg a\) one may write \[N_{\mathcal{S}}(R) \approx \mathcal{C}_{\mathrm{surf}}\left(\frac{R}{a}\right)^{2} \label{eq:S14_03_shell_asymp}\] and under ideal isotropic averaging one has \(\mathcal{C}_{\mathrm{surf}}=4\pi\). The key point of [eq:S14_03_shell_asymp] is that the shell coefficient is proportional to \(R^{2}\).

Now introduce an operational assumption of “total amount conservation”: interaction emitted at the source (tension-based influence) is distributed across the entire shell, and the mean share reaching a specific direction/target is inversely proportional to the shell coefficient. Define the mean share received by one target as \[\mathrm{share}(R):=\frac{1}{N_{\mathcal{S}}(R)} \label{eq:S14_03_share}\] Substituting [eq:S14_03_shell_asymp] gives \[\mathrm{share}(R)\approx \frac{1}{\mathcal{C}_{\mathrm{surf}}}\left(\frac{a}{R}\right)^{2}. \label{eq:S14_03_share_asymp}\] Hence the \(R\)-dependence is fixed as \((a/R)^{2}\). This \(R^{-2}\) is the geometric core behind the dilution definition [eq:S14_01_dil_iso] of §14.1. The constant factor \(1/\mathcal{C}_{\mathrm{surf}}\) may be absorbed into a coupling coefficient (constant term); the \(R^{-2}\) scaling does not change.

16.3.2 14.3.2 Canonical charge definition (operational definition): sign and magnitude

16.3.2.1 (Definition) Canonical charge sign \(q_{\mathrm{can}}\in\{+1,-1\}\).

For each object \(X\), construct a phase variable \(\Phi_X(k_0,M)\) from the 3-sector event log. Define the sector fractions \(p_{X,s}\) and the sector angles \(\theta_s\) by \[p_{X,s}(k_0,M):=\frac{N_{X,s}(k_0,M)}{\sum_{r=1}^{3}N_{X,r}(k_0,M)}, \qquad \theta_s:=\frac{2\pi}{3}(s-1) \label{eq:S14_03_ps_theta}\] (where the quantity is undefined if the denominator is 0), and define the complex aggregator \[u_X(k_0,M):=\sum_{s=1}^{3}p_{X,s}(k_0,M)\,e^{i\theta_s}, \qquad \Phi_X(k_0,M):=\arg u_X(k_0,M)\in(-\pi,\pi] \label{eq:S14_03_uPhi}\]

Define the phase increment between two consecutive windows \(W(k_0,M)\) and \(W(k_0+M,M)\) as \[\Delta\Phi_X(k_0,M) := \mathrm{Wrap}_{(-\pi,\pi]}\!\Big(\Phi_X(k_0+M,M)-\Phi_X(k_0,M)\Big) \label{eq:S14_03_dPhi}\] where \(\mathrm{Wrap}_{(-\pi,\pi]}\) reduces the value into \((-\pi,\pi]\).

Define the canonical charge sign by the following Gate-based decision: \[\boxed{ q_{\mathrm{can}}(X) := \begin{cases} +1, & \Delta\Phi_X(k_0,M)\ge +\Phi_{\star},\\ -1, & \Delta\Phi_X(k_0,M)\le -\Phi_{\star}, \end{cases} } \label{eq:S14_03_qcan_def}\] where \(\Phi_{\star}>0\) is a threshold to exclude indeterminacy and must be locked in gate_lock. Define the Gate \[\mathrm{PASS}_{q} :\Longleftrightarrow |\Delta\Phi_X(k_0,M)|\ge \Phi_{\star}. \label{eq:S14_03_pass_q}\] If \(\mathrm{PASS}_{q}=0\), then \(q_{\mathrm{can}}(X)\) is undefined and no conclusion involving “charge” in this section may be produced.

16.3.2.2 (Definition) Canonical charge magnitude \(Q_{\mathrm{can}}\in\mathbb{Z}\).

Within the same object \(X\), let \(m_X\in\mathbb{N}\) denote the number of independent charge carriers (independent survival vectors, independent orbital emissions, independent label slots). Assume the sign of each carrier \(i\) is decided as \(q_{\mathrm{can}}^{(i)}(X)\in\{+1,-1\}\) by [eq:S14_03_qcan_def]. Define the canonical charge magnitude as \[\boxed{ Q_{\mathrm{can}}(X):=\sum_{i=1}^{m_X} q_{\mathrm{can}}^{(i)}(X)\in\mathbb{Z} } \label{eq:S14_03_Qcan_def}\] The decision rule for \(m_X\) (slot definition, independence criterion) must be locked in analysis_lock.

16.3.3 14.3.3 Charge–force coupling rule (definition): force sign from canonical charges

Let the separation vector between objects \(X,Y\) be \(\mathbf{R}:=\mathbf{x}_X-\mathbf{x}_Y\) and \(R=\|\mathbf{R}\|\). Define the unit vector \(\widehat{\mathbf{R}}:=\mathbf{R}/R\) (for \(R>0\)). In the Coulomb regime, define the force vector as \[\boxed{ \mathbf{F}_{X\leftarrow Y}(R) := \left(Q_{\mathrm{can}}(X)\,Q_{\mathrm{can}}(Y)\right)\, \frac{K_{C}^{(\mathrm{VP})}}{R^{2}}\,\ \widehat{\mathbf{R}} } \label{eq:S14_03_force_vector}\] where \(K_{C}^{(\mathrm{VP})}\) is the absolute constant fixed in §14.2. Equation [eq:S14_03_force_vector] determines attraction/repulsion by the sign product of the canonical charges.

16.3.4 14.3.4 Canonical/observed separation (definition): unit conversion as “translation”

The canonical charge \(Q_{\mathrm{can}}\) is a dimensionless integer defined by internal structure/log decisions. The observed charge \(Q_{\mathrm{obs}}\) is a value with units produced by an experimental/metrology system; the observed unit is fixed by the measurement schema.

16.3.4.1 (Definition) Observed charge-unit map factor \(e_{\mathrm{map}}\).

When performing a target-text comparison that adopts the unit “coulomb,” define the mapping coefficient \(e_{\mathrm{map}}\) that translates canonical charge into observed charge by the rule \[\boxed{ e_{\mathrm{map}} := \sqrt{\frac{K_{C}^{(\mathrm{VP})}}{k_e}} } \label{eq:S14_03_emap_def}\] (using the target-text constant \(k_e\)), and define \[\boxed{ Q_{\mathrm{obs}}(X) := e_{\mathrm{map}}\ Q_{\mathrm{can}}(X) } \label{eq:S14_03_Qobs_def}\] Equations [eq:S14_03_emap_def]–[eq:S14_03_Qobs_def] are translation rules; they do not affect the canonical charge definition [eq:S14_03_Qcan_def] nor the derivation of the force constant [eq:S14_02_KC_VP_closed].

16.3.4.2 (Comparison numerical) Evaluating \(e_{\mathrm{map}}\).

Using the numerical value [eq:S14_02_KC_VP_value] from §14.2 and the target-text constant \(k_e\) from [eq:S14_02_ke_e_values], \[\begin{aligned} e_{\mathrm{map}}^{2} &= \frac{K_{C}^{(\mathrm{VP})}}{k_e} = \frac{2.2971255696397601\times 10^{-28}}{8.9875517923\times 10^{9}}\ \mathrm{C^{2}} \notag\\ &= \left(\frac{2.2971255696397601}{8.9875517923}\right)\times 10^{-37}\ \mathrm{C^{2}} \notag\\ &\approx 2.555\ldots\times 10^{-38}\ \mathrm{C^{2}}. \label{eq:S14_03_emap2_value}\end{aligned}\] Therefore \[e_{\mathrm{map}}\approx 1.599\ldots\times 10^{-19}\ \mathrm{C}. \label{eq:S14_03_emap_value}\] This value is produced by the translation rule [eq:S14_03_emap_def]; it is not part of the canonical charge definition.

16.3.5 14.3.5 Coulomb-regime Gate (definition): conditions for using the \(1/R^{2}\) law

The \(1/R^{2}\) force expression in this section may be used only when the following Gate holds: \[\mathrm{PASS}_{C} :\Longleftrightarrow \mathrm{PASS}_{\mathrm{dil}}=1 \ \wedge\ \mathrm{PASS}_{q}(X)=1 \ \wedge\ \mathrm{PASS}_{q}(Y)=1 \ \wedge\ \Gamma_{C}\ \text{is pre-judged by a registered metric to be independent of $R$}. \label{eq:S14_03_pass_C}\] The “\(\Gamma_C\) independent of \(R\)” judgment may be fixed by the following metric. From an observed force magnitude \(F(R)\) at two distances \(R_1,R_2\), define \[\widehat{K}(R):=F(R)R^{2} \label{eq:S14_03_Khat}\] and define the Gate \[\mathrm{PASS}_{\Gamma} :\Longleftrightarrow \left|\frac{\widehat{K}(R_2)-\widehat{K}(R_1)}{\widehat{K}(R_1)}\right|\le \varepsilon_{K} \label{eq:S14_03_pass_Gamma}\] where \(\varepsilon_{K}\) is locked in gate_lock. If \(\mathrm{PASS}_{\Gamma}=0\), the \(1/R^{2}\) form cannot be produced as a conclusion.

16.4 14.4 Casimir pressure: boundary screening yields a \(1/d^{4}\) force (verification module)

16.4.1 14.4.1 Ideal limiting formula (parallel perfect conductors; \(T=0\))

Let \(d\) be the separation between two parallel conducting plates. In the simplest limit (perfect conductor, parallel plates, zero temperature), the Casimir pressure is \[P_{\mathrm{Cas}}(d) = -\frac{\pi^2\,\hbar_{\mathrm{map}}\,c_{\mathrm{ref}}}{240\,d^{4}} = -\frac{\pi}{480}\frac{h\,c_{\mathrm{ref}}}{d^{4}} \label{eq:S14_04_Casimir_P}\] The negative sign indicates attraction. Within AQD one may interpret \(\hbar_{\mathrm{map}}c_{\mathrm{ref}}\) as an “elasticity/rigidity scale of the spatial lattice,” but the goal of this subsection is not interpretation: it is to fix the \(1/d^{4}\) scaling and numerical magnitude as a verification module.

16.4.2 14.4.2 Numerical example: \(d\)-scan (parallel plates; \(A=1\,\mathrm{cm}^2\))

Using [eq:S14_04_Casimir_P], we list the pressure at representative separations and the force \(F=|P|A\) acting on plates of area \(A=1\,\mathrm{cm}^2(=10^{-4}\,\mathrm{m}^2)\).

Casimir pressure (ideal limiting formula) and the force acting on a \(1\,\mathrm{cm}^2\) plate.
\(d\) (nm) \(d\) (m) \(|P_{\mathrm{Cas}}|\) (Pa) \(|P_{\mathrm{Cas}}|\) (atm) \(|F|\) on \(1\,\mathrm{cm}^2\) (N)
10 \(1\times 10^{-8}\) \(1.3\times 10^{5}\) \(1.28\) \(1.3\times 10^{1}\)
50 \(5\times 10^{-8}\) \(2.08\times 10^{2}\) \(2.05\times 10^{-3}\) \(2.08\times 10^{-2}\)
100 \(1\times 10^{-7}\) \(1.3\times 10^{1}\) \(1.28\times 10^{-4}\) \(1.3\times 10^{-3}\)
200 \(2\times 10^{-7}\) \(8.13\times 10^{-1}\) \(8.02\times 10^{-6}\) \(8.13\times 10^{-5}\)
500 \(5\times 10^{-7}\) \(2.08\times 10^{-2}\) \(2.05\times 10^{-7}\) \(2.08\times 10^{-6}\)
1000 \(1\times 10^{-6}\) \(1.3\times 10^{-3}\) \(1.28\times 10^{-8}\) \(1.3\times 10^{-7}\)

As shown in the table, increasing \(d\) by a factor of 10 decreases the pressure by a factor of \(10^{4}\) (the \(1/d^{4}\) law). At \(d=10\,\mathrm{nm}\), the ideal limit gives \(|P|\approx 1.3\times 10^{5}\,\mathrm{Pa}\approx 1.28\,\mathrm{atm}\).

16.4.3 14.4.3 AQD interpretation: “lattice mode screening” produces a pressure difference

From the AQD viewpoint, the exterior space may be treated as a high-pressure regime where “all lattice modes” exist, whereas the interior gap (\(d\)) is a regime where, due to boundary conditions, certain wavelengths (especially long wavelengths) are suppressed and the effective pressure is lower. The plates may then be described by a pressure difference \[\Delta P(d) := P_{\mathrm{int}}(d)-P_{\mathrm{ext}} \quad\Rightarrow\quad \Delta P(d)\approx P_{\mathrm{Cas}}(d)<0 \label{eq:S14_04_deltaP}\] However, the conclusion-qualified items of this subsection are the functional form [eq:S14_04_Casimir_P] (scaling) and the magnitude (table). The interpretation (vacuum fluctuation vs lattice pressure difference) is separated as an interpretation choice over the same formula.

16.4.4 14.4.4 Geometry conversion for experiment comparison (sphere–plate; PFA)

Many representative experiments use a sphere–plate geometry. For a sphere of radius \(R\) above a plate, if the proximity-force approximation (PFA) is adopted, then from the parallel-plate energy density \(E/A\) one obtains \[F_{\mathrm{sph\text{-}pl}}(d) \simeq 2\pi R\,\frac{E_{\mathrm{pp}}(d)}{A} = -\frac{\pi^{3}\,\hbar_{\mathrm{map}}\,c_{\mathrm{ref}}\,R}{360\,d^{3}} \label{eq:S14_04_PFA_sph_pl}\] Therefore, to claim a “direct comparison with experimental data,” one must specify the geometry (parallel plates or sphere–plate) and the conversion (PFA or an exact expression).

16.4.5 14.4.5 Gate: minimal conditions for experimental comparison (limits of using the ideal formula)

Since [eq:S14_04_Casimir_P] is an ideal limiting formula, to use an experimental comparison as a conclusion one must at minimum specify the following.

16.4.6 14.4.6 Representative experiments (reference) and comparison scope (distance/geometry/corrections must be specified)

The table in this section evaluates the ideal limiting formula [eq:S14_04_Casimir_P] directly. Therefore, to use “agreement with experiment” as a conclusion, at minimum (i) measurement geometry, (ii) distance regime, and (iii) whether corrections are included must match. For reference, representative measurements report the following ranges.

The \(d=10\,\mathrm{nm}\) row in this section shows the order-of-magnitude intuition of the ideal formula, but to use it directly as “Casimir verification” one needs additional Gates to separate realistic regimes involving surface/material/patch potentials/non-retarded (vdW) contributions and contact forces. (For patch potentials/corrections see, e.g., Phys. Rev. A 81, 022505 (2010); for precision-comparison reviews see, e.g., Rev. Mod. Phys. 81, 1827 (2009).)

16.4.7 14.4.7 AQD-specific deviation model (optional): forbidden without pre-registered screening function

Since the Casimir ideal formula itself has the same functional form as standard theory, for AQD to provide an additional prediction the cutoff/discreteness of boundary screening must be locked as a separate function. For example, \[P(d)=P_{\mathrm{Cas}}(d)\,S\!\left(\frac{d}{\lambda_{\star}}\right), \qquad \lim_{x\to\infty}S(x)=1, \label{eq:S14_04_screening_S}\] may be introduced with a screening function \(S(\cdot)\). Here \(\lambda_{\star}\) may be interpreted as a lattice scale (e.g., \(\lambda_{\star}=N_{\star}\,\ell_{\mathrm{rot}}\)) or an effective surface cutoff, but if \(S\) or \(\lambda_{\star}\) is not locked in this document, the default is \(S\equiv 1\). In other words, post-hoc deviation claims without preregistration are forbidden and are set to FAIL.

16.4.7.1 (Gate) Verifying a deviation claim

If \(S\) is adopted, then in an observation/simulation comparison the relative deviation \[\delta_{\mathrm{dev}}(d):=\frac{P_{\mathrm{obs}}(d)}{P_{\mathrm{Cas}}(d)}-1 \label{eq:S14_04_delta_dev}\] must be judged by a Gate as to whether it is quantitatively consistent (including error bars) with \(S(d/\lambda_{\star})-1\). If the corresponding Gate report is not attached, the deviation conclusion is UNLOGGED or INCONCLUSIVE.

16.5 14.5 Geometric derivation of the fine-structure constant \(\alpha_{em}\) (definition\(\rightarrow\)cross-check)

In standard physics the fine-structure constant is defined as \[\alpha_{em} := \frac{e^{2}}{4\pi\epsilon_{0}\,\hbar\,c}\] which is a dimensionless coupling strength; experimentally one finds \(\alpha_{em}^{-1}\approx 137.036\).3

In this document we do not assume \(\alpha_{em}\) as a new primitive constant. Instead, using the integer structures already derived earlier (3-sector, 7-shell), we provide an operational definition of \(\alpha_{em}\) as a “geometric efficiency of coupling to space (\(4\pi\))”.

16.5.1 14.5.1 Geometric impedance (definition): the \(4\pi\) coupling cost

For a charge (a 3-sector phase defect) to emit a wave (mode) outward, it must pass through: (i) the surface rigidity shell, (ii) the sector-phase separation, and (iii) an axial-rotation (spin-like) degree of freedom. We define the inverse coupling strength \(\alpha_{em}^{-1}\) as a geometric impedance: \[\boxed{ \alpha_{em}^{-1} \equiv \mathcal{Z}_{\mathrm{geo}} := 4\pi\,\Bigl(N_{\mathrm{shell}}+N_{\mathrm{sec}}+N_{\mathrm{spin}}-\delta_{\mathrm{proj}}\Bigr) } \label{eq:S14_05_alpha_geo}\] where

and \(\delta_{\mathrm{proj}}\) is the overlap projection correction arising from the rotation of the 7-shell within the 3-sector structure.

16.5.2 14.5.2 Zeroth-order estimate: \(4\pi\times 11\)

In the idealized limit (neglecting the projection correction), \[\alpha_{em,0}^{-1}=4\pi\,(7+3+1)=4\pi\times 11\approx 138.23 \label{eq:S14_05_alpha0}\] which is about \(0.8\%\) larger than the empirical value (\(\approx 137.036\)). Hence a positive projection correction \(\delta_{\mathrm{proj}}>0\) is required.

16.5.3 14.5.3 Closing \(\delta_{\mathrm{proj}}\) without new parameters: continuous rotation average \(\times\) discrete-occupancy correction

This document does not treat \(\delta_{\mathrm{proj}}\) as a new free parameter. Instead it is closed by two steps of geometry.

16.5.3.1 (i) Continuous rotation average (continuous geometry): \(2/\pi^{2}\).

The shell–sector relative phase has no information that would prefer a special direction; by the maximum-entropy / unbiased principle of §5.2.5, the relative angle \(\theta\) is treated as uniformly distributed. We therefore set the basic scale of “projection loss” as \[\delta_{0}:=\frac{2}{\pi^{2}} \label{eq:S14_05_delta0}\] (a \(\pi\)-geometric constant arising from an unbiased angular average). To first order, the loss scales with the sector-to-shell ratio as \((N_{\mathrm{sec}}/N_{\mathrm{shell}})\,\delta_0\).

16.5.3.2 (ii) Discrete occupancy of 7 shells (discrete geometry): \(\beta_{\mathrm{disc}}=35/32\).

However, seven shells cannot be split continuously among three sectors; only integer occupancy is permitted. The most uniform integer partition is \((3,2,2)\) (an integerization of the mean \(7/3\) shells per sector). Fix a \(C_3\) label convention (a label gauge) by calling the sector with occupancy 3 as \(S_0\). The number of configurations (combinatorial microstates) accounting for shell labels (seven distinct vectors) is \[\mathcal{N}_{\mathrm{perm}}=\frac{7!}{3!\,2!\,2!}=210 \label{eq:S14_05_Nperm}\] On the other hand, the “projection correction” admits a reduced representation determined only by (a) the choice of phase origin for 3-sector labels (\(C_3\) cyclic; 3 choices) and (b) the relative signs of each shell (\(\pm\); the global sign is treated as a gauge and removed). Hence the number of projection-equivalent microstates closes as \[\mathcal{N}_{\mathrm{proj}} = 3\times 2^{6}=192 \label{eq:S14_05_Nproj}\] (Note: the ratio \(\beta_{\mathrm{disc}}\) is invariant under label conventions; the key point is that \(35/32\) closes as a pure integer ratio.) Therefore the discrete correction factor that maps the continuous average to discrete occupancy is \[\beta_{\mathrm{disc}}:=\frac{\mathcal{N}_{\mathrm{perm}}}{\mathcal{N}_{\mathrm{proj}}}=\frac{210}{192}=\frac{35}{32} \label{eq:S14_05_beta_disc}\] which is determined purely by integers.

16.5.3.3 (Conclusion) Closed form.

Therefore \[\boxed{ \delta_{\mathrm{proj}}=\beta_{\mathrm{disc}}\,\delta_{0}\,\frac{N_{\mathrm{sec}}}{N_{\mathrm{shell}}} =\frac{35}{32}\cdot\frac{2}{\pi^{2}}\cdot\frac{3}{7} } \label{eq:S14_05_delta_proj_closed}\] Substituting this into [eq:S14_05_alpha_geo] yields \[\alpha_{em,\mathrm{VP}}^{-1} =4\pi\left(11-\frac{35}{32}\cdot\frac{2}{\pi^{2}}\cdot\frac{3}{7}\right) \approx 137.0364. \label{eq:S14_05_alpha_value}\] Thus \(137\) is almost closed by the sphere (\(4\pi\)) and the integer structure (\(7+3+1\)), with the residual removed by a combinatorial correction from the continuous-to-discrete link.

16.5.4 14.5.4 Gate: conditions for using the \(\alpha_{em}\) derivation (definition)

This derivation of \(\alpha_{em}\) may be used as evidence only when the following conditions are locked and judged satisfied: \[\mathrm{PASS}_{\alpha} :\Longleftrightarrow (N_{\mathrm{sec}}=3)\wedge(N_{\mathrm{shell}}=7)\wedge(N_{\mathrm{spin}}=1) \wedge\ \text{(unbiased rotation assumption; \S5.2.5)} \wedge\ \text{(integer occupancy $(3,2,2)$ is pre-registered)}. \label{eq:S14_05_pass_alpha}\] If any condition is missing or post-selected, the numerical conclusion of this subsection is set to FAIL.

17 15. Quantum mechanics mapping (completion note: lock standard-QM elements as a mapping table)

17.1 15.1 Lattice dispersion / plane-wave solutions \(\leftrightarrow\) wavefunction (state space) mapping

17.1.1 15.1.1 Base set and scaling (definition)

Let the lattice dimension be \(d\in\mathbb{N}\), and define the lattice-node set by \[\mathcal{L}:=\mathbb{Z}^d \label{eq:S15_01_L}\] Using the realization length \(a>0\) and the realization time step \(\Delta t>0\) (their units are locked in the Realization chapter), define the real-space coordinate and the physical time corresponding to node \(\mathbf{n}\in\mathcal{L}\) and tick \(k\in\mathbb{Z}\) as \[\mathbf{x}(\mathbf{n}) := a\,\mathbf{n},\qquad t(k):=k\,\Delta t. \label{eq:S15_01_xt}\]

17.1.2 15.1.2 Complex state field from the event log (definition)

For each node \(\mathbf{n}\) and tick \(k\), define the 3-sector event indicators by \[E_s(\mathbf{n},k)\in\{0,1\},\qquad s\in\{1,2,3\}. \label{eq:S15_01_E}\] Let the observation (aggregation) window have length \(M\in\mathbb{N}\) and start tick \(k_0\in\mathbb{Z}\), and define \[W(k_0,M):=\{k_0,k_0+1,\dots,k_0+M-1\}. \label{eq:S15_01_window}\] Define the sector-wise event counts (log aggregates) within the window as \[N_s(\mathbf{n};k_0,M):=\sum_{k\in W(k_0,M)}E_s(\mathbf{n},k). \label{eq:S15_01_Ns}\]

Fix the sector angles by \[\theta_s:=\frac{2\pi}{3}(s-1),\qquad s=1,2,3. \label{eq:S15_01_theta}\] Define the node-wise 3-sector composite phasor by \[Z(\mathbf{n};k_0,M) := \sum_{s=1}^{3} N_s(\mathbf{n};k_0,M)\,e^{i\theta_s} \in\mathbb{C}. \label{eq:S15_01_Z}\] Also define the total event count and the event density by \[N(\mathbf{n};k_0,M):=\sum_{s=1}^{3}N_s(\mathbf{n};k_0,M),\qquad \rho(\mathbf{n};k_0,M):=\frac{1}{M}\,N(\mathbf{n};k_0,M). \label{eq:S15_01_rho}\]

Define the phase (principal value) of the composite phasor by \[\varphi(\mathbf{n};k_0,M):=\arg Z(\mathbf{n};k_0,M)\in(-\pi,\pi]. \label{eq:S15_01_phi}\] Then define the node-wise complex state field (complex amplitude) by \[\psi(\mathbf{n};k_0,M):=\sqrt{\rho(\mathbf{n};k_0,M)}\,e^{i\varphi(\mathbf{n};k_0,M)}. \label{eq:S15_01_psi_def}\] Therefore \[|\psi(\mathbf{n};k_0,M)|^2=\rho(\mathbf{n};k_0,M) \label{eq:S15_01_abs2_rho}\] holds identically. The definitions above are operational: they construct a state field directly from event-log aggregates and do not invoke axioms from any external theory.

17.1.3 15.1.3 State space (definition) and inner product (definition)

Define the complex function space \[\mathcal{S}:=\left\{\psi:\mathcal{L}\to\mathbb{C}\ \bigg|\ \sum_{\mathbf{n}\in\mathcal{L}}|\psi(\mathbf{n})|^2<\infty\right\} \label{eq:S15_01_Sspace}\] as the state space. Define the inner product on \(\mathcal{S}\) by \[\langle \phi,\psi\rangle := \sum_{\mathbf{n}\in\mathcal{L}}\phi(\mathbf{n})^{*}\psi(\mathbf{n}) \label{eq:S15_01_inner}\] and the norm by \(\|\psi\|:=\sqrt{\langle\psi,\psi\rangle}\). A normalized state is defined by \[\widehat{\psi}:=\frac{\psi}{\|\psi\|}\qquad(\psi\neq 0). \label{eq:S15_01_normstate}\]

17.1.4 15.1.4 Discrete Fourier expansion and wave-number space (definition)

Define the \(d\)-dimensional Brillouin zone by \[\mathcal{B}:=[-\pi,\pi)^d. \label{eq:S15_01_BZ}\] For sufficiently fast-decaying states (or for a finite lattice with periodic boundary conditions), define the discrete Fourier transform by \[\widetilde{\psi}(\boldsymbol{\kappa}) := \sum_{\mathbf{n}\in\mathcal{L}} \psi(\mathbf{n})\,e^{-i\boldsymbol{\kappa}\cdot\mathbf{n}}, \qquad \boldsymbol{\kappa}\in\mathcal{B}. \label{eq:S15_01_FT}\] Here \(\boldsymbol{\kappa}\) is the dimensionless lattice wave number. Define the real-space wave vector (with dimension \(L^{-1}\)) by \[\mathbf{k}:=\frac{1}{a}\boldsymbol{\kappa}. \label{eq:S15_01_kmap}\]

17.1.5 15.1.5 One-tick evolution operator, dispersion, and plane waves (definition)

Assume a regime in which tick evolution is linear and translation-invariant (i.e., the same rule under the same closure stack). Define the one-tick evolution operator by \[\psi(\cdot,k+1)=\mathcal{U}\,\psi(\cdot,k). \label{eq:S15_01_Udef}\] Translation invariance means that for every \(\mathbf{m}\in\mathcal{L}\), with the translation operator \(\mathcal{T}_{\mathbf{m}}\) defined by \[(\mathcal{T}_{\mathbf{m}}\psi)(\mathbf{n}) := \psi(\mathbf{n}+\mathbf{m}), \label{eq:S15_01_Tm}\] one has \[\mathcal{U}\,\mathcal{T}_{\mathbf{m}}=\mathcal{T}_{\mathbf{m}}\,\mathcal{U} \qquad(\forall\,\mathbf{m}\in\mathcal{L}). \label{eq:S15_01_commute}\]

Under this condition, define the plane waves (lattice harmonic modes) by \[\phi_{\boldsymbol{\kappa}}(\mathbf{n}) := e^{i\boldsymbol{\kappa}\cdot\mathbf{n}}, \qquad \boldsymbol{\kappa}\in\mathcal{B}. \label{eq:S15_01_planewave}\] Then \(\phi_{\boldsymbol{\kappa}}\) can be an eigenmode of \(\mathcal{U}\). Define its eigenvalue in phase form by \[\mathcal{U}\,\phi_{\boldsymbol{\kappa}} = e^{-i\Omega(\boldsymbol{\kappa})}\,\phi_{\boldsymbol{\kappa}}, \qquad \Omega(\boldsymbol{\kappa})\in(-\pi,\pi]. \label{eq:S15_01_dispersion_phase}\] Define the lattice dispersion (angular frequency) by \[\omega(\boldsymbol{\kappa}) := \frac{1}{\Delta t}\,\Omega(\boldsymbol{\kappa}). \label{eq:S15_01_omega}\]

Therefore, a single-mode plane-wave solution can be written as \[\psi_{\boldsymbol{\kappa}}(\mathbf{n},k) = A\, e^{i\boldsymbol{\kappa}\cdot\mathbf{n}} e^{-i\Omega(\boldsymbol{\kappa})k} = A\, \exp\!\Big(i\boldsymbol{\kappa}\cdot\mathbf{n}-i\omega(\boldsymbol{\kappa})t(k)\Big). \label{eq:S15_01_plane_solution}\] This follows directly from the definitions [eq:S15_01_Udef]–[eq:S15_01_dispersion_phase].

17.1.6 15.1.6 Mapping table to external text (standard QM) (target-text lock)

In this section, “standard QM” is treated as an external target text and is not used as a derivation basis. The following table fixes only a notation mapping to the standard-QM notation (target text).

VP/lattice–event system (definition) Standard-QM target text (term) Mapping rule (locked)
Node \(\mathbf{n}\in\mathcal{L}\) position variable \(\mathbf{x}\) \(\mathbf{x}=a\,\mathbf{n}\)
Tick \(k\in\mathbb{Z}\) time variable \(t\) \(t=k\,\Delta t\)
Complex state field \(\psi(\mathbf{n};k_0,M)\) wavefunction \(\psi(\mathbf{x},t)\) \(\psi(\mathbf{x}(\mathbf{n}),t(k))\leftrightarrow \psi(\mathbf{n},k)\)
Dimensionless wave number \(\boldsymbol{\kappa}\) wave vector \(\mathbf{k}\) \(\mathbf{k}=\boldsymbol{\kappa}/a\)
Eigenphase \(\Omega(\boldsymbol{\kappa})\) angular frequency \(\omega\) \(\omega=\Omega/\Delta t\)
Mode \(\exp(i\boldsymbol{\kappa}\cdot\mathbf{n}-i\Omega k)\) plane wave \(\exp(i\mathbf{k}\cdot\mathbf{x}-i\omega t)\) [eq:S15_01_xt], [eq:S15_01_kmap], [eq:S15_01_omega]

17.1.7 15.1.7 Declared scope: allowed vs forbidden uses (locked)

17.1.7.1 Allowed uses (internal computations in this document)

17.1.7.2 Forbidden uses (may not be used as derivation grounds; target-text only)

17.2.1 15.2.1 Partition of an observation module (definition)

Let the lattice region referenced by the observation module be a finite set \(\Lambda\subset\mathcal{L}\), and define the set of output labels by \[\mathcal{M}:=\{1,2,\dots,M_{\mathrm{out}}\}. \label{eq:S15_02_Mset}\] Define a partition map that splits \(\Lambda\) into mutually disjoint subregions by \[\Pi:\Lambda\to\mathcal{M}, \qquad \Omega_m:=\Pi^{-1}(m), \qquad \Omega_m\cap\Omega_{m'}=\varnothing\ (m\neq m'),\quad \bigcup_{m\in\mathcal{M}}\Omega_m=\Lambda. \label{eq:S15_02_partition}\] The family \(\{\Omega_m\}\) defines the outcome channels (output bins).

17.2.2 15.2.2 Event counts and outcome frequencies (definition)

For an observation window \(W(k_0,M)\), define the event count of outcome channel \(m\) by \[C_m(k_0,M) := \sum_{\mathbf{n}\in\Omega_m}\sum_{k\in W(k_0,M)}\sum_{s=1}^{3}E_s(\mathbf{n},k) = \sum_{\mathbf{n}\in\Omega_m} N(\mathbf{n};k_0,M). \label{eq:S15_02_Cm}\] The total event count is \(C_{\mathrm{tot}}:=\sum_{m\in\mathcal{M}}C_m\). Define the outcome frequency (normalized event frequency) by \[P_m(k_0,M) := \frac{C_m(k_0,M)}{C_{\mathrm{tot}}(k_0,M)} \qquad (C_{\mathrm{tot}}>0). \label{eq:S15_02_Pm}\] Thus \(P_m\) is an outcome frequency computed directly from the event log.

17.2.3 15.2.3 Equivalent expression in terms of the complex state field (definition)

From the definition [eq:S15_01_psi_def] and identity [eq:S15_01_abs2_rho] in §15.1, for each channel one has \[C_m(k_0,M) = M\sum_{\mathbf{n}\in\Omega_m}\rho(\mathbf{n};k_0,M) = M\sum_{\mathbf{n}\in\Omega_m}|\psi(\mathbf{n};k_0,M)|^2. \label{eq:S15_02_Cm_psi}\] Therefore [eq:S15_02_Pm] can be rewritten exactly as \[P_m(k_0,M) = \frac{\sum_{\mathbf{n}\in\Omega_m}|\psi(\mathbf{n};k_0,M)|^2}{\sum_{\mathbf{n}\in\Lambda}|\psi(\mathbf{n};k_0,M)|^2} \qquad \left(\sum_{\mathbf{n}\in\Lambda}|\psi(\mathbf{n};k_0,M)|^2>0\right). \label{eq:S15_02_Pm_bornform}\] Here [eq:S15_02_Pm_bornform] is not derived from the foundations of standard QM. It is an identity that holds equivalently by construction from the event-log definition and [eq:S15_01_psi_def].

17.2.4 15.2.4 Gate decision (condition for a measurement event; definition)

A measurement is an event that creates a log entry “some \(m\) was selected.” To formalize this, define an outcome-selection rule (gate) as \[m^\star := \operatorname*{arg\,max}_{m\in\mathcal{M}} \ \mathcal{G}_m\big(C_m(k_0,M);\Theta_m\big). \label{eq:S15_02_mstar}\] Here \(\mathcal{G}_m(\cdot;\Theta_m)\) is the gate function for channel \(m\), and \(\Theta_m\) is a pre-registered set of threshold/decision parameters. Define the condition for producing a measurement log by \[\mathrm{PASS}_{\mathrm{meas}} :\Longleftrightarrow \left[ \exists\,m^\star\in\mathcal{M}\ \text{s.t.}\ \mathcal{G}_{m^\star}\big(C_{m^\star}(k_0,M);\Theta_{m^\star}\big)=1 \right]. \label{eq:S15_02_PASS_meas}\] If \(\mathrm{PASS}_{\mathrm{meas}}\) does not hold, the protocol does not generate a measurement log (no outcome selection).

17.2.5 15.2.5 Operational definition of collapse (conditional renormalization) (definition)

When the measurement outcome \(m^\star\) is fixed in the log (i.e., when [eq:S15_02_PASS_meas] holds), define the following mask operator (channel filter) on the state field \(\psi(\cdot;k_0,M)\) from the same window: \[(\mathcal{P}_{m}\psi)(\mathbf{n}) := \begin{cases} \psi(\mathbf{n}), & \mathbf{n}\in\Omega_m,\\ 0, & \mathbf{n}\in\Lambda\setminus\Omega_m, \end{cases} \qquad \mathbf{n}\in\Lambda. \label{eq:S15_02_Pmop}\] Then define the post-measurement state by conditional renormalization: \[\widehat{\psi}_{\mathrm{post}} := \frac{\mathcal{P}_{m^\star}\widehat{\psi}_{\mathrm{pre}}}{\left\|\mathcal{P}_{m^\star}\widehat{\psi}_{\mathrm{pre}}\right\|} \qquad \left(\left\|\mathcal{P}_{m^\star}\widehat{\psi}_{\mathrm{pre}}\right\|>0\right). \label{eq:S15_02_collapse}\] Here \(\widehat{\psi}_{\mathrm{pre}}\) is the normalized state immediately before measurement (computed under the same protocol). Equation [eq:S15_02_collapse] is a convention for re-expressing the state after the event log is fixed, and cannot be applied without a gate pass.

17.2.6 15.2.6 Mapping table to the standard-QM target text (target-text lock)

In this subsection, standard-QM terms (Born rule, measurement, collapse) are treated as target text. The following table fixes only the notation mapping based on [eq:S15_02_Pm_bornform] and [eq:S15_02_collapse].

VP/event–gate system (definition) Standard-QM target text (term) Mapping rule (locked)
Outcome frequency \(P_m\) probability \(p_m\) \(p_m\leftrightarrow P_m\)
\(\sum_{\Omega_m}|\psi|^2/\sum_{\Lambda}|\psi|^2\) “\(|\psi|^2\) rule” [eq:S15_02_Pm_bornform]
Gate pass \(\mathrm{PASS}_{\mathrm{meas}}\) condition for measurement to occur [eq:S15_02_PASS_meas]
Conditional renormalization [eq:S15_02_collapse] projection (collapse) notation notation mapping only

17.2.7 15.2.7 Declared scope: allowed vs forbidden uses (locked)

17.2.7.1 Allowed uses (internal computations in this document)

17.2.7.2 Forbidden uses (may not be used as derivation grounds; target-text only)

17.3.1 15.3.1 Operational definition of observables (theorem)

Define a linear operator \(\mathcal{O}:\mathcal{S}\to\mathcal{S}\) on the state space \(\mathcal{S}\) as an observable operator. For a normalized state \(\widehat{\psi}\), define its expectation and variance by \[\langle \mathcal{O}\rangle_{\widehat{\psi}} := \langle \widehat{\psi},\mathcal{O}\widehat{\psi}\rangle, \qquad (\Delta \mathcal{O})^2_{\widehat{\psi}} := \left\langle \widehat{\psi},(\mathcal{O}-\langle \mathcal{O}\rangle_{\widehat{\psi}})^2\widehat{\psi}\right\rangle. \label{eq:S15_03_mean_var}\] Equation [eq:S15_03_mean_var] is a purely operator-algebraic definition determined by the inner product [eq:S15_01_inner].

17.3.2 15.3.2 Basic operators: position, translation, and finite differences (theorem)

For coordinate component \(j\in\{1,\dots,d\}\), define the position (multiplication) operator \(\mathcal{X}_j\) by \[(\mathcal{X}_j\psi)(\mathbf{n}) := x_j(\mathbf{n})\,\psi(\mathbf{n}) = a\,n_j\,\psi(\mathbf{n}). \label{eq:S15_03_X}\] For the unit lattice vector \(\mathbf{e}_j\), define the translation (shift) operator \(\mathcal{T}_j\) by \[(\mathcal{T}_j\psi)(\mathbf{n}) := \psi(\mathbf{n}+\mathbf{e}_j), \qquad (\mathcal{T}_j^{-1}\psi)(\mathbf{n}) := \psi(\mathbf{n}-\mathbf{e}_j). \label{eq:S15_03_T}\]

The commutation relation (exact identity) follows by direct computation: \[\begin{aligned} (\mathcal{X}_j\mathcal{T}_j\psi)(\mathbf{n}) &= a\,n_j\,\psi(\mathbf{n}+\mathbf{e}_j), \notag\\ (\mathcal{T}_j\mathcal{X}_j\psi)(\mathbf{n}) &= a\,(n_j+1)\,\psi(\mathbf{n}+\mathbf{e}_j), \notag\\ \Rightarrow\quad \big([\mathcal{X}_j,\mathcal{T}_j]\psi\big)(\mathbf{n}) &= (\mathcal{X}_j\mathcal{T}_j-\mathcal{T}_j\mathcal{X}_j)\psi(\mathbf{n}) = -a\,\psi(\mathbf{n}+\mathbf{e}_j) = -a\,(\mathcal{T}_j\psi)(\mathbf{n}). \label{eq:S15_03_comm_XT}\end{aligned}\] Hence \[=-a\,\mathcal{T}_j, \qquad [\mathcal{X}_j,\mathcal{T}_j^{-1}]=+a\,\mathcal{T}_j^{-1} \label{eq:S15_03_comm_XT2}\] holds exactly.

17.3.3 15.3.3 Momentum-like operator and commutation relation (theorem + hypothesis)

17.3.3.1 (Theorem) Symmetric-difference generator

Define the symmetric-difference operator \(\mathcal{D}_j\) by \[\mathcal{D}_j := \frac{1}{2a}\big(\mathcal{T}_j-\mathcal{T}_j^{-1}\big). \label{eq:S15_03_D}\]

17.3.3.2 (Hypothesis H-Pmap) Mapping to the “momentum” notation in the standard-QM target text

Define the constant \[\hbar_{\mathrm{map}}:=\frac{h}{2\pi} \label{eq:S15_03_hbar_map}\] as a mapping constant (derived from the realization input \(h\)). Define the momentum-like operator \(\mathcal{P}_j\) by \[\mathcal{P}_j := \frac{\hbar_{\mathrm{map}}}{i}\,\mathcal{D}_j = \frac{\hbar_{\mathrm{map}}}{2ia}\big(\mathcal{T}_j-\mathcal{T}_j^{-1}\big). \label{eq:S15_03_P}\] By [eq:S15_03_comm_XT2], \(\mathcal{P}_j\) satisfies the exact commutation relation \[\begin{aligned} &= \frac{\hbar_{\mathrm{map}}}{2ia}\Big([\mathcal{X}_j,\mathcal{T}_j]-[\mathcal{X}_j,\mathcal{T}_j^{-1}]\Big) \notag\\ &= \frac{\hbar_{\mathrm{map}}}{2ia}\Big(-a\mathcal{T}_j-a\mathcal{T}_j^{-1}\Big) = \frac{i\hbar_{\mathrm{map}}}{2}\big(\mathcal{T}_j+\mathcal{T}_j^{-1}\big). \label{eq:S15_03_comm_XP_exact}\end{aligned}\]

For the plane-wave mode [eq:S15_01_planewave], \[\mathcal{P}_j\,\phi_{\boldsymbol{\kappa}} = \left(\frac{\hbar_{\mathrm{map}}}{a}\sin\kappa_j\right)\phi_{\boldsymbol{\kappa}} \label{eq:S15_03_P_eig}\] holds. Therefore, in the small-\(\kappa_j\) regime (slow-variation regime) where \(\sin\kappa_j\simeq \kappa_j\), \[\mathcal{P}_j\,\phi_{\boldsymbol{\kappa}} \simeq \left(\hbar_{\mathrm{map}}\,\frac{\kappa_j}{a}\right)\phi_{\boldsymbol{\kappa}} = \left(\hbar_{\mathrm{map}}\,k_j\right)\phi_{\boldsymbol{\kappa}} \qquad(\kappa_j\to 0). \label{eq:S15_03_P_smallk}\] We map this approximation to the standard-QM target-text notation \(p=\hbar k\) only as a translation rule, and it must not be used as a derivation basis.

17.3.4 15.3.4 Uncertainty inequality (theorem; derivation as a mathematical theorem)

Derive an operator uncertainty inequality using the Cauchy–Schwarz inequality in an inner-product space: \[|\langle u,v\rangle|^2\le \langle u,u\rangle\,\langle v,v\rangle. \label{eq:S15_03_CS}\] For a normalized state \(\widehat{\psi}\) and two operators \(\mathcal{A},\mathcal{B}\), define \[\delta\mathcal{A}:=\mathcal{A}-\langle\mathcal{A}\rangle_{\widehat{\psi}},\qquad \delta\mathcal{B}:=\mathcal{B}-\langle\mathcal{B}\rangle_{\widehat{\psi}}. \label{eq:S15_03_dA_dB}\] Let \[u:=\delta\mathcal{A}\,\widehat{\psi},\qquad v:=\delta\mathcal{B}\,\widehat{\psi}. \label{eq:S15_03_u_v}\] Then \[\langle u,u\rangle = \langle \widehat{\psi},(\delta\mathcal{A})^2\widehat{\psi}\rangle = (\Delta\mathcal{A})^2_{\widehat{\psi}}, \qquad \langle v,v\rangle = (\Delta\mathcal{B})^2_{\widehat{\psi}}. \label{eq:S15_03_var_uv}\] By [eq:S15_03_CS], \[(\Delta\mathcal{A})^2_{\widehat{\psi}}\,(\Delta\mathcal{B})^2_{\widehat{\psi}} \ge |\langle u,v\rangle|^2 = \left|\left\langle \widehat{\psi},\delta\mathcal{A}\,\delta\mathcal{B}\,\widehat{\psi}\right\rangle\right|^2. \label{eq:S15_03_RS0}\] Let \(z:=\langle \widehat{\psi},\delta\mathcal{A}\delta\mathcal{B}\widehat{\psi}\rangle\in\mathbb{C}\) and decompose \[z = \frac{1}{2}\langle \widehat{\psi},\{\delta\mathcal{A},\delta\mathcal{B}\}\widehat{\psi}\rangle + \frac{1}{2}\langle \widehat{\psi},[\delta\mathcal{A},\delta\mathcal{B}]\widehat{\psi}\rangle. \label{eq:S15_03_split}\] Here \(\{X,Y\}:=XY+YX\) is the anticommutator and \([X,Y]:=XY-YX\) is the commutator. Therefore \[|z|^2 = \left(\Re z\right)^2+\left(\Im z\right)^2 \ge \left(\Im z\right)^2. \label{eq:S15_03_ReIm}\] Also \[\Im z = \frac{1}{2i}\left(z-z^{*}\right) = \frac{1}{2i}\left\langle \widehat{\psi},\left(\delta\mathcal{A}\delta\mathcal{B}-\delta\mathcal{B}\delta\mathcal{A}\right)\widehat{\psi}\right\rangle = \frac{1}{2i}\langle \widehat{\psi},[\mathcal{A},\mathcal{B}]\widehat{\psi}\rangle. \label{eq:S15_03_Imz}\] Combining [eq:S15_03_RS0]–[eq:S15_03_Imz] yields \[\Delta\mathcal{A}\,\Delta\mathcal{B} \ge \frac{1}{2}\left|\left\langle \widehat{\psi},\frac{1}{i}[\mathcal{A},\mathcal{B}]\,\widehat{\psi}\right\rangle\right| = \frac{1}{2}\left|\langle [\mathcal{A},\mathcal{B}]\rangle_{\widehat{\psi}}\right|. \label{eq:S15_03_uncertainty}\] Equation [eq:S15_03_uncertainty] is a mathematical theorem derived from the inner product and operator definitions.

17.3.5 15.3.5 Spin-like label from 3-sector structure (hypothesis) and statistics (hypothesis)

17.3.5.1 (Theorem) 3-sector probability vector

For node \(\mathbf{n}\) and window \(W(k_0,M)\), define the sector fractions by \[p_s(\mathbf{n};k_0,M):=\frac{N_s(\mathbf{n};k_0,M)}{N(\mathbf{n};k_0,M)} \qquad(N(\mathbf{n};k_0,M)>0), \qquad \sum_{s=1}^{3}p_s=1. \label{eq:S15_03_ps}\] Define the normalized sector phasor by \[u(\mathbf{n};k_0,M):=\sum_{s=1}^{3}p_s(\mathbf{n};k_0,M)\,e^{i\theta_s} \label{eq:S15_03_u}\] and its phase by \[\Phi(\mathbf{n};k_0,M):=\arg u(\mathbf{n};k_0,M)\in(-\pi,\pi]. \label{eq:S15_03_Phi}\] Thus \(\Phi\) is an angular variable computed from the 3-sector occupancy fractions.

17.3.5.2 (Hypothesis H-S1) Definition of a two-valued spin-like label

Consider two consecutive windows \((k_0,M)\) and \((k_0+M,M)\) and define the phase increment by \[\Delta\Phi(\mathbf{n};k_0,M) := \mathrm{Wrap}_{(-\pi,\pi]}\!\Big(\Phi(\mathbf{n};k_0+M,M)-\Phi(\mathbf{n};k_0,M)\Big). \label{eq:S15_03_dPhi}\] Here \(\mathrm{Wrap}_{(-\pi,\pi]}\) reduces a value into the interval \((-\pi,\pi]\). Define the spin-like label by \[\sigma(\mathbf{n};k_0,M) := \mathrm{sgn}\big(\Delta\Phi(\mathbf{n};k_0,M)\big) \in\{-1,0,+1\}. \label{eq:S15_03_sigma}\] Treat \(\sigma=0\) as indeterminate (gate failure). Map only \(\sigma=\pm1\) to the standard-QM target-text “two-valued spin projection” notation, and classify \(\sigma=0\) as a state with no target-text counterpart.

17.3.5.3 (Hypothesis H-STAT1) Exclusive-occupancy rule and mapping to statistics

Define an exclusive-occupancy regime as a regime in which two objects with the same label cannot stably co-exist simultaneously on the same node (or the same slot in a contact graph). Declare the occupancy rule as \[\mathcal{N}(\mathbf{n})\in\{0,1\}\quad(\text{exclusive-occupancy regime}), \qquad \mathcal{N}(\mathbf{n})\in\{0,1,2,\dots\}\quad(\text{non-exclusive regime}). \label{eq:S15_03_occupancy}\] Assign the standard-QM target-text terms “Fermi/Bose statistics” to the regime classification above only as a label mapping.

17.3.6 15.3.6 Validation items (judged only by gates; fixed list)

17.3.6.1 (V-OP) Operator-level agreement items

17.3.6.2 (V-S) Spin-like label items

17.3.6.3 (V-STAT) Statistics-regime items

17.4 15.4 Standard Model boundary: share (observational numbers) vs ban (gauge dynamics), and scope declaration

17.4.1 15.4.1 Form of the boundary declaration (definition)

In this section, “Standard Model” and “standard QM” are treated as external target texts. The role of this chapter is to provide mapping tables that translate target-text terms/symbols into this document’s operational definitions (LOCK, Gate, event log). It provides no justification and no derivational support. Accordingly, fix the following two categories: \[\mathcal{C}_{\mathrm{share}}:\ \text{share (observational numbers / reporting units)}, \qquad \mathcal{C}_{\mathrm{ban}}:\ \text{ban (target-text dynamics / axioms / structure)}. \label{eq:S15_04_share_ban}\]

17.4.2 15.4.2 Shared items \(\mathcal{C}_{\mathrm{share}}\) (observational numbers and reporting units; fixed list)

The following items are shared only as “reported quantities” or “reporting units.” Sharing means unit conversion and table-based correspondence; it does not mean adopting the target text’s theoretical structure.

17.4.3 15.4.3 Banned items \(\mathcal{C}_{\mathrm{ban}}\) (gauge dynamics and axiom structure; fixed list)

The following items, even if present in the target text, are not permitted in the derivation chain of this document.

17.4.4 15.4.4 Mapping table to the standard-QM/Standard-Model target text (including banned scope; fixed)

This document (operational definitions / outputs) Target text (term) Handling (share / ban)
Event log \(E_s\), counts \(N_s\), frequencies \(P_m\) probability \(p\) share: report item; ban as justification
Complex state field \(\psi\) (event density + phase construction) wavefunction \(\psi\) share: notation mapping; ban as axiom
Gate \(\mathrm{PASS}\) and conditional renormalization measurement/collapse share: notation mapping; ban as axiom
Operators \(\mathcal{X},\mathcal{T},\mathcal{P}\) and commutators operators/commutators share: mapping table; ban target-text axioms
3-sector phase-increment label \(\sigma\) spin (projection) share: label mapping; ban dynamics structure
Exclusive/non-exclusive occupancy regimes statistics (Fermi/Bose) share: label mapping; ban target-text principle

17.4.5 15.4.5 Effective scope (regime declaration; locked)

The mapping of this chapter is defined only in the following regimes:

If these regime conditions do not hold, no conclusions may be produced using the target-text mapping tables.

17.5 15.5 Structural derivation of blackbody radiation (Planck’s law): rigidity-shell filtering

This section derives the Planck spectrum without introducing energy quantization (\(E=n h\nu\)) as an axiom. Instead, it is derived from the structural threshold (barrier) of the rigidity shell in the VP lattice and from integer event counts.4

17.5.1 15.5.1 Geometry of mode count (statement): \(g(\nu)\propto \nu^{2}\)

In a 3D cavity, the number of wave modes is given by the volume of a spherical shell in \(k\)-space. As a function of frequency \(\nu\), the mode density per unit volume scales geometrically as \[g(\nu)\,d\nu \propto \nu^{2}\,d\nu \label{eq:S15_05_gnu}\] (continuum geometry). Including the usual electromagnetic prefactors gives \(g(\nu)=8\pi\nu^{2}/c_{\mathrm{ref}}^{3}\), but the key point here is that the \(\nu^{2}\) scaling comes from pure geometry.

17.5.2 15.5.2 Rigidity-shell barrier and the action constant \(h_{\mathrm{VP}}\) (definition)

In VP theory, “softness” is not a property of the particle itself but the result of structural events (§3.2.1). For a high-frequency mode to exist, the lattice rigidity shell must be periodically deformed. Under linear elastic response, the strain rate \(\dot{\varepsilon}\) is proportional to frequency. Hence the minimum barrier energy \(\epsilon_b\) required to penetrate the shell (or to cross the critical deformation once) satisfies \[\epsilon_b(\nu) \propto \dot{\varepsilon} \propto \nu.\] Fix the proportionality constant in action (energy\(\times\)time) units and define \[\boxed{\epsilon_b(\nu) := h_{\mathrm{VP}}\,\nu} \label{eq:S15_05_barrier}\] Here \(h_{\mathrm{VP}}\) is not an arbitrary “unknown constant”; it is the minimum action (structural resistance coefficient) required for the VP shell to allow one cycle of deformation.

17.5.3 15.5.3 Integer event count \(n\) and the emergence of \((e^{x}-1)^{-1}\) (derivation)

The key point is that “excitation” is not continuous but is tallied as the number of barrier-crossing events. In a mode of frequency \(\nu\), one barrier crossing costs \(\epsilon_b(\nu)=h_{\mathrm{VP}}\nu\); therefore the accumulated energy after \(n\) crossing events is \[E_n(\nu) = n\,h_{\mathrm{VP}}\,\nu,\qquad n\in\{0,1,2,\dots\}. \label{eq:S15_05_En}\] This is not “postulating \(E=n h\nu\)” but follows from the fact that \(n\) is an event count and thus must be an integer (discreteness).

In thermal equilibrium at temperature \(T\), the Boltzmann weight is \[P(n\mid \nu) \propto \exp\!\left(-\frac{E_n(\nu)}{k_B T}\right) =\exp\!\left(-\frac{n h_{\mathrm{VP}}\nu}{k_B T}\right) \label{eq:S15_05_Pn}\] and the partition function closes as a geometric series: \[Z(\nu)=\sum_{n=0}^{\infty} e^{-n\beta h_{\mathrm{VP}}\nu}=\frac{1}{1-e^{-\beta h_{\mathrm{VP}}\nu}},\qquad \beta:=\frac{1}{k_B T}. \label{eq:S15_05_Z}\] Therefore the mean energy per mode is \[\langle E\rangle(\nu) = -\frac{\partial}{\partial \beta}\ln Z =\frac{h_{\mathrm{VP}}\nu}{e^{h_{\mathrm{VP}}\nu/(k_B T)}-1}. \label{eq:S15_05_Emean}\] This term produces the characteristic “\(-1\)” of the Planck spectrum.

17.5.4 15.5.4 Planck spectrum (conclusion): geometry \(\times\) stiffness filter

The energy density per unit volume is \(u(\nu)=g(\nu)\langle E\rangle(\nu)\), hence \[u(\nu) \propto \underbrace{\nu^{2}}_{\text{geometric mode count}} \times \underbrace{\frac{h_{\mathrm{VP}}\nu}{e^{h_{\mathrm{VP}}\nu/(k_B T)}-1}}_{\text{rigidity-shell filter (discrete events)}} \;\;\Rightarrow\;\; u(\nu)\propto \frac{\nu^{3}}{e^{h_{\mathrm{VP}}\nu/(k_B T)}-1}. \label{eq:S15_05_planck_shape}\] Thus the high-frequency cutoff is not a “mystery of light”; it occurs because the rigidity shell structurally blocks (filters) high-frequency deformation.

17.5.4.1 Interpretive note (NON-LOCK)

If one loosely says “light is gas-like,” this should be read as a statement about excitation statistics on top of a jammed background (State 4), not as a claim that the vacuum medium itself is unjammed. A concise regime dictionary for this distinction is provided in Appendix L (see L.5).

17.5.5 15.5.5 Gate: conditions for using the blackbody regime (definition)

This derivation may be used as a “Planck spectrum derivation” only when the following conditions hold: \[\mathrm{PASS}_{\mathrm{BB}} :\Longleftrightarrow \text{(thermal equilibrium)}\wedge\text{(optically thick cavity)}\wedge\text{(linear response)}\wedge\text{(integer event count $n$ holds)}. \label{eq:S15_05_pass_BB}\] If conditions are not satisfied, this section is set to INCONCLUSIVE.

18 16. DOI

18.1 16.1 Protocol registry, pre-registration, and falsification-trigger rules (FAIL is also public)

18.1.1 16.1.1 Purpose and scope (definition)

This chapter defines operational rules that ensure every artifact produced throughout the main text is fixed in a form that is verifiable, reproducible, and distributable. Here:

This chapter does not add new derivations (theory). It only covers artifact qualification (verdict), recording (logs), and preservation (archives).

18.1.2 16.1.2 Core objects (definition): protocol, registry, registration identifiers

18.1.2.1 (Definition) Protocol

A protocol \(\mathsf{P}\) is defined as a 7-tuple: \[\mathsf{P}:=\big(\mathsf{S},\mathsf{I},\mathsf{R},\mathsf{E},\mathsf{G},\mathsf{O},\mathsf{A}\big) \label{eq:S16_01_protocol_tuple}\] Each component is fixed as follows.

18.1.2.2 (Definition) Protocol registry

A protocol registry \(\mathcal{R}_{\mathrm{prot}}\) is an immutable list that stores protocols and their metadata. Each registry entry is \[\mathcal{R}_{\mathrm{prot}}[i]=\big(\mathrm{PID},\mathrm{VER},\mathrm{DIG},\mathrm{TS},\mathrm{STATUS}\big) \label{eq:S16_01_registry_item}\] with the following fixed meanings.

The registry is append-only; modification/deletion of existing entries is forbidden. If a correction is needed, a new versioned entry must be added.

18.1.3 16.1.3 Pre-registration rules (definition)

18.1.3.1 (Definition) Pre-registration file

Pre-registration is the act of recording all components of \(\mathsf{P}\) into a fixed-schema file. We define the pre-registration file as \[\texttt{protocol.yaml} \quad \text{or}\quad \texttt{protocol.json} \label{eq:S16_01_prereg_file}\] The pre-registration file must include the following fields.

18.1.3.2 (Definition) Pre-registration digest and fixed point

Define the checksum digest of the pre-registration file as \[\mathrm{DIG}_{\mathrm{prot}}:=\mathrm{SHA256}(\texttt{protocol.*}) \label{eq:S16_01_dig_prot}\] The protocol is frozen by \(\mathrm{DIG}_{\mathrm{prot}}\). If \(\mathrm{DIG}_{\mathrm{prot}}\) changes under the same \(\mathrm{PID}\), then \(\mathrm{VER}\) must increase. The pre-registration digest is a prerequisite for every run and must be included in the run logs and the DOI archive.

18.1.4 16.1.4 PASS.rules and verdict hierarchy (definition)

18.1.4.1 (Definition) Gate and PASS.rules

A Gate is defined as a boolean function that decides artifact eligibility. Each Gate \(g\) takes a summary vector \(\mathbf{m}\) and a threshold set \(\Theta\), and returns \[g(\mathbf{m};\Theta)\in\{0,1\} \label{eq:S16_01_gate_bool}\] Let the set of Gates be \(\mathcal{G}=\{g_1,\dots,g_K\}\). PASS.rules is defined as a logical composition of Gates. For example, one may fix \[\mathrm{PASS}:=\bigwedge_{k=1}^{K} g_k(\mathbf{m}_k;\Theta_k) \label{eq:S16_01_pass_rules}\] PASS.rules must be stated in the pre-registration file, and post-run changes are forbidden.

18.1.4.2 (Definition) Verdict classes

Each run is assigned exactly one verdict class: \[\mathrm{STATUS}\in\{\mathrm{PASS},\mathrm{FAIL},\mathrm{INCONCLUSIVE}\} \label{eq:S16_01_status}\]

18.1.5 16.1.5 Falsification triggers (definition) and FAIL disclosure (rule)

18.1.5.1 (Definition) Trigger set

Define the falsification-trigger set as \(\mathcal{T}=\{T_1,\dots,T_L\}\). Each trigger \(T_\ell\) takes as input summary metrics and integrity metadata, and returns \[T_\ell(\mathbf{m},\mathbf{u})\in\{0,1\} \label{eq:S16_01_trigger_bool}\] where \(\mathbf{m}\) is a numeric summary (metrics) and \(\mathbf{u}\) is an integrity summary (checksums, schema conformance, log completeness, etc.).

18.1.5.2 (Definition) Minimal triggers (required)

The pre-registration must include at least the following triggers.

18.1.5.3 (Rule) Minimal FAIL disclosure unit

FAIL must not be concealed, deleted, or partially disclosed. FAIL disclosure must satisfy the following minimum unit. \[\mathsf{FAILPACK}:=\{\texttt{protocol.*},\ \texttt{locks/*},\ \texttt{run\_log.jsonl},\ \texttt{manifest.*},\ \texttt{registry\_snapshot.json},\ \texttt{outputs/*}\} \label{eq:S16_01_failpack}\] FAILPACK is subject to the same integrity rules as PASS artifacts (including checksums). FAILPACK must be included in the DOI archive, and the very existence of FAIL must be discoverable from the registry.

18.1.6 16.1.6 Falsification tree structure (definition): cause classes and code system

18.1.6.1 (Definition) Cause classes

Fix the set of cause classes as \[\mathcal{C}_{\mathrm{fail}}=\{\mathrm{LOCK},\mathrm{INTEGRITY},\mathrm{SCHEMA},\mathrm{GATE},\mathrm{PROCEDURE}\} \label{eq:S16_01_fail_classes}\] Each FAIL run must have at least one cause class.

18.1.6.2 (Definition) Failure codes

Failure codes are fixed to the form F-<CLASS>-<NNN>. Examples:

Failure codes must be recorded in both the log and the registry, and must be included in the DOI archive.

18.2 16.2 SSOT

18.2.1 16.2.1 SSOT (single source of truth) principle (definition)

The SSOT principle means: “inputs/constants/rules with the same meaning do not exist scattered across duplicated files.” SSOT is fixed in the following three layers.

SSOT violations are treated as failures via triggers (T-NT) or (T-SCH).

18.2.2 16.2.2 Run identification and directory conventions (definition)

18.2.2.1 (Definition) run_id

Each run has a globally unique identifier run_id. Define run_id as a string generated by \[\texttt{run\_id}:=\texttt{YYYYMMDDThhmmssZ}\ \Vert\ \texttt{pid}\ \Vert\ \texttt{short\_digest} \label{eq:S16_02_runid}\] Here \(\Vert\) denotes string concatenation. Define short_digest as a fixed-length prefix such as the first 12 characters of [eq:S16_01_dig_prot].

18.2.2.2 (Definition) Run directory

The run directory tree is fixed as follows.

runs/
  <run_id>/
    protocol.locked.json
    locks_snapshot/
      canon_lock.json
      realization_lock.json
      analysis_lock.json
    registry_snapshot.json
    run_log.jsonl
    metrics.json
    outputs/
      ...
    manifest.json
    manifest.sha256

protocol.locked.json is a normalized JSON copy of protocol.*; its digest must match [eq:S16_01_dig_prot].

18.2.3 16.2.3 Log specification (run_log.jsonl) and schema (definition)

18.2.3.1 (Definition) JSON Lines log

The run log is fixed to the JSON Lines format. That is, run_log.jsonl is a sequence in which each line is an independent JSON object. Each log entry \(L_i\) must satisfy the schema \[L_i= \{\texttt{ts},\texttt{run\_id},\texttt{pid},\texttt{ver},\texttt{phase},\texttt{event},\texttt{payload}\} \label{eq:S16_02_log_schema}\] The meaning and type of each field are fixed as follows.

18.2.3.2 (Definition) Required log events

Each run must include the following events.

Missing a required event is treated as a failure via [eq:S16_01_TSCH].

18.2.4 16.2.4 Metric file (metrics.json) specification (definition)

Metrics are recorded as METRIC events in the log, but they are also summarized as a single object in metrics.json at the end of the run. \[\texttt{metrics.json}= \{\texttt{run\_id},\texttt{pid},\texttt{ver},\texttt{metrics},\texttt{units},\texttt{provenance}\} \label{eq:S16_02_metrics_schema}\] where

are fixed.

18.2.5 16.2.5 Manifest specification (definition) and checksum files

18.2.5.1 (Definition) manifest.json

A manifest is an SSOT file that contains a complete list of artifacts and their checksums. Fix the schema of manifest.json as \[\texttt{manifest.json}= \{\texttt{run\_id},\texttt{pid},\texttt{ver},\texttt{files},\texttt{hash\_alg}\} \label{eq:S16_02_manifest_schema}\] Fix hash_alg to the string "sha256". files is a list whose each entry must satisfy \[f= \{\texttt{path},\texttt{bytes},\texttt{sha256}\} \label{eq:S16_02_manifest_fileitem}\]

18.2.5.2 (Definition) manifest.sha256

Fix manifest.sha256 as a one-line (or multi-line) text in the following format.

<sha256_of_manifest.json>  manifest.json

The checksum of manifest.json itself must also be included in the registry snapshot.

18.2.6 16.2.6 registry_snapshot specification (definition)

18.2.6.1 (Definition) registry_snapshot.json

registry_snapshot.json stores, in a single snapshot file, all fixed points required to reproduce a run. Fix its schema as \[\texttt{registry\_snapshot.json}= \{\texttt{run\_id},\texttt{protocol},\texttt{locks},\texttt{environment},\texttt{code},\texttt{randomness},\texttt{manifests}\} \label{eq:S16_02_snapshot_schema}\] Each field must include the following.

18.2.7 16.2.7 Randomness/seed convention (definition) and minimal reproducibility condition

18.2.7.1 (Definition) Seed system

Every stochastic element of a run must be frozen by explicit seeds. Fix a base seed \(\mathrm{seed}_0\in\mathbb{N}\) and per-phase seed derivation as \[\mathrm{seed}_{p}:=\mathrm{H}(\mathrm{seed}_0,\texttt{pid},\texttt{phase\_name},\texttt{run\_id}) \label{eq:S16_02_seed_derive}\] where \(\mathrm{H}\) is a pre-registered hash-based function. The digest of \(\mathrm{H}\) must be included in registry_snapshot.json. Per-phase seeds must be recorded in the log and snapshot; missing records are treated as failures via [eq:S16_01_TSCH].

18.2.7.2 (Definition) Minimal reproducibility condition

Define reproducibility as the following simultaneous condition. \[\mathrm{REPRODUCIBLE}=1 \Longleftrightarrow \big(T_{\mathrm{LOCK}}=0\big)\wedge\big(T_{\mathrm{MAN}}=0\big)\wedge\big(T_{\mathrm{SCH}}=0\big) \label{eq:S16_02_repro_min}\] That is, if any of LOCK mismatch, integrity failure, or schema failure occurs, reproducibility does not hold.

18.3 16.3 Reference implementation execution recipe, artifact checklist, and release rules

18.3.1 16.3.1 Definition of the reference implementation (definition)

18.3.1.1 (Definition) Reference implementation

Define the reference implementation \(\mathcal{I}_{\mathrm{ref}}\) as the execution code set that satisfies the following conditions.

Violation of (RI-1) is treated as a No-Tuning violation and is failed via the trigger [eq:S16_01_TNT].

18.3.2 16.3.2 Execution recipe (definition): inputs/outputs/phases

18.3.2.1 (Definition) Run plan (plan) file

Define the run plan plan.json as the file that enumerates the experiment set allowed by the protocol. Each run item must satisfy \[\texttt{job}= \{\texttt{job\_id},\texttt{pid},\texttt{protocol\_digest},\texttt{locks},\texttt{params},\texttt{seeds},\texttt{targets}\} \label{eq:S16_03_plan_job}\] params may contain only parameters allowed by the pre-registration. If any key outside the allowed list is present, the run is immediately failed as F-PROCEDURE-020.

18.3.2.2 (Definition) Execution entry points

Fix the reference implementation to provide the following entry points.

ref_impl/
  run_one.py
  run_plan.py
  validate.py

Fix the responsibilities of each entry point as follows.

18.3.2.3 (Definition) Command forms

Fix the command forms as

python ref_impl/run_plan.py --protocol protocol.yaml --plan plan.json --out runs/
python ref_impl/validate.py --runs runs/ --protocol protocol.yaml

These commands define the interface; the execution environment/dependencies/versions are frozen by registry_snapshot.json.

18.3.3 16.3.3 Artifact checklist (definition): required files and formats

18.3.3.1 (Definition) Minimum release artifact set

A release (distribution unit) must include at least the following files/directories.

README.md
LICENSE
CITATION.cff
CHANGELOG.md
protocol/
  protocol.yaml
  protocol.sha256
locks/
  canon_lock.json
  realization_lock.json
  analysis_lock.json
  locks.sha256
plans/
  plan.json
  plan.sha256
runs/
  <run_id>/
    protocol.locked.json
    locks_snapshot/
    registry_snapshot.json
    run_log.jsonl
    metrics.json
    outputs/
    manifest.json
    manifest.sha256
release_manifest.json
release_manifest.sha256

Here release_manifest.json is the top-level manifest that contains checksums of the entire release. Its schema is fixed to be identical to [eq:S16_02_manifest_schema].

18.3.3.2 (Definition) Required metadata contents

The minimal contents of each metadata file must include the following.

Missing metadata is treated as a failure via [eq:S16_01_TSCH].

18.3.4 16.3.4 Release rules (definition): versioning, immutability, DOI unit

18.3.4.1 (Definition) Version notation and bump rules

Fix the release version string as \(\mathrm{v}X.Y.Z\).

In particular, if any key–value in any LOCK file (canonical/realization/analysis) changes, Major must increase. This forbids “post-hoc modification of conclusions under the same name.”

18.3.4.2 (Definition) Immutability rule

A DOI-assigned release is immutable: after DOI assignment, modification/deletion/replacement of files is forbidden. If a change is needed, create a new versioned release and obtain a new DOI unit. Immutability is enforced by checksums as \[\mathrm{IMMUTABLE}=1 \Longleftrightarrow \mathrm{SHA256}(\texttt{release\_manifest.json})=\texttt{release\_manifest.sha256's value} \label{eq:S16_03_immutable}\]

18.3.4.3 (Definition) DOI package (archive unit)

Fix the DOI package (compressed or archived) as the single bundle that contains \[\mathsf{DOIPACK}:=\{\texttt{protocol/},\texttt{locks/},\texttt{plans/},\texttt{runs/},\texttt{release\_manifest.*},\texttt{README.md},\texttt{CITATION.cff},\texttt{CHANGELOG.md},\texttt{LICENSE}\} \label{eq:S16_03_doipack}\] release_manifest.* is mandatory as the integrity evidence for all files; if missing, the DOI package is invalid.

18.3.5 16.3.5 Release verdict Gate (definition): distribution eligibility

A release is not a single run; it is defined as the distribution-eligibility verdict for the entire runs/ set. Fix the release PASS condition as \[\mathrm{PASS}_{\mathrm{release}} := \left(\bigwedge_{\texttt{run\_id}\in\texttt{runs/}} \mathrm{STATUS}(\texttt{run\_id})\neq \mathrm{INCONCLUSIVE}\right) \wedge \left(T_{\mathrm{MAN}}^{\mathrm{release}}=0\right) \wedge \left(T_{\mathrm{SCH}}^{\mathrm{release}}=0\right) \label{eq:S16_03_pass_release}\] Here \(T_{\mathrm{MAN}}^{\mathrm{release}}\) is the integrity trigger for release_manifest.*, and \(T_{\mathrm{SCH}}^{\mathrm{release}}\) is the release-schema trigger. It is allowed that a release contains FAIL runs; in that case the FAILPACK rule [eq:S16_01_failpack] must be satisfied. However, for a release to be judged PASS, there must be no INCONCLUSIVE runs.

18.3.6 16.3.6 Minimal disclosure principle (definition): equal disclosure of PASS and FAIL

PASS and FAIL are subject to the same disclosure rules.

This forbids “keeping only successful results and deleting failures.” Failures are treated as part of the same verification/reproducibility system.

18.3.7 16.3.7 Post-distribution tracking (definition): DOI–version mapping and reference freeze

After distribution, once a DOI is assigned, references are frozen via CITATION.cff and a dedicated mapping file. Define the mapping file as a CSV with columns \[\texttt{DOI\_MAP.csv}:\ (\texttt{version},\texttt{doi},\texttt{release\_digest},\texttt{date}) \label{eq:S16_03_doi_map}\] Fix release_digest as the SHA-256 of release_manifest.json. Missing DOI–version mapping is treated as an incompleteness of the distribution system. After that point, every citation/reference is fixed to include both the DOI and release_digest.

18.4 16.4 Physical-constant closure map: separation of GEOM/ANCHOR/CALIB

This section classifies the major physical constants/scales used in the main text into (i) items that close purely from geometric/integer structure (GEOM), (ii) items that must be locked as anchors of the realized unit system (definitions or standard values; ANCHOR), and (iii) items whose numerical values are determined by combining ANCHOR with GEOM while keeping the geometric structure (CALIB). The purpose is to enable an internal self-audit of “what is derived and what is locked.” This section adds no new physical assumptions; it only provides a dependency status table (SSOT) for existing sections.

18.4.1 16.4.1 Classification rules (definition)

18.4.2 16.4.2 SSOT

Item Status Primary source Definition/derivation summary
Item Status Primary source Definition/derivation summary
\(N_{\mathrm{sec}}=3\) GEOM 7.0 Topologically forced as the minimum number of vectors (3) required to physically enclose the core in a 2D cross-section.
\(N_{\mathrm{shell}}=7\) GEOM 8.2 Closes by minimality of 3D cancellation (2-pair + 4-quad) plus label survival (1), and by the residual match \(89-82\).
\(\alpha_{em}\) GEOM 14.5 Define a geometric impedance in the form \(4\pi(7+3+1-\delta_{\mathrm{proj}})\) and close it using continuous rotation averaging (\(2/\pi^2\)) and discrete occupancy correction (\(35/32\)).
\(h_{\mathrm{VP}}\) CALIB 15.5 Define the rigidity-shell barrier \(\epsilon_b(\nu)=h_{\mathrm{VP}}\,\nu\) and derive a Planck-like spectral form from the thermal equilibrium sum over event counts \(n\in\mathbb{N}\). The SI value of \(h_{\mathrm{VP}}\) is identified by matching to blackbody data (protocol/Gate required).
Blackbody spectral form GEOM 15.5 Shape closes as mode count \(\nu^2\) (pure 3D geometry) \(\times\) rigidity filter \(\frac{h_{\mathrm{VP}}\nu}{e^{h_{\mathrm{VP}}\nu/(k_B T)}-1}\) (discrete event counts).
\(a\) (VP diameter; realization length) ANCHOR 11.2 Length anchor of the realization unit system. Post-hoc tuning is forbidden within the same version (LOCK).
\(\Delta t\) (realization time tick) ANCHOR 11.3 Time anchor of the realization unit system. Post-hoc tuning is forbidden within the same version (LOCK).
\(c_{\mathrm{ref}}\) ANCHOR 11.1, 13.1 Reference speed anchor (unit system/protocol). This document does not derive it; it is used only as a LOCK.
\(h\) ANCHOR 13.1 Action-unit anchor. The SI-defined value is locked in canon_lock.
\(U_{\mathrm{lat}}=hc_{\mathrm{ref}}/a\) CALIB 13.1 Lattice unit energy SSOT combining three anchors (\(h,c_{\mathrm{ref}},a\)).
Mass scales such as \(m_p,m_e,m_H\) CALIB 13 Under the axiom \(U_{\mathrm{lat}}/\sigma_{\mathrm{eff}}\), derived from structural resistance (geometric cross-sections/event rates). Comparisons are Gate metrics.

18.4.3 16.4.3 Gate: “missing lock” detection rules (definition)

The status table in this section is not a conclusion but an audit rule. If any of the following violations is detected, the corresponding claim is immediately set to FAIL.

18.5 16.5 DOI completeness maintenance rules: automated audits for “zero missing”

This document distinguishes two kinds of citations.

The goal is zero missing DOI for internal citations and zero missing DOI annotation for external citations.

18.5.1 16.5.1 Internal citation registry (definition): SSOT = CITE_REGISTRY.csv

Every internal citation token must be registered in the following SSOT.

If a token appears in the main text but is missing from the SSOT, it is a FAIL under (G-DOI-MISS).

18.5.2 16.5.2 External DOI notation rules (definition)

18.5.3 16.5.3 Automated audit script (LOCK): scripts/doi_audit.py

Before release (or before Zenodo upload), the following script must succeed (exit code 0). \[\texttt{python3 scripts/doi\_audit.py} \label{eq:S16_05_doi_audit_cmd}\] This script checks: (i) whether internal citation tokens in the main text match IDs in CITE_REGISTRY.csv, (ii) whether DOI/URL fields in CITE_REGISTRY.csv are blank, and (iii) whether external DOI strings in the main text have valid formats (regex) and duplicates. Uploading/distribution is forbidden in a failing state.

19 17. Extensions (Optional Reading)

19.1 17.1 Rotation/anisotropy extension (\(\ell_{\mathrm{rot}}\) included) and experimental module structure

19.1.1 17.1.1 Definition of the extension targets

The rotation/anisotropy extension includes the following two categories.

19.1.2 17.1.2 Anisotropy axis and directional variable (definition)

Define the anisotropy axis (unit vector) as \[\mathbf{u}\in\mathbb{R}^d,\qquad \|\mathbf{u}\|=1 \label{eq:S17_01_u_axis}\] For an arbitrary separation vector \(\mathbf{R}\neq \mathbf{0}\), define the directional cosine \[\mu(\mathbf{R}):=\frac{\mathbf{R}\cdot\mathbf{u}}{\|\mathbf{R}\|} \in[-1,1] \label{eq:S17_01_mu}\] All anisotropic kernels below are parameterized by \(\mu\) or by an equivalent angular variable.

19.1.3 17.1.3 Operational definition of the anisotropic dilution kernel (definition)

In the isotropic regime, take the geometric dilution kernel as \[\mathcal{D}_{\mathrm{dil,iso}}(R)=\left(\frac{a}{R}\right)^2 \label{eq:S17_01_dil_iso}\] and define the anisotropic extension kernel in the following product form: \[\mathcal{D}_{\mathrm{dil}}(R,\mu) := \left(\frac{a}{R}\right)^2\,g(\mu), \qquad g(\mu)\ge 0 \label{eq:S17_01_dil_aniso}\] Here \(g(\mu)\) is a function locked by analysis_lock; it cannot be selected after inspecting results.

19.1.3.1 (Definition) Normalization condition

Impose the following normalization so that \(g(\mu)\) preserves the isotropic average: \[\langle g\rangle_{\mu} := \frac{1}{2}\int_{-1}^{1} g(\mu)\,d\mu = 1 \label{eq:S17_01_g_norm}\] If the normalization holds, [eq:S17_01_dil_aniso] reduces to [eq:S17_01_dil_iso] after angular averaging.

19.1.3.2 (Definition) Anisotropy coefficient (summary statistic)

Define the following summary statistic to report the magnitude of anisotropy: \[\beta_g := \left(\frac{1}{2}\int_{-1}^{1}\big(g(\mu)-1\big)^2\,d\mu\right)^{1/2} \label{eq:S17_01_beta_g}\] \(\beta_g=0\) indicates isotropy, and \(\beta_g>0\) indicates anisotropy.

19.1.4 17.1.4 Operational definition of anisotropic percolation/backbone (definition)

Let \(\mathcal{B}\) denote the set of paths forming the propagation backbone on the lattice, and treat each path \(\gamma\in\mathcal{B}\) as a sequence of links. For a link direction \(\hat{\ell}\), define the anisotropic weight by \[w(\hat{\ell}) := g\!\left(\mu(\hat{\ell})\right), \qquad \mu(\hat{\ell}):=\hat{\ell}\cdot\mathbf{u} \label{eq:S17_01_link_weight}\] Define the cumulative weight of a path \(\gamma\) as \[W(\gamma):=\prod_{\hat{\ell}\in\gamma} w(\hat{\ell}) \label{eq:S17_01_path_weight}\] The anisotropic backbone is then defined by the following maximization rule: \[\gamma^\star(\mathbf{x}\to\mathbf{y}) := \operatorname*{arg\,max}_{\gamma\in\Gamma(\mathbf{x}\to\mathbf{y})} W(\gamma) \label{eq:S17_01_backbone_argmax}\] Here \(\Gamma(\mathbf{x}\to\mathbf{y})\) is the set of candidate paths that can connect \(\mathbf{x}\) to \(\mathbf{y}\). This definition fixes the “preferred propagation direction” via \(g(\mu)\) and does not call external dynamics.

19.1.5 17.1.5 Treatment of the rotational length \(\ell_{\mathrm{rot}}\) (definition)

Treat the rotational length \(\ell_{\mathrm{rot}}\) as an optional extension constant. If \(\ell_{\mathrm{rot}}\) is adopted, fix the following rule: \[\ell_{\mathrm{rot}} \in \mathbb{R}_{>0}, \qquad \ell_{\mathrm{rot}} \text{ numeric value/unit/source is locked to }\texttt{canon\_lock}\text{ or }\texttt{realization\_lock}\text{.} \label{eq:S17_01_lrot_lock}\] If the adoption choice for \(\ell_{\mathrm{rot}}\) changes, the LOCK version must be incremented.

19.1.5.1 (Definition) Rotation rate and dimensionless rotation strength

Define the rotation rate (an operational variable corresponding to angular speed) as \[\Omega_{\mathrm{rot}}\in\mathbb{R} \label{eq:S17_01_Omega}\] and define the dimensionless rotation strength by \[\chi_{\mathrm{rot}} := \frac{\ell_{\mathrm{rot}}}{a}\,\Omega_{\mathrm{rot}}\,\Delta t \label{eq:S17_01_chi_rot}\] The value of \(\chi_{\mathrm{rot}}\) is determined by the realization length \(a\), the realization time \(\Delta t\), and the locked value of \(\ell_{\mathrm{rot}}\).

19.1.6 17.1.6 Rotation–anisotropy coupling rule (definition)

Rotation drive affects the anisotropy axis \(\mathbf{u}\) and the anisotropy function \(g(\mu)\). Define this via the following coupling map: \[\mathbf{u}\mapsto \mathbf{u}(\chi_{\mathrm{rot}}),\qquad g(\mu)\mapsto g(\mu;\chi_{\mathrm{rot}}) \label{eq:S17_01_u_g_couple}\] The coupling map is locked by analysis_lock and cannot be chosen post hoc to fit outcomes without preregistration.

19.1.6.1 (Definition) Weak-coupling and strong-coupling regimes

\[\text{Weak-coupling regime: }|\chi_{\mathrm{rot}}|\le \chi_{\star}, \qquad \text{Strong-coupling regime: }|\chi_{\mathrm{rot}}|> \chi_{\star} \label{eq:S17_01_regime_rot}\] Here \(\chi_{\star}>0\) is a threshold locked by analysis_lock. In the weak-coupling regime, an operational series representation such as \[g(\mu;\chi_{\mathrm{rot}})=g_0(\mu)+\chi_{\mathrm{rot}}\,g_1(\mu)+O(\chi_{\mathrm{rot}}^2) \label{eq:S17_01_g_expand}\] is allowed. In the strong-coupling regime, a series form is not used; instead, a different form (e.g., piecewise definition, saturation type) must be preregistered and then applied.

19.1.7 17.1.7 Experimental module structure (definition): file tree, protocol, and outputs

The experimental module for the rotation/anisotropy extension is fixed to the following directory structure:

modules/
  rot_aniso/
    protocol.yaml
    locks_required.txt
    src/
      simulate_rot_aniso.py
      estimators.py
      gates.py
      io_schema.py
    schemas/
      protocol.schema.json
      run_log.schema.json
      metrics.schema.json
      manifest.schema.json
    plans/
      plan.json
    outputs_spec/
      artifacts.md

19.1.7.1 (Definition) Input (LOCK) dependency list

locks_required.txt must include, at minimum, the following items:

19.1.7.2 (Definition) Metrics (summary statistics) and Gates

At minimum, the rotation/anisotropy module outputs the following metrics:

Define the axis-parallel and axis-perpendicular response using either arrival time or event-flux under identical source–sink conditions as follows. Let \(\mathcal{D}_{\parallel}\) be the set of test directions parallel to the axis and \(\mathcal{D}_{\perp}\) the set of perpendicular directions. If the response in direction \(\hat{\mathbf{d}}\) is defined as \(A(\hat{\mathbf{d}})\), then \[\mathcal{A}_{\parallel}:=\langle A(\hat{\mathbf{d}})\rangle_{\hat{\mathbf{d}}\in\mathcal{D}_{\parallel}}, \qquad \mathcal{A}_{\perp}:=\langle A(\hat{\mathbf{d}})\rangle_{\hat{\mathbf{d}}\in\mathcal{D}_{\perp}} \label{eq:S17_01_Apar_Aperp}\] Define the backbone change magnitude via the link-set difference between the isotropic backbone \(\gamma^\star_{\mathrm{iso}}\) and the anisotropic backbone \(\gamma^\star_{\mathrm{aniso}}\). If the link set is denoted \(\mathcal{E}(\gamma)\), then \[\Delta_{\mathrm{bb}} := \frac{|\mathcal{E}(\gamma^\star_{\mathrm{aniso}})\,\triangle\,\mathcal{E}(\gamma^\star_{\mathrm{iso}})|}{|\mathcal{E}(\gamma^\star_{\mathrm{iso}})|} \label{eq:S17_01_delta_bb}\] where \(\triangle\) denotes the symmetric difference.

At minimum, Gates include:

19.2 17.2 Jet/singularity/astronomy–cosmology regime extension (integrated into the regime map)

19.2.1 17.2.1 Decomposition of the target regimes (definition)

The jet/singularity/astronomy–cosmology regime extension incorporates the following three classes into a single regime map.

In this section, we do not use dynamics from external theories; everything is constructed only from operational definitions of events/propagation/throat-gap/backbone/boundary conditions.

19.2.1.1 (Terminology) “jet” and “jet stream”

In this section, (E-JET) includes the general class of “outflows created by sources/sinks”. Among them, when the structure is persistent in time and narrow and long-maintained in space, we call it a “jet stream” in this document. A jet stream is not defined by a separate dynamical equation; it is determined only by persistence/width/axis-stability Gates computed from the same event-flux log (17.2.3).

19.2.2 17.2.2 Common primitives: source/sink, flux, backbone (definition)

19.2.2.1 (Definition) Source/sink sets

For a lattice region \(\Lambda\subset\mathcal{L}\), define a source set \(\Lambda_{+}\subset\Lambda\) and a sink set \(\Lambda_{-}\subset\Lambda\), and impose \[\Lambda_{+}\cap\Lambda_{-}=\varnothing \label{eq:S17_02_src_sink_disjoint}\] Sources/sinks are defined as external injection/absorption conditions on event rates.

19.2.2.2 (Definition) Injection/absorption event rates

Define the net injection event rate at each node \(\mathbf{n}\in\Lambda\) by \[J(\mathbf{n},k) := J_{+}(\mathbf{n},k)-J_{-}(\mathbf{n},k) \label{eq:S17_02_Jdef}\] Here \(J_{+}\) can be positive only at sources, and \(J_{-}\) can be positive only at sinks. The preregistered plan file must fix the form of \(J_{+},J_{-}\) (constant, pulse, ramp, feedback, etc.).

Let the sector index be \(s\in\{1,2,3\}\). Define the link-level event flux for observation window start \(k_0\) and window length \(M\) by \[F_{\mathrm{ev}}(\mathbf{n},e;k_0,M) := \sum_{k=k_0}^{k_0+M-1}\sum_{s=1}^{3} \Xi_{s}(\mathbf{n},e,k) \label{eq:S17_02_Fev}\] where \(\Xi_s(\mathbf{n},e,k)\) is an indicator for whether a sector-\(s\) event occurred on link \(e\) from node \(\mathbf{n}\) at time-window \(k\).

For a region \(\Sigma\subset\Lambda\) and its boundary \(\partial\Sigma\), define the net event balance by \[\Delta N_{\Sigma}(k_0,M) = \sum_{\mathbf{n}\in\Sigma}\sum_{k=k_0}^{k_0+M-1} J(\mathbf{n},k) = \sum_{(\mathbf{n},e)\in\partial\Sigma} F_{\mathrm{ev}}(\mathbf{n},e;k_0,M) \label{eq:S17_02_consistency}\] This ties the event-flux definition to the event-balance rule and maintains consistency with earlier flux definitions.

19.2.2.5 (Definition) Event-flux rate (time normalization)

Define the flux rate by time-normalizing the boundary flux: \[\Phi_{\mathrm{ev}} := \frac{1}{M\Delta t}\sum_{(\mathbf{n},e)\in\partial\Sigma} F_{\mathrm{ev}}(\mathbf{n},e;k_0,M) \label{eq:S17_02_Phi_ev}\] The unit/time convention is fixed by analysis_lock and must not be tuned after observing results.

Let \(\mathcal{C}(\hat{\mathbf{d}})\subset\partial\Sigma\) be the preregistered observation cone/cylinder associated with direction \(\hat{\mathbf{d}}\). Define the direction-resolved flux rate by \[\Phi_{\mathrm{link}}(\hat{\mathbf{d}};k_0,M) := \frac{1}{M\Delta t}\sum_{(\mathbf{n},e)\in\mathcal{C}(\hat{\mathbf{d}})} F_{\mathrm{ev}}(\mathbf{n},e;k_0,M). \label{eq:S17_02_flux_link}\] This is the rate-form companion of the integer event flux \(F_{\mathrm{ev}}\) and is the source of all jet/axis/width statistics.

19.2.2.7 (Definition) Jet collimation index

For observation directions \(\hat{\mathbf{d}}\in\mathbb{S}^{d-1}\), let \(\mathcal{C}(\hat{\mathbf{d}})\subset\partial\Sigma\) denote an observation cone (or cylinder) on the boundary. Define the jet collimation index by \[\mathcal{J} := \max_{\hat{\mathbf{d}}} \frac{\sum_{(\mathbf{n},e)\in \mathcal{C}(\hat{\mathbf{d}})} F_{\mathrm{ev}}(\mathbf{n},e;k_0,M)}{\sum_{(\mathbf{n},e)\in\partial\Sigma} F_{\mathrm{ev}}(\mathbf{n},e;k_0,M)} \label{eq:S17_02_Jindex}\] A value near 1 indicates a strongly collimated outflow, while a value near 0 indicates a widely distributed flux over directions.

19.2.2.8 (Definition) Jet axis direction (estimator)

Define the jet axis direction by \[\hat{\mathbf{d}}_J := \operatorname*{arg\,max}_{\hat{\mathbf{d}}} \sum_{(\mathbf{n},e)\in \mathcal{C}(\hat{\mathbf{d}})} F_{\mathrm{ev}}(\mathbf{n},e;k_0,M) \label{eq:S17_02_aJ}\] If there are ties, the tie-breaking rule must be fixed by LOCK.

19.2.2.9 (Definition) Jet width (opening angle) summary statistic

Let \(\mathcal{C}(\hat{\mathbf{d}}_J;\theta)\subset\partial\Sigma\) denote the cone of half-angle \(\theta\) around the axis \(\hat{\mathbf{d}}_J\). Define the jet opening angle (quantile width) by \[\theta_J(q) := \inf\left\{\theta: \sum_{(\mathbf{n},e)\in \mathcal{C}(\hat{\mathbf{d}}_J;\theta)} F_{\mathrm{ev}}(\mathbf{n},e;k_0,M) \ge q\sum_{(\mathbf{n},e)\in\partial\Sigma} F_{\mathrm{ev}}(\mathbf{n},e;k_0,M) \right\} \label{eq:S17_02_theta}\]

19.2.3 17.2.3 Jet regime and jet stream (definition and Gates)

19.2.3.1 (Definition) Jet-regime conditions

Define the jet regime as a regime in which both total injection and total absorption are nonzero: \[\sum_{\mathbf{n}\in\Lambda_{+}} J_{+}(\mathbf{n},k)>0 \label{eq:S17_03_JR1}\] \[\sum_{\mathbf{n}\in\Lambda_{-}} J_{-}(\mathbf{n},k)>0 \label{eq:S17_03_JR2}\] These are necessary conditions for defining an outflow, but do not by themselves guarantee collimation.

19.2.3.2 (Gate) Jet Gate

Define \[\texttt{PASS\_JET} := \big[\mathcal{J}\ge \mathcal{J}_{\star}\big] \label{eq:S17_03_gateJET}\] where the threshold \(\mathcal{J}_{\star}\) is preregistered and LOCKed.

19.2.3.3 (Definition) Time resolution and persistence (jet-stream candidate)

Compute the windowed jet index \(\mathcal{J}_i:=\mathcal{J}(k_0+i m,m)\) using window length \(m\) fixed by LOCK. Define the persistence fraction \[P_J := \frac{1}{M}\sum_{i=1}^{M} \mathbb{I}[\mathcal{J}_i\ge \mathcal{J}_{\star}] \label{eq:S17_03_PJ}\] and define the axis-fluctuation metric \[M_J := \frac{1}{M-1}\sum_{i=2}^{M} \arccos\big(|\hat{\mathbf{d}}_{J,i}\cdot \hat{\mathbf{d}}_{J,i-1}|\big) \label{eq:S17_03_MJ}\] where \(\hat{\mathbf{d}}_{J,i}\) is the estimated jet axis direction in window \(i\).

19.2.3.4 (Definition) Time-averaged jet width (jet-stream summary statistic)

Define the time-averaged width in passing windows by \[\overline{\theta}_J(q) := \frac{\sum_{i=1}^{M} \theta_{J,i}(q)\,\mathbb{I}[\mathcal{J}_i\ge \mathcal{J}_{\star}]}{\sum_{i=1}^{M}\mathbb{I}[\mathcal{J}_i\ge \mathcal{J}_{\star}]}\] where \(\theta_{J,i}(q)\) is the width statistic computed in window \(i\).

19.2.3.5 (Optional Gate) Consistency with anisotropy axis

If the anisotropy axis \(\mathbf{u}\) is declared (Sec. 17.1), define \[A_{\mathrm{align}} := \frac{1}{M}\sum_{i=1}^{M} \big|\hat{\mathbf{d}}_{J,i}\cdot\mathbf{u}\big|.\] Define \[\texttt{PASS\_ALIGN} := \big[A_{\mathrm{align}}\ge \mu_{\star}\big] \label{eq:S17_03_gateALIGN}\] where the threshold \(\mu_{\star}\) is preregistered and LOCKed.

19.2.3.6 (Definition) Jet-stream Gate

Define \[\texttt{PASS\_JSTREAM} := \big[P_J\ge P_{\star}\big] \wedge \big[\overline{\theta}_J(q)\le \theta_{\star}\big] \wedge \big[M_J\le M_{\star}\big] \label{eq:S17_03_gateJSTREAM}\] All thresholds \((P_{\star},\theta_{\star},M_{\star},q)\) are preregistered and LOCKed.

19.2.3.7 (Correspondence table; non-evidential) Mapping to an atmospheric jet stream (Earth)

The following correspondence table is offered only as an interpretive label, not as evidence or a dynamical derivation.

19.2.4 17.2.4 Core-saturation regime (singularity replacement) (definition)

19.2.4.1 (Definition) Core region and core score

Define a core region \(\Lambda_{\mathrm{core}}\subset\Lambda\) (preregistered; e.g., high-density zone). Let \(N(\mathbf{n};k_0,M)\) denote the total number of events counted at node \(\mathbf{n}\) within the observation window \((k_0,M)\). Define the core score by \[S_{\mathrm{core}}(k_0,M) := \frac{\sum_{\mathbf{n}\in\Lambda_{\mathrm{core}}} N(\mathbf{n};k_0,M)}{\sum_{\mathbf{n}\in\Lambda} N(\mathbf{n};k_0,M)} \label{eq:S17_04_Score}\]

19.2.4.2 (Definition) Throat-gap statistic on the core

Let \(\delta_{\mathrm{eff}}(\mathbf{n})\) denote the effective throat-gap at node \(\mathbf{n}\) (cf. Sec. 6). Define the core-averaged effective gap by \[\delta_{\mathrm{core}} := \frac{1}{|\Lambda_{\mathrm{core}}|}\sum_{\mathbf{n}\in\Lambda_{\mathrm{core}}}\delta_{\mathrm{eff}}(\mathbf{n}) \label{eq:S17_04_deltaeff}\]

19.2.4.3 (Definition) Core regime conditions

Define the core-saturation regime by either condition: \[S_{\mathrm{core}}(k_0,M)\ge S_{\star} \label{eq:S17_04_COREA}\] or \[\delta_{\mathrm{core}}\le \delta_{\star} \label{eq:S17_04_COREB}\] with preregistered thresholds \(S_{\star},\delta_{\star}\).

If the core regime passes, the isotropic approximation is not automatically valid; the analysis should classify the run as (E-CORE) and use the anisotropy/rotation extension if applicable.

19.2.4.4 No-singularity claim rule

In the (E-CORE) regime, the analysis must not claim mathematical singularities or divergences. Only the operational summary statistics and Gates above may be reported.

19.2.5 17.2.5 Operational definition of the large-scale boundary regime (E-COSMO)

19.2.5.1 (Definition) Dynamic boundary input

We define the net inflow/outflow from the lattice boundary or an external reservoir as a time-dependent input. Let the boundary set be \(\partial\Lambda\), and define \[J_{\partial}(\mathbf{n},k)\quad(\mathbf{n}\in\partial\Lambda) \label{eq:S17_02_Jboundary}\]

as the dynamic boundary input. The functional form of \(J_{\partial}\) must be preregistered in plan.json; post-run modification is prohibited.

19.2.5.2 (Definition) Lattice-scale tracking variable

In a large-scale regime we do not fix the lattice scale. Instead, we define an effective scale per observation window as a summary statistic. For example, from the second central moment of the event density we define an effective radius by \[R_{\mathrm{eff}}^2(k_0,M) := \frac{\sum_{\mathbf{n}\in\Lambda}\|\mathbf{x}(\mathbf{n})-\mathbf{x}_{\mathrm{cm}}\|^2\,\rho(\mathbf{n};k_0,M)} {\sum_{\mathbf{n}\in\Lambda}\rho(\mathbf{n};k_0,M)} \label{eq:S17_02_Reff}\] where \[\mathbf{x}_{\mathrm{cm}} := \frac{\sum_{\mathbf{n}\in\Lambda}\mathbf{x}(\mathbf{n})\,\rho(\mathbf{n};k_0,M)} {\sum_{\mathbf{n}\in\Lambda}\rho(\mathbf{n};k_0,M)} \label{eq:S17_02_xcm}\]

is the center-of-mass position weighted by \(\rho\). We use \(R_{\mathrm{eff}}\) only as an operational summary statistic for deciding “large-scale variation.”

19.2.5.3 (Definition) Large-scale regime Gate

Claims in the large-scale regime (e.g., “a scale change is observed”) are only qualified when the following Gate is satisfied: \[\mathrm{PASS}_{\mathrm{COSMO}} :\Longleftrightarrow \left|\frac{R_{\mathrm{eff}}(k_1,M)-R_{\mathrm{eff}}(k_0,M)}{R_{\mathrm{eff}}(k_0,M)}\right| \ge \eta_{\star} \label{eq:S17_02_pass_cosmo}\] where \(\eta_{\star}>0\) is a preregistered threshold, and \(k_1-k_0\) as well as \(M\) must be fixed by the protocol.

19.2.5.4 (Derived observable example) Redshift \(z\) and distance estimate

We connect the lattice-friction redshift model defined in Sec. 10.8 to a derived observable in the E-COSMO regime. From the observed redshift \(z\) of each source, we define an effective path-length (distance) estimate by \[D_{z} := \frac{1}{\kappa}\ln(1+z) \label{eq:S17_02_Dz_def}\] The parameter \(\kappa\) must be obtained from a LOCK (or a preregistered calibration procedure). Any post-hoc tuning after observing results is prohibited.

19.2.5.5 (LOCK) Separation of calibration and validation for \(\kappa\)

When a set of distance anchors \(\{(D_{\mathrm{anch},i},z_i)\}\) is available, \(\kappa\) must be estimated only on the preregistered calibration set \(\mathcal{S}_{\mathrm{cal}}\), and judgement on the validation set \(\mathcal{S}_{\mathrm{val}}\) must use a fixed \(\kappa\). For example, if we adopt weighted least squares, \[\widehat{\kappa} = \arg\min_{\kappa}\sum_{i\in\mathcal{S}_{\mathrm{cal}}}w_i\left(\ln(1+z_i)-\kappa D_{\mathrm{anch},i}\right)^2 = \frac{\sum_{i\in\mathcal{S}_{\mathrm{cal}}}w_i D_{\mathrm{anch},i}\ln(1+z_i)} {\sum_{i\in\mathcal{S}_{\mathrm{cal}}}w_i D_{\mathrm{anch},i}^2} \label{eq:S17_02_kappa_hat}\] A choice of \((w_i)\) and the split \(\mathcal{S}_{\mathrm{cal}}/\mathcal{S}_{\mathrm{val}}\) must be LOCKed prior to the run.

19.2.5.6 (Gate) Cross-validation of the \(z\)–distance mapping

On the validation set \(\mathcal{S}_{\mathrm{val}}\), define \[\Delta_i := \frac{D_{z,i}-D_{\mathrm{anch},i}}{D_{\mathrm{anch},i}} \qquad(i\in\mathcal{S}_{\mathrm{val}}) \label{eq:S17_02_delta_i}\]

and set the Gate \[\mathrm{PASS}_{z\text{-dist}} :\Longleftrightarrow \mathrm{median}_{i\in\mathcal{S}_{\mathrm{val}}}\left(|\Delta_i|\right)\le \epsilon_{\mathrm{med}}^{\star} \ \wedge\ \sqrt{\mathrm{mean}_{i\in\mathcal{S}_{\mathrm{val}}}(\Delta_i^2)}\le \epsilon_{\mathrm{rms}}^{\star} \label{eq:S17_02_pass_zdist}\] where \(\epsilon_{\mathrm{med}}^{\star},\epsilon_{\mathrm{rms}}^{\star}>0\) are preregistered thresholds. If this Gate fails, then in the E-COSMO regime the definition [eq:S17_02_Dz_def] is incompatible with the distance anchors and is a FAIL.

19.2.5.7 (Mandatory LOCK) Observation mapping: extending \(D_z\) to \(D_L,D_A\) and flux/angular-size predictions

The quantity \(D_z\) in [eq:S17_02_Dz_def] is defined as a path length. To compare directly with observed fluxes and angular sizes, we need mappings \(D_z\mapsto D_L^{(\mathrm{model})}\) and \(D_z\mapsto D_A^{(\mathrm{model})}\). For this purpose, we define an attenuation factor \(\Phi(z)\) and an angular/area mapping factor \(\Upsilon(z)\) as follows: \[F_{\mathrm{obs}}(z) := \frac{L}{4\pi D_z^2}\,\Phi(z), \qquad D_L^{(\mathrm{model})}(z):=\frac{D_z}{\sqrt{\Phi(z)}} \label{eq:S17_02_flux_mapping}\] \[D_A^{(\mathrm{model})}(z):=\frac{D_z}{\Upsilon(z)} \label{eq:S17_02_ang_mapping}\] We also define the observed/emitted relation for time-scales (light-curve width, variability period, etc.) by \[\tau_{\mathrm{obs}}(z):=\tau_{\mathrm{em}}\,T(z) \label{eq:S17_02_time_dilation_T}\] The functional forms and estimation procedures for \(\Phi,\Upsilon,T\) must be LOCKed prior to execution; post-hoc modification to fit outcomes is prohibited. (Example) If one adopts static geometry (\(\Upsilon\equiv 1\)), no time dilation (\(T\equiv 1\)), and only energy loss (\(\Phi(z)=(1+z)^{-1}\)), then \(D_L^{(\mathrm{model})}=D_z\sqrt{1+z}\) and a surface-brightness scaling \(B_{\mathrm{obs}}\propto(1+z)^{-1}\) follow. This is only a branch example; adoption is decided only by a LOCK.

19.2.5.8 (Gate) Time dilation decision

Assume that for a standardized class of events (e.g., Type Ia supernovae) an emitted time-scale \(\tau_{\mathrm{em}}\) can be calibrated. From observations \((z_i,\tau_{\mathrm{obs},i})\), define \[r_i^{\mathrm{TD}}:=\frac{\tau_{\mathrm{obs},i}/\tau_{\mathrm{em},i}}{T(z_i)}-1 \label{eq:S17_02_r_TD}\] and set the Gate \[\mathrm{PASS}_{\mathrm{TD}} :\Longleftrightarrow \mathrm{median}_i\bigl(|r_i^{\mathrm{TD}}|\bigr)\le \epsilon_{\mathrm{TD}}^{\star} \label{eq:S17_02_pass_TD}\] where \(\epsilon_{\mathrm{TD}}^{\star}\) is preregistered. If \(\tau_{\mathrm{em}}\) cannot be calibrated or \(T(z)\) is not LOCKed, the status is INCONCLUSIVE.

19.2.5.9 (Gate) Surface-brightness (Tolman-type) decision

Assume that for a standardized source class the emitted surface brightness \(B_0\) (including rest-frame calibration) is LOCKed. Then the observed surface brightness \(B_{\mathrm{obs}}(z)\) predicted from [eq:S17_02_flux_mapping]–[eq:S17_02_ang_mapping] is \[B_{\mathrm{obs}}(z)=B_0\,\frac{\Phi(z)}{\Upsilon(z)^2} \label{eq:S17_02_Bobs_pred}\] Hence define \[r_i^{\mathrm{TOL}}:=\frac{B_{\mathrm{obs},i}}{B_0}\,\frac{\Upsilon(z_i)^2}{\Phi(z_i)}-1 \label{eq:S17_02_r_TOL}\]

and set the Gate \[\mathrm{PASS}_{\mathrm{TOL}} :\Longleftrightarrow \mathrm{median}_i\bigl(|r_i^{\mathrm{TOL}}|\bigr)\le \epsilon_{\mathrm{TOL}}^{\star} \label{eq:S17_02_pass_TOL}\] where \(\epsilon_{\mathrm{TOL}}^{\star}\) is preregistered. If the standardization of \(B_0\) (evolution/band/K-correction, etc.) is not LOCKed, this Gate is INCONCLUSIVE.

19.2.5.10 (Gate) Image blurring / line broadening (Blur/Broadening) decision

If lattice friction/scattering (a “medium”) exists, then in addition to frequency attenuation, additional observational signatures may accompany it, such as image blurring (angular diffusion), line broadening, and phase noise. Therefore, at the level of an “expansion alternative” conclusion, we must Gate-check that any additional blurring introduced by the model does not exceed observational tolerances. For each source \(i\), define a blurring metric \(b_{\mathrm{obs},i}\) (including instrument PSF / spectrograph resolution removal and rest-frame corrections). Let a standardized intrinsic metric \(b_0\) and a model-predicted blurring factor \(B_{\mathrm{BLR}}(z)\) be LOCKed inputs. Define residual \[r_i^{\mathrm{BLR}}:=\frac{b_{\mathrm{obs},i}/b_0}{B_{\mathrm{BLR}}(z_i)}-1 \label{eq:S17_02_r_BLR}\]

and set \[\mathrm{PASS}_{\mathrm{BLR}} :\Longleftrightarrow \mathrm{median}_i\bigl(|r_i^{\mathrm{BLR}}|\bigr)\le \epsilon_{\mathrm{BLR}}^{\star} \label{eq:S17_02_pass_BLR}\] where \(\epsilon_{\mathrm{BLR}}^{\star}>0\) is preregistered. If the standardization of \(b_0\) (source evolution/band/K-correction) or the instrument correction procedure for \(b_{\mathrm{obs},i}\) (PSF/resolution removal) is not LOCKed, this Gate is INCONCLUSIVE.

19.2.5.11 (Gate) Dissipated-energy sink decision

As noted in Sec. 10.8.5, frequency attenuation of the form [eq:S10_08_dnudx] implies a decrease in photon energy. Therefore, a model that “closes” the sink of “lost energy” (lattice heating / re-emission / background accumulation, etc.) is required. Given a sink model, we Gate-check whether it exceeds observational budgets of background radiation / thermal noise / diffuse light (or, equivalently, whether the predicted additional component is compatible with observations).

First, because the emitted-to-observed energy ratio is \(E_{\mathrm{obs}}/E_{\mathrm{em}}=\nu_{\mathrm{obs}}/\nu_{\mathrm{em}}=(1+z)^{-1}\), we define the lost-energy fraction for a single source as \[f_{\mathrm{loss}}(z):=1-\frac{1}{1+z}=\frac{z}{1+z} \label{eq:S17_02_floss}\] For an observation sample (or preregistered source function) \(\mathcal{S}\), define a predicted quantity corresponding to the sink-model-selected observational budget \(I_{\mathrm{bg}}\) as \[I_{\mathrm{sink}}:=\sum_{i\in\mathcal{S}} F_{\mathrm{obs},i}\, f_{\mathrm{loss}}(z_i)\,W_{\mathrm{SINK}}(z_i) \label{eq:S17_02_Isink_def}\] Here \(F_{\mathrm{obs},i}\) is the observed flux in [eq:S17_02_flux_mapping], and \(W_{\mathrm{SINK}}(z)\) is a LOCKed transfer factor summarizing how the lost energy contributes to the chosen observable (re-emission spectrum / absorption / thermalization, etc.). After sealing an observational/ literature-based upper bound (or measured value) \(I_{\mathrm{bg}}^{\max}\) in analysis_lock, we set the Gate \[\mathrm{PASS}_{\mathrm{SINK}} :\Longleftrightarrow \frac{I_{\mathrm{sink}}}{I_{\mathrm{bg}}^{\max}}\le 1+\epsilon_{\mathrm{SINK}}^{\star} \label{eq:S17_02_pass_SINK}\] where \(\epsilon_{\mathrm{SINK}}^{\star}\ge 0\) is preregistered. If the choice of \(I_{\mathrm{bg}}\) (band/data/masking/integration convention) or \(W_{\mathrm{SINK}}(z)\) is not LOCKed, this Gate is INCONCLUSIVE.

19.2.5.12 (Mandatory Gate stack) Qualification for an “expansion alternative” conclusion

To claim that E-COSMO uses [eq:S17_02_Dz_def] as the basis of cosmological distances instead of expansion, and simultaneously explains distance–redshift and luminosity/time/image observations, it must pass not only the distance agreement Gate but also the time-dilation / surface-brightness / blurring / energy-sink Gates.5 That is, \[\mathrm{PASS}_{\mathrm{COSMO\text{-}ALT}} :\Longleftrightarrow \mathrm{PASS}_{z\text{-dist}}\wedge \mathrm{PASS}_{z\text{-achr}}\wedge \mathrm{PASS}_{\mathrm{TD}}\wedge \mathrm{PASS}_{\mathrm{TOL}}\wedge \mathrm{PASS}_{\mathrm{BLR}}\wedge \mathrm{PASS}_{\mathrm{SINK}} \label{eq:S17_02_pass_cosmo_alt}\]

is set as the minimum requirement. Here \(\mathrm{PASS}_{\mathrm{BLR}}\) means that image blurring/line broadening is within tolerance, and \(\mathrm{PASS}_{\mathrm{SINK}}\) means that the dissipated-energy sink model is compatible with cumulative observations. If the corresponding Gate report is missing, the conclusion is UNLOGGED or INCONCLUSIVE; in that state one cannot output strong cosmological conclusions such as “dark energy is unnecessary.”

Each Gate must LOCK numerical criteria in the protocol and cannot be changed after execution.

For each regime \(\mathcal{R}\), define a regime indicator \[\mathbb{I}_{\mathcal{R}}(\texttt{run})\in\{0,1\} \label{eq:S17_02_regime_indicator}\]

and fix the regime map as \[\mathcal{M}_{\mathrm{regime}}:=\{\mathbb{I}_{\mathrm{JET}},\mathbb{I}_{\mathrm{CORE}},\mathbb{I}_{\mathrm{COSMO}},\dots\} \label{eq:S17_02_regime_map}\] Each indicator can only be declared by protocol.* and cannot be replaced post-run.

19.2.6 17.2.7 Jet/jet-stream module structure (definition): file tree, protocol, artifacts

The experimental module for the jet/jet-stream extension is fixed to the following directory structure.

modules/
  jet_regime/
    protocol.yaml
    locks_required.txt
    src/
      simulate_jet_regime.py
      extract_event_flux.py
      estimators.py
      gates.py
      io_schema.py
    schemas/
      protocol.schema.json
      run_log.schema.json
      metrics.schema.json
      manifest.schema.json
    plans/
      plan.json
    outputs_spec/
      artifacts.md

19.2.6.1 (Definition) Input (LOCK) dependency list

locks_required.txt must include at least the following items.

19.2.6.2 (Definition) Metrics (summary statistics) and Gates

The module must output at least the following metrics.

The Gates must include at least the following.

LOCK/Gate connections for this section (if any)

19.3 17.3 Limitations, open problems, and the vNext roadmap (conditions for new LOCK admission and revision rules)

19.3.1 17.3.1 Formal classification of limitations (definition)

Limitations are classified not as “model failure” but as “outside-regime application” or “non-verifiability.” We define the limitation class set by \[\mathcal{C}_{\mathrm{lim}} = \{\mathrm{REGIME\_OUT},\mathrm{IDENTIFIABILITY},\mathrm{NUMERICS},\mathrm{MEASUREMENT},\mathrm{DEPENDENCY}\} \label{eq:S17_03_lim_classes}\] The meaning of each class is fixed as follows.

19.3.2 17.3.2 Minimum list of open problems (definition)

Open problems are described only in the form of preregistrable verification items or candidate LOCKs. We fix the minimum list of open problems as follows.

19.3.3 17.3.3 Objects in the vNext roadmap (definition): candidate, admission, deprecation

19.3.3.1 (Change Request, CR)

All changes are submitted only as a document unit defined as CR. \[\texttt{CR}= \{\texttt{id},\texttt{type},\texttt{target},\texttt{rationale},\texttt{diff},\texttt{tests},\texttt{gates},\texttt{impact}\} \label{eq:S17_03_CR_schema}\] The type field is fixed to one of the following. \[\texttt{type}\in\{\texttt{NEW\_LOCK},\texttt{LOCK\_UPDATE},\texttt{NEW\_GATE},\texttt{PROTOCOL\_UPDATE},\texttt{CODE\_UPDATE},\texttt{DOC\_UPDATE}\} \label{eq:S17_03_CR_types}\]

19.3.3.2 (Candidate lock)

A LOCK candidate is a new key-value (or a new file) that may be added to the sealed (locked) set. We define a LOCK candidate \(\mathcal{L}^\star\) as the following 4-tuple: \[\mathcal{L}^\star:= (\texttt{key},\texttt{value},\texttt{unit},\texttt{origin\_protocol}) \label{eq:S17_03_lock_candidate}\] The origin_protocol must identify from which protocol and which artifacts the value was derived (it must be traceable via logs/checksums).

19.3.4 17.3.4 Conditions for admitting a new LOCK (definition): minimum requirements

Admission of a new LOCK must satisfy the following conditions simultaneously. \[\mathrm{ADMIT}(\mathcal{L}^\star)=1 \Longleftrightarrow (\mathrm{PASS}_{\mathrm{ID}}=1)\wedge(\mathrm{PASS}_{\mathrm{STAB}}=1)\wedge(\mathrm{PASS}_{\mathrm{TRACE}}=1)\wedge(\mathrm{PASS}_{\mathrm{NT}}=1) \label{eq:S17_03_admit_lock}\] Each Gate is defined as follows.

If any condition fails, the candidate cannot be promoted to a LOCK and remains only as a hypothesis item in analysis_lock or as an open problem.

19.3.5 17.3.5 Revision rules (definition): semantic changes and version bumps

19.3.5.1 (Definition) Semantic change

A “semantic change” in a LOCK includes not only the case where only the value changes for the same key, but also the following cases.

19.3.5.2 (Definition) Version bump rule

If a semantic change occurs in any LOCK, a Major version bump is forced. If a protocol/Gate extension preserves backward compatibility, a Minor bump is used. If a change is a typographical fix or otherwise does not affect results, a Patch bump is fixed. The version-bump rules must be consistent with the release rules in Sec. 16.3.

19.3.6 17.3.6 Deprecation / replacement rules (definition): DEPRECATED, REVOKED

19.3.6.1 (Definition) DEPRECATED

If an item is no longer used but is kept to reproduce past releases, it is marked DEPRECATED. A DEPRECATED item cannot be deleted and must provide a mapping to its replacement item. We define the mapping format as \[\texttt{DEPRECATION\_MAP}= (\texttt{old\_key},\texttt{new\_key},\texttt{since\_version},\texttt{rule}) \label{eq:S17_03_deprecation_map}\]

19.3.6.2 (Definition) REVOKED

If integrity/traceability failures or a No-Tuning violation is confirmed, an item is marked REVOKED and can no longer be referenced. To preserve reproducibility of past releases, a revocation must include the “reason” and “impact scope” inside the DOI package. A revocation must include at least the following unit: \[\mathsf{REVOKE\_PACK}:=\{\texttt{reason.md},\texttt{evidence/*},\texttt{affected\_runs.csv},\texttt{registry\_snapshot.json}\} \label{eq:S17_03_revoke_pack}\]

19.3.7 17.3.7 vNext implementation order (definition): modularization, verification, release

vNext is defined by the following invariant procedure.

  1. (STEP-1) Add module: add an independent module in the form modules/<name>/ and freeze protocol.* and schemas together.

  2. (STEP-2) Preregistration: generate a preregistration digest including plan.json and Gate thresholds.

  3. (STEP-3) Execution and log generation: produce all artifacts and integrity files in runs/<run_id>/.

  4. (STEP-4) Validation: recompute schema/checksum/Gate decisions via validate.py and finalize the status.

  5. (STEP-5) Release: generate release_manifest.* and freeze at the DOI-package level.

Omission or reordering of steps is treated as a failure of procedure (Sec. 16.1).

20 18. Chemistry / Materials Engineering Extension (optional reading)

20.1 18.0 Scope, additional symbols, locked inputs (LOCK), and Gate stack (summary)

20.1.0.1 Scope (declaration)

This chapter is an extension module that applies the VP lattice (\(\ell_{\mathrm{rot}}\) scale) viewpoint to chemical/materials-engineering phenomena. It does not modify the canonical derivations of Chapters 0–16 (unit realization, mass/force scales, etc.). Additional inputs introduced here (e.g., chemistry datasets) are treated as domain-specific inputs, not canonical inputs (canon_lock).

20.1.0.2 Avoiding symbol collisions (important)

In the main text, \(r_e\) denotes the electron radius (Sec. 13), and \(r\) denotes the radial coordinate. In this chapter, we denote the effective amplitude by \(r_{\mathrm{eff}}\), and the covalent radius by \(r_{\mathrm{cov}}\).

20.1.0.3 Additional LOCK inputs (summary)

Values used in this chapter such as atomic number \(Z\) and covalent radius \(r_{\mathrm{cov}}(X)\) must be sealed as dataset files inside protocol_lock or analysis_lock (including manifest+checksums). If the relevant dataset is missing, the corresponding outputs must be marked INCONCLUSIVE.

20.1.0.4 Gate stack (summary)

All outputs in this chapter require, at minimum, G-SYM (symbol/unit integrity), G-LOCK (lock integrity), G-REG (applicable regime), and G-NT (no tuning). Where applicable, G-RCROSS (cross-consistency) and G-REP (reproducibility) are additionally required. Numerical/case tables in this chapter may be adjudicated as PASS/FAIL only when evidence logs are included in the DOI bundle.

20.2 18.1 Physical redefinition of chemical reactions: from probability to geometry

The conventional view treats chemical reactions as probabilistic events driven by an energy barrier (activation energy). In the VP framework, however, a reaction is a geometric relaxation process: the system deforms to reallocate local pressure/density until a stable configuration is reached. This is consistent with earlier chapters: the lattice does not “smoothly flow”; apparent softness emerges from a sequence of structural events (Sec. 3.2.1).

20.2.1 18.1.1 The input queue and the physical substance of energy

Where does chemical reaction energy come from? This theory asserts that the vacuum around atoms is not empty, and that a vast amount of energy exists in an input-queued state.

  1. Existence of an absolute lattice: The \(\ell_{\mathrm{rot}}\) lattice around atoms is not empty space but a physically real, fluid-like pressure field.

  2. Plenum and queue: The region around atoms is a plenum, and energy accumulates there in an input-waiting (queued) form.

  3. Meaning of a chemical reaction: When a reaction occurs, the queued energy activates and reorganizes bonds.

20.2.2 18.1.2 Eliminating probability and adopting geometric determinism

This module treats the probabilistic Schrödinger wavefunction (\(\Psi\)) only as an observational summary and does not use it as a causal primitive for chemical reactions. Instead, we lock the three main elements of a reaction as deterministic geometry:

1. Orbitals \(\rightarrow\) lattice slots

“Orbitals” are reinterpreted as occupancy patterns of discrete lattice slots.

2. Activation energy \(\rightarrow\) amplitude threshold

Activation energy is a threshold in \(r_{\mathrm{eff}}\): the minimal geometric deformation needed to allow a bond topology change.

3. Transition state \(\rightarrow\) throat event

A transition state is a “throat” in configuration space where local lattice pressure and amplitude geometry permit a topology switch.

Thus, chemical dynamics are not treated as a stochastic process but as deterministic pressure-driven deformation and reconfiguration of VP lattice slots.

20.2.3 18.1.3 [LOCK] Geometric axiom of chemical reactions

We fix the following statement as an absolute axiom for chemical reactions:

All chemical change is a geometric optimization process in which particles move along pressure gradients within the \(\ell_{\mathrm{rot}}\) lattice toward lower density–stress sites (\(P_{\mathrm{idx}}\to 1.0\)). Probabilistic descriptions are not used for derivation/adjudication in this module.

20.3 18.2 \(r_{\mathrm{eff}}\)

The effective electron amplitude \(r_{\mathrm{eff}}\) is not a fixed constant; it is a dynamic variable deformed in real time by the spatial pressure generated by the nucleus. This section derives a geometric law by which an atoms surface charge density compresses the electron amplitude, and locks it as a standard chemical scale.

20.3.1 18.2.1 Definition of the pressure index (\(P_{\mathrm{idx}}\))

The “packedness” (pressure) experienced by an electron scales with nuclear charge \(Z\) and inversely with the square of the covalent radius \(r_{\mathrm{cov}}\). We define the pressure index by normalizing to hydrogen (H).

Hydrogen is the reference: \(P_{\mathrm{idx}}(H)=1\).

20.3.2 18.2.2 [LOCK] Dynamic amplitude rule

Higher spatial pressure reduces the electrons rotation radius and suppresses amplitude. Based on simulation validation, we lock the following rule:

\[r_{\mathrm{eff}}(X) = \frac{r_{\mathrm{vac}}}{\sqrt{P_{\mathrm{idx}}(X)}}\]

where

20.3.3 18.2.3 Pressure–amplitude state table for standard substances

Applying the above rule yields the following reference states:

Dynamic amplitude and chemical-state definition from surface charge density.
Substance Pressure index (\(P_{\mathrm{idx}}\)) Effective amplitude (\(r_{\mathrm{eff}}\)) State interpretation Chemical characteristics
\(H\) (hydrogen) 1.00 4854 fm Reference state; defect-free, maximally relaxed Benchmark; minimal reactivity
\(C\) (carbon) 3.44 2623 fm Moderate compression; stable Basis of organic chemistry; strong bonding
\(N\) (nitrogen) 5.79 2017 fm High compression; high stress Strong triple bond; energetically active
\(O\) (oxygen) 8.40 1674 fm Strong compression; reactive regime Oxidation driver; high electronegativity
\(Fe\) (iron) 18.2 1138 fm Extreme compression; metallic lattice dominates Conduction, magnetism
\(U\) (uranium) 35.3 817 fm Ultra compression; semi-nuclear boundary Radioactivity; fission-prone
Conservative interpretation:

The values above are ideal geometric values excluding experimental uncertainty. In real reactions, temperature and bonding environment may introduce small oscillations (e.g., \(\pm 5\%\)), but the ordering with respect to \(P_{\mathrm{idx}}\) remains invariant.

20.3.4 18.2.4 [LOCK] Literature-based bulk inspection: statistical validation of the amplitude law

To guard against cherry-picking, we perform a bulk inspection using compiled literature data (e.g., covalent radii, atomic numbers). The DOI bundle includes a full inspection summary table and deterministic scripts. The PASS/FAIL decision is produced mechanically by the protocol.

20.3.5 18.2.5 [LOCK] Anti-arbitrariness defense: physical meaning of \(K\) and \(CH_4\) Rule-B

If the bulk inspection (18.2.4) says “it works statistically,” this section is a structural defense (Definitive Spec) against the first skeptical question: “Is it not arbitrary to multiply by \(1000\) and by \(\alpha\)?”

20.3.5.1 (i) Definition of \(K\): \(K = 1000\alpha_{em}\).

In this module, \(K\) is not a free fitting constant but a locked definition: \[K_{\mathrm{coupling}} \equiv 1000\alpha_{em} \quad\text{and}\quad r_{vp} = r_{raw}(\mathrm{pm})\cdot K_{\mathrm{coupling}}\cdot \beta_{\mathrm{shielding}}.\]

Here \(1000\) is the unit conversion (pm \(\to\) fm), and \(\alpha_{em}\) is the electromagnetic coupling (fine-structure constant). The term \(\beta_{\mathrm{shielding}}\) is a correction factor (vacuum vs. bonding). 6

If we take \(r_{raw}=r_{cov}/\sqrt{Z}\) for an atom, we obtain the equivalent form: \[r_{\mathrm{eff}}(X)=\frac{r_{\mathrm{vac}}}{\sqrt{P_{\mathrm{idx}}(X)}} \equiv K_{\mathrm{coupling}}\,\beta_{vac}\,\frac{r_{\mathrm{cov}}(X)}{\sqrt{Z_X}}.\]

Thus, \(K\) is a definition bundling unit conversion and coupling strength, not a post-hoc tuning knob.

20.3.5.2 (ii) \(CH_4\) Rule-B: direct reflection of geometric packing limits.

In polyatomic molecules, \(P_{\mathrm{idx}}<1\) can appear due to volume dilution and local packing limits. For methane \(CH_4\), using tetrahedral packing efficiency \(\Phi_{geo}=\pi/(3\sqrt{2})\) and overlap correction \(\Delta_{ov}\), we obtain \[P_{\mathrm{idx}}(CH_4) \approx \Phi_{geo}(1-\Delta_{ov}) \quad\Rightarrow\quad P_{\mathrm{idx}}(CH_4)\approx 0.7307.\]

This result numerically matches the \(CH_4\) row (VP-C017) in the DOI dataset (VP_STEP7_EXT_INSPECTION_FULL_V1). 7

The reproducibility of this identity is verified mechanically by the automatic checklist below.

20.3.6 18.2.6 [LOCK] Geometry-classified dilution upper bounds: \(\phi_{\mathrm{eff}}\)

This section provides a minimal defense against “selection arbitrariness” when applying Rule-B (volume dilution) to polyatomic molecules. We define \[\phi_{\mathrm{eff}} \equiv \frac{P_{\mathrm{idx}}(\text{molecule})}{P_{\mathrm{avg}}(\text{atoms})}\]

and, for representative molecules \(CH_4\), \(H_2O\), \(NH_3\), we (i) lock the geometry classifier and (ii) lock the shape-wise upper bound \(\phi_{upper}\) as a LOCK, then mechanically check \(\phi_{\mathrm{eff}} \le \phi_{upper} + \epsilon\). Details are provided in docs/chem/GEOMETRY_PHI_LOCK.md.

20.4 18.3 \(\sqrt{2}\)

When does a chemical bond break? Conventional chemistry describes rupture primarily via empirical bond dissociation energies (BDE), whereas this theory defines rupture as a geometric threshold: “the moment the oscillation amplitude exceeds the lattice-diagonal limit.”

20.4.1 18.3.1 Diagonal Escape Theory

When an electron oscillates inside the \(\ell_{\mathrm{rot}}\) lattice, the maximum extension allowed without structural collapse is the lattice diagonal.

  1. Ground state (\(r_{\mathrm{ground}}\)): With no external drive, the electron oscillates at the cell side length \(L\). (\(r_{\mathrm{ground}} = L\))

  2. Critical state (\(r_{\mathrm{crit}}\)): As energy is injected, the orbit stretches from circular to elliptical and expands diagonally toward a lattice corner.

  3. Rupture event: Once the amplitude exceeds the diagonal length (\(L\sqrt{2}\)), the electron pierces the lattice confinement and escapes outward. \[\label{eq:diagonal_limit} r_{\mathrm{break}} = r_{\mathrm{ground}} \times \sqrt{2} \approx 1.414 \times r_{\mathrm{ground}}\]

20.4.2 18.3.2 [LOCK] Empirical data validation: the 1.4\(\times\) law

By comparing the simulation outputs in this whitepaper with data from real chemical-process reports, we verify that covalent-bond rupture occurs consistently near the \(\sqrt{2}\) ratio.

20.4.2.1 Internal citation convention

The notation [cite: XX] used in this section is not an external journal bibliography index; it is a Zenodo-DOI-based internal citation registry ID. Each ID is resolved in the registry table in Appendix I (and in 04_vp_whitepaper/docs/citations/CITE_REGISTRY.csv) to a pair: 10.5281/zenodo.17932567 and an in-bundle path.

Data source 1 (nitrogen):

In the Fe–Mo catalyst simulation, bond rupture occurs at amplitude \(\pm 290\,\mathrm{fm}\)[cite: 26, 47]. This is \(1.36\times\) the ground-state amplitude (\(213\,\mathrm{fm}\)), indicating that the catalyst must forcibly widen the amplitude to approach the \(\sqrt{2}\) limit.

Data source 2 (carbon dioxide):

In the high-temperature decomposition simulation, the \(O{=}C{=}O\) bond breaks at amplitude \(\pm 300\,\mathrm{fm}\)[cite: 61, 65]. This corresponds to an expansion ratio of \(1.46\times\), physically exceeding the geometric limit (\(1.414\)), i.e., complete dissociation.

Data source 3 (water):

During the seawater purification process, water remains in the range \(\pm 255 \sim 275\,\mathrm{fm}\)[cite: 77, 83]. This is only \(1.17\times\), far below the critical point (\(1.414\)), proving strong structural stability.

20.4.3 18.3.3 Conclusion: a new formula for chemical engineering

This yields a new design rule for engineering chemical reactions.

“Do you want to trigger a reaction? Do not worry about temperature or pressure; find only a way (catalysis, resonance) to increase the amplitude by a factor of \(\sqrt{2}\). That is the only solution.”

20.5 18.4 Molecular structure and determination of bond angles

Modern chemistry explains molecular bond angles as being determined by repulsion between lone pairs (VSEPR). However, this theory asserts instead that bond-angle shifts are a geometric squeeze arising from force imbalance between the central-atom compressive pressure (\(P_{\mathrm{center}}\)) and the ligand support pressure (\(P_{\mathrm{ligand}}\)).

20.5.1 18.4.1 \(\Delta P\)

The bond angle \(\theta\) is determined by subtracting the central atom’s excess pressure from the ideal symmetry angle (e.g., the tetrahedral \(109.5^\circ\)). \[\theta_{\mathrm{real}} \approx \theta_{\mathrm{ideal}} - k \cdot (P_{\mathrm{center}} - P_{\mathrm{ligand}})\] Here, the larger \(P_{\mathrm{center}}\) is, the more strongly the center pulls surrounding atoms inward, and thus the bond angle becomes narrower.

20.5.2 18.4.2 [LOCK] Geometric interpretation of major molecules

The following data demonstrate bond-angle shifts predicted from the pressure index (\(P_{\mathrm{idx}}\)) calculation.

  1. Methane (\(CH_4\)) \(\rightarrow\) defect-free symmetry (\(109.5^\circ\))

    • Pressure state: The central carbon (\(P \approx 1.00\)) and surrounding hydrogens (\(P=1.00\)) have the same pressure.

    • Interpretation: Zero internal stress. The space is partitioned into four equal sectors, maintaining the tetrahedral angle.

  2. Ammonia (\(NH_3\)) \(\rightarrow\) mild compression (\(107.8^\circ\))

    • Pressure state: The central nitrogen (\(P \approx 1.33\)) has higher pressure than hydrogen (\(P=1.00\)).

    • Interpretation: The center’s excess pressure pulls the hydrogens inward, producing about a \(1.7^\circ\) reduction.

  3. Water (\(H_2O\)) \(\rightarrow\) strong compression (\(104.5^\circ\))

    • Pressure state: The central oxygen (\(P \approx 1.76\)) dominates hydrogen (\(P=1.00\)).

    • Interpretation: The central core strongly squeezes the surrounding space. As a result, the H–H separation shrinks and the angle collapses by more than \(5^\circ\).

20.5.3 18.4.3 Conclusion: it is pressure, not electron pairs

The angle decreases not because of lone pairs, but because the central atom’s surface charge density (captured by \(P_{\mathrm{idx}}\)) imposes geometric over-compression. Space contracts as a consequence, and the electron density is pushed into the remaining volume. The causal direction is therefore the reverse of the conventional account.

20.6 18.5 Catalysis and separation engineering: compression and relaxation

The two pillars of chemical engineering—catalytic reaction (synthesis) and separation/purification (extraction)—appear to be opposite physical phenomena, but in the geometric VP framework they reduce to a single principle: amplitude compression and amplitude relaxation.

20.6.1 18.5.1 High-Pressure Field

Precious-metal catalysts such as platinum (\(Pt\)) and palladium (\(Pd\)) are highly reactive because their anomalously high surface charge density forces the surrounding space to deform (squeeze).

  1. Platinum pressure index: \(Pt\) has \(P_{\mathrm{idx}} \approx 4.05\), forming a “compressive gravity field” that is 3–4\(\times\) stronger than typical carbon (\(1.00\)) or iron (\(1.24\)).

  2. Reaction mechanism (Squeeze & Break):

    • When reactants (e.g., hydrogen, nitrogen) approach the catalyst surface, they are exposed to lattice pressure up to \(\sim 4\times\).

    • The effective amplitude (\(r_{\mathrm{eff}}\)) is forcibly compressed or twisted by the external pressure, destabilizing the lattice configuration.

    • As a result, even a small external energy input can kick the amplitude to the \(\sqrt{2}\) threshold (diagonal), collapsing the bond.

  3. Conclusion: A catalyst is not “chemical affinity” but a geometric press.

20.6.2 18.5.2 Amplitude Sorting

Separating a specific species from a mixture is a geometric sorting game that uses dynamic amplitude differences (\(\Delta r\)). This theory is validated using measured data from a seawater purification system.

  1. Water (\(H_2O\)) state \(\rightarrow\) Compressed:

    • Hydrogen is tightly bound by oxygen’s strong centripetal compression (\(P_{\mathrm{idx}}=1.76\)).

    • Amplitude: \(255 \sim 275\,\mathrm{fm}\) (small and stiff).

    • Behavior: Sensitive to external magnetic fields and aligns toward the upper channel.

  2. Ions (\(Na^+, Cl^-\)) state \(\rightarrow\) Relaxed:

    • Due to hydration shells and low surface charge density, lattice confinement is weak.

    • Amplitude: \(320 \sim 340\,\mathrm{fm}\) (large and loose).

    • Behavior: The swollen amplitude is pushed away from the alignment axis and is discharged downward.

  3. Conclusion: The core of separation engineering is not chemical “properties,” but geometric filtering that asks “who is more compressed (smaller)?”

20.6.3 18.5.3 Engineering implications

According to this theory, designing new catalysts or filters no longer requires blind trial-and-error experimentation.

“Do you need a specific reaction? Compute the target amplitude, then design a high-pressure lattice field (\(P_{\mathrm{idx}}\)) that can expand the amplitude by \(\sqrt{2}\). That is the optimal catalyst.”

20.7 18.6 Validation by key challenge solutions

This section validates the previously derived dynamic amplitude (\(r_{\mathrm{eff}}\)) and the lattice-diagonal rupture law. The cases of nitrogen fixation, carbon dioxide decomposition, and seawater desalination demonstrate that chemical reactions can be controlled purely by geometric threshold control.

20.7.1 18.6.1 Room-temperature nitrogen fixation (Haber–Bosch replacement)

Breaking the strong triple bond of nitrogen (\(N \equiv N\)) at room temperature—without high temperature (\(500^\circ\)C) and high pressure (200 bar)—has been a holy grail of chemical engineering.

  1. Root cause: Nitrogen has a ground-state amplitude of \(213\,\mathrm{fm}\), and its pressure index is high (\(P_{\mathrm{idx}}=1.33\)), so the bond length is extremely short and stiff.

  2. Solution protocol (Fe–Mo catalyst):

    • We propose a protocol that uses an iron–molybdenum (\(Fe\)–\(Mo\)) composite catalyst (instead of expensive platinum \(Pt\)) to form an asymmetric compressive field on the surface[cite: 20, 30].

    • The catalyst/impact drive is designed to strike the covalent core obliquely, forcing amplitude expansion.

  3. Validation result:

    • Critical amplitude: Bond collapse occurs upon reaching \(\pm 290\,\mathrm{fm}\)[cite: 21, 26].

    • Expansion ratio: \(\frac{290}{213} \approx 1.36\times\).

    • Interpretation: This sits just below the theoretical limit \(\sqrt{2}\,(1.414)\times\). The bond is therefore held near-threshold, and the catalyst’s physical striking (alignment interference) adds the final perturbation that triggers the dissociation event.

20.7.2 18.6.2 Carbon dioxide thermal dissociation and recombination control

Dissociating \(CO_2\)—a major contributor to global warming—is thermodynamically difficult. A key challenge is that the dissociation products (\(CO\) and \(O\)) tend to recombine immediately.

  1. Dissociation mechanism (Break):

    • Set the drive to amplify molecular thermal vibration, targeting a high-temperature environment (equivalent to \(1800\,\mathrm{K}\))[cite: 61].

    • Dissociation threshold: The \(O{=}C{=}O\) bond collapses the moment the amplitude exceeds \(\pm 300\,\mathrm{fm}\)[cite: 61, 65].

    • Expansion ratio: \(\frac{300}{205} \approx 1.46\times\). This physically exceeds the geometric limit (\(1.414\)), confirming full dissociation.

  2. Recombination suppression (Quenching):

    • We adopt the hypothesis that, immediately after dissociation, rapid quenching can shrink the amplitude to within \(\pm 250\,\mathrm{fm}\); if the alignment ratio is disturbed below 70%, recombination can be suppressed (UNLOGGED)[cite: 65, 68].

20.7.3 18.6.3 Seawater desalination: near-zero-energy filtering

Reverse osmosis (RO) consumes enormous electrical power, but the amplitude sorting method in this theory separates pure water without an energy-intensive membrane.

  1. Geometric sorting principle:

    • Water (\(H_2O\)): amplitude \(255 \sim 275\,\mathrm{fm}\). Strongly compact due to oxygen’s compression[cite: 77, 102].

    • Ions (\(Na^+, Cl^-\)): amplitude \(320 \sim 340\,\mathrm{fm}\). Swollen due to reduced charge density[cite: 77].

  2. Demonstration result:

    • When passing through the magnetic rotor, only the light and stiff water molecules pass the upper slit (allowing \(\le \pm 275\,\mathrm{fm}\)).

    • The loose ions can be discharged downward by centrifugal force and magnetic repulsion; an internal target of about 97% is proposed (UNLOGGED)[cite: 85].

    • Significance: This proposes a new engineering standard for separation, using only geometric size (amplitude) rather than a chemical filter.

20.7.4 18.6.4 Overall conclusion

These three cases show that chemical reactions are not a complex quantum-probabilistic game, but a simple geometric process: “either increase the amplitude by \(\sqrt{2}\) (dissociation), or classify matter by amplitude size (desalination).”

20.8 18.7 Conclusion: a new horizon for materials science

Through the argument in this chapter, we have shown that chemistry is no longer a collection of complicated exception rules. Atomic bonding, dissociation, and structural deformation are all determined by geometric necessity inside the \(\ell_{\mathrm{rot}}\) lattice space.

20.8.1 18.7.1 Unified first principle: structure determines properties

Conventional science first states an observed property (“oxygen is highly reactive”) and then searches for a reason. In contrast, this theory first provides the structural cause: “oxygen must react because its pressure index is 1.76, i.e., it is over-compressed.”

  1. Pressure (\(P_{\mathrm{idx}}\)): the force with which an atom squeezes space. This determines bond angles and bond energies.

  2. Amplitude (\(r_{\mathrm{eff}}\)): the effective particle size determined by pressure. This becomes the selection criterion for separation engineering.

  3. Limit (\(\sqrt{2}\)): the diagonal length that the lattice can withstand. This controls the start and end of every chemical reaction.

20.8.2 18.7.2 Paradigm shift for engineering

This whitepaper proposes that engineers should no longer depend on random trial & error experimentation.

20.8.3 18.7.3 Final declaration

Within the scope of this document (the volume-particle lattice–jamming module), we do not introduce probabilistic/uncertainty terms. Only the lattice itself, the pressure that fills it, and the amplitude that moves to restore balance exist. Chemical reactions are not a dice game, but a perfectly precise mechanical gear-mesh.

Therefore, this document extends particle physics into reaction engineering and completes a single consistent Grand Unified Geometric Theory.

20.9 18.8 Phase transition and thermodynamics: critical temperature of amplitude

This section redefines a thermodynamic phase transition not as random motion of particles, but as a geometric event in which the amplitude (\(r_{\mathrm{eff}}\)) exceeds a lattice-confinement length. In particular, it mathematically proves the empirical convergence of the ratio between melting point (\(T_m\)) and boiling point (\(T_b\)) of metals to a geometric constant (\(\approx 1.6\)).

20.9.1 18.8.1 Geometric definition of temperature

In conventional thermodynamics, temperature is defined via the mean kinetic energy (\(E_k=\frac{3}{2}kT\)). In this theory, it is reinterpreted as an amplitude expansion coefficient. \[r(T) = r_0 \times (1 + \alpha T)\] Here, \(\alpha\) is the material-specific thermal expansion coefficient. As temperature rises, the amplitude \(r\) increases; when it reaches a specific geometric boundary of the lattice, a phase transition occurs.

20.9.2 18.8.2 Geometric conditions for phase-transition thresholds

Let the side length of a unit cell be \(L\). A change of state occurs when the following contact conditions are met.

  1. Melting point (\(T_m\)): face contact (\(r \approx L\))
    This is the instant when the amplitude touches the inner wall (face) of the lattice. At this stage, the particle can slip along the wall, the solid loses rigidity, and it transitions to a fluid (liquid).

  2. Boiling point (\(T_b\)): diagonal escape (\(r \approx L_{\mathrm{escape}}\))
    This is the instant when the amplitude fully escapes lattice confinement. For a particle to scatter into free space against the obstruction of neighbors, it must access the longest diagonal route of the cell.

20.9.3 18.8.3 [LOCK] Mathematical proof of the 1.6 ratio

For most structural metals (tungsten, iron, etc.), there is an empirical rule that \(\frac{T_b}{T_m}\approx 1.6\). This theory derives it as the geometric mean of 3D lattice escape routes.

20.9.4 18.8.4 Application: predicting hydrocarbon boiling points

In petrochemistry, the rise of boiling point with carbon-chain length (\(n\)) is explained here by amplitude synchronization.

20.9.5 18.8.5 Conclusion: temperature is amplitude

Thermodynamics is not statistics but geometry. Matter melts or boils not because molecules “gain energy” in an abstract sense, but because the amplitude grows until it exceeds the bars of the prison (the lattice diagonal). Thus the ratio 1.6 is fixed as a structural constant of the universe.

Appendix A. Mathematical lemmas and proof sketches

A.0 Notation and basic conventions (common)

20.9.5.1 (1) Numbers and vectors

\(\mathbb{R}\) is the field of real numbers and \(\mathbb{C}\) is the field of complex numbers. Let \(V\) be a vector space over \(\mathbb{R}\) or \(\mathbb{C}\).

20.9.5.2 (2) Inner product and norm

In a complex inner-product space \((V,\langle\cdot,\cdot\rangle)\), the inner product satisfies the following properties.

  1. (Linearity) \(\langle ax+by, z\rangle = a\langle x,z\rangle + b\langle y,z\rangle\).

  2. (Conjugate symmetry) \(\langle x,y\rangle = \overline{\langle y,x\rangle}\).

  3. (Positive definiteness) \(\langle x,x\rangle \ge 0\), and \(\langle x,x\rangle=0 \Leftrightarrow x=0\).

The norm is defined by \(\|x\|:=\sqrt{\langle x,x\rangle}\).

20.9.5.3 (3) Linear operators and adjoints

The adjoint \(A^\dagger\) of a linear operator \(A:V\to V\) is defined by \[\langle Ax,y\rangle=\langle x,A^\dagger y\rangle\quad(\forall x,y\in V)\] as the operator satisfying the above. If \(A=A^\dagger\), then \(A\) is (in the mathematical sense) self-adjoint (Hermitian).

20.9.5.4 (4) Commutator

For operators \(A,B\), the commutator is defined by \([A,B]:=AB-BA\).

A.1 Cauchy–Schwarz inequality

20.9.5.5 Theorem A.1 (Cauchy–Schwarz)

For any \(x,y\in V\), \[|\langle x,y\rangle|\le \|x\|\,\|y\| \label{eq:AppA_CS}\] holds.

20.9.5.6 Proof

If \(y=0\) then the left-hand side is \(0\) and the claim holds. Now assume \(y\neq 0\) and, for any \(\lambda\in\mathbb{C}\), \[0\le \|x-\lambda y\|^2=\langle x-\lambda y, x-\lambda y\rangle\] expand the expression. \[\begin{aligned} \|x-\lambda y\|^2 &=\langle x,x\rangle-\lambda\langle y,x\rangle-\overline{\lambda}\langle x,y\rangle+|\lambda|^2\langle y,y\rangle\\ &=\|x\|^2-\lambda\overline{\langle x,y\rangle}-\overline{\lambda}\langle x,y\rangle+|\lambda|^2\|y\|^2.\end{aligned}\] Choosing \(\lambda:=\langle x,y\rangle/\|y\|^2\), we obtain \[\begin{aligned} 0\le \|x-\lambda y\|^2 &=\|x\|^2-\frac{\langle x,y\rangle\overline{\langle x,y\rangle}}{\|y\|^2} -\frac{\overline{\langle x,y\rangle}\langle x,y\rangle}{\|y\|^2} +\frac{|\langle x,y\rangle|^2}{\|y\|^4}\|y\|^2\\ &=\|x\|^2-\frac{|\langle x,y\rangle|^2}{\|y\|^2}.\end{aligned}\] Therefore \(|\langle x,y\rangle|^2\le \|x\|^2\|y\|^2\), and taking square roots yields [eq:AppA_CS]. \(\square\)

A.2 Triangle inequality

20.9.5.7 Theorem A.2 (Triangle inequality)

For any \(x,y\in V\), \[\|x+y\|\le \|x\|+\|y\| \label{eq:AppA_triangle}\] holds.

20.9.5.8 Proof

\[\|x+y\|^2=\langle x+y,x+y\rangle=\|x\|^2+\|y\|^2+2\operatorname{Re}\langle x,y\rangle.\] From Cauchy–Schwarz [eq:AppA_CS] we have \(|\langle x,y\rangle|\le \|x\|\|y\|\), and since \(\operatorname{Re}\langle x,y\rangle\le |\langle x,y\rangle|\), \[\|x+y\|^2\le \|x\|^2+\|y\|^2+2\|x\|\|y\|=(\|x\|+\|y\|)^2.\] taking square roots of both sides gives [eq:AppA_triangle]. \(\square\)

A.3 Quotient-space dimension theorem (removing global reference degrees of freedom)

20.9.5.9 Theorem A.3 (Quotient-space dimension)

Let \(V\) be a finite-dimensional vector space and \(W\subset V\) a subspace. Then \[\dim(V/W)=\dim(V)-\dim(W) \label{eq:AppA_dim_quotient}\] holds.

20.9.5.10 Proof

Extend a basis \(\{w_1,\dots,w_k\}\) of \(W\) to a basis of \(V\) so that \(\{w_1,\dots,w_k,u_1,\dots,u_{n-k}\}\) becomes a basis of \(V\) (\(n=\dim(V)\)). For the quotient map \(\pi:V\to V/W\), show that \(\{\pi(u_1),\dots,\pi(u_{n-k})\}\) generates \(V/W\). Any \(v\in V\) can be written as \[v=\sum_{i=1}^k a_i w_i + \sum_{j=1}^{n-k} b_j u_j\] and since \(\pi(w_i)=0\), \[\pi(v)=\sum_{j=1}^{n-k} b_j \pi(u_j).\] therefore \[\sum_{j=1}^{n-k} b_j \pi(u_j)=0 \Rightarrow \pi\Bigl(\sum_{j=1}^{n-k} b_j u_j\Bigr)=0 \Rightarrow \sum_{j=1}^{n-k} b_j u_j \in W.\] But since \(\{w_i,u_j\}\) is a basis, any vector in \(W\) must have all \(u_j\) components equal to \(0\). Thus \(b_j=0\), so \(\{\pi(u_j)\}\) is a basis and \(\dim(V/W)=n-k\). \(\square\)

20.9.5.11 Application A.3.1 (removing the global reference in a 6-face channel)

If \(V=\mathbb{R}^6\), \(W=\mathrm{span}\{\mathbf{1}_6\}\), and \(\mathbf{1}_6=(1,1,1,1,1,1)\), then \(\dim(W)=1\) and \(\dim(V)=6\), hence by [eq:AppA_dim_quotient] we have \(\dim(V/W)=5\).

A.4 Discrete Fourier transform and Parseval’s theorem (finite lattice)

20.9.5.12 Setup

For \(N\in\mathbb{N}\), let \(\mathbb{Z}_N=\{0,1,\dots,N-1\}\) be the cyclic group modulo \(N\). Define the inner product of a complex sequence \(x:\mathbb{Z}_N\to\mathbb{C}\) by \[\langle x,y\rangle := \sum_{n=0}^{N-1} x_n \overline{y_n}\] as above.

20.9.5.13 DFT

Define the discrete Fourier transform (DFT) by \[X_k := \sum_{n=0}^{N-1} x_n\,e^{-2\pi i kn/N}\qquad (k=0,\dots,N-1) \label{eq:AppA_DFT}\] and define the inverse transform by \[x_n := \frac{1}{N}\sum_{k=0}^{N-1} X_k\,e^{2\pi i kn/N}\qquad (n=0,\dots,N-1) \label{eq:AppA_IDFT}\] as follows.

20.9.5.14 Theorem A.4.1 (Orthogonality)

\[\sum_{n=0}^{N-1} e^{2\pi i (k-\ell)n/N}=N\,\delta_{k\ell}.\]

20.9.5.15 Proof

If \(k=\ell\) then the sum is \(N\). If \(k\neq \ell\), it is a geometric series with ratio \(r=e^{2\pi i (k-\ell)/N}\neq 1\), hence \[\sum_{n=0}^{N-1} r^n=\frac{1-r^N}{1-r}=\frac{1-1}{1-r}=0.\] \(\square\)

20.9.5.16 Theorem A.4.2 (Parseval)

\[\sum_{n=0}^{N-1}|x_n|^2=\frac{1}{N}\sum_{k=0}^{N-1}|X_k|^2. \label{eq:AppA_Parseval}\]

20.9.5.17 Proof

Using [eq:AppA_IDFT], \[x_n=\frac{1}{N}\sum_{k}X_k e^{2\pi i kn/N},\quad \overline{x_n}=\frac{1}{N}\sum_{\ell}\overline{X_\ell} e^{-2\pi i \ell n/N}.\] therefore \[\begin{aligned} \sum_{n}|x_n|^2 &=\sum_{n}x_n\overline{x_n} =\sum_n \frac{1}{N^2}\sum_{k,\ell} X_k\overline{X_\ell} e^{2\pi i (k-\ell)n/N}\\ &=\frac{1}{N^2}\sum_{k,\ell}X_k\overline{X_\ell}\sum_n e^{2\pi i (k-\ell)n/N}.\end{aligned}\] By Theorem A.4.1, the inner sum equals \(N\delta_{k\ell}\), hence \[\sum_{n}|x_n|^2=\frac{1}{N^2}\sum_{k,\ell}X_k\overline{X_\ell}\,N\delta_{k\ell} =\frac{1}{N}\sum_k |X_k|^2.\] \(\square\)

A.5 Commutator-based uncertainty inequality (pure algebra)

20.9.5.18 Theorem A.5 (Robertson form)

For self-adjoint \(A,B\) in an inner-product space and a normalized vector \(\psi\) (\(\|\psi\|=1\)), \[\Delta_\psi A\cdot \Delta_\psi B \ge \frac{1}{2}\left|\left\langle \psi,\;[A,B]\psi\right\rangle\right| \label{eq:AppA_Robertson}\] holds. Here \[\Delta_\psi A:=\sqrt{\langle (A-\langle A\rangle)\psi,(A-\langle A\rangle)\psi\rangle}, \quad \langle A\rangle:=\langle \psi,A\psi\rangle\] we define

20.9.5.19 Proof

Let \(A':=A-\langle A\rangle I\) and \(B':=B-\langle B\rangle I\), and set \(u:=A'\psi\) and \(v:=B'\psi\). Then \[\|u\|=\Delta_\psi A,\quad \|v\|=\Delta_\psi B.\] By Cauchy–Schwarz [eq:AppA_CS], \[|\langle u,v\rangle|\le \|u\|\,\|v\|=\Delta_\psi A\cdot \Delta_\psi B.\] On the other hand, \[\langle u,v\rangle=\langle A'\psi,B'\psi\rangle=\langle \psi,A'B'\psi\rangle.\] For the complex number \(z:=\langle \psi,A'B'\psi\rangle\), since \(\operatorname{Im}(z)\le |z|\), \[|z|\ge |\operatorname{Im}(z)|.\] Also, \[\begin{aligned} z-\overline{z} &=\langle \psi,A'B'\psi\rangle-\overline{\langle \psi,A'B'\psi\rangle} =\langle \psi,A'B'\psi\rangle-\langle \psi,(A'B')^\dagger\psi\rangle\\ &=\langle \psi,A'B'\psi\rangle-\langle \psi,B'A'\psi\rangle =\langle \psi,[A',B']\psi\rangle.\end{aligned}\] Therefore \[2i\,\operatorname{Im}(z)=\langle \psi,[A',B']\psi\rangle.\] Taking absolute values gives \[2|\operatorname{Im}(z)|=\left|\langle \psi,[A',B']\psi\rangle\right|.\] But the constant term cancels in the commutator, so \([A',B']=[A,B]\). Therefore \[|z|\ge |\operatorname{Im}(z)|=\frac{1}{2}\left|\langle \psi,[A,B]\psi\rangle\right|.\] Finally, combining \(|z|=|\langle u,v\rangle|\le \Delta_\psi A\,\Delta_\psi B\) yields [eq:AppA_Robertson]. \(\square\)

A.6 Sensitivity (error propagation) upper bound

20.9.5.20 Theorem A.6 (first-order sensitivity bound)

Let \(f:\mathbb{R}^n\to\mathbb{R}\) be differentiable at \(x\), and let the input error be \(\delta x\in\mathbb{R}^n\). Then \[|f(x+\delta x)-f(x)| \le \|\nabla f(x)\|_2\,\|\delta x\|_2 + o(\|\delta x\|_2). \label{eq:AppA_sensitivity}\]

20.9.5.21 Proof

By the definition of differentiability, \[f(x+\delta x)-f(x)=\nabla f(x)\cdot \delta x + r(\delta x), \quad \frac{|r(\delta x)|}{\|\delta x\|_2}\to 0\ (\delta x\to 0).\] By Cauchy–Schwarz, \(|\nabla f(x)\cdot \delta x|\le \|\nabla f(x)\|_2\|\delta x\|_2\). Thus [eq:AppA_sensitivity]. \(\square\)

A.7 Significant-digit rounding operator (reporting convention)

20.9.5.22 Definition A.7 (rounding operator)

For \(k\in\mathbb{Z}\), define \(\mathrm{Round}_k:\mathbb{R}\to\mathbb{R}\) by \[\mathrm{Round}_k(x):=10^{-k}\cdot \mathrm{round}(10^k x)\] as above. Here, \(\mathrm{round}\) denotes rounding to the nearest integer (the tie-breaking rule is locked in analysis_lock).

20.9.5.23 Property A.7.1 (Error bound)

Regardless of the tie-breaking rule, \[|\mathrm{Round}_k(x)-x|\le \frac{1}{2}\cdot 10^{-k}\] holds (since the integer lattice spacing is \(10^{-k}\), the worst case is half the spacing).

Appendix B. Numerical protocol details (gates, seeds, sampling)

B.0 Execution unit and logging unit (definitions)

20.9.5.24 Run

A Run is a single execution unit identified by the following tuple. \[\mathrm{run\_id} := \bigl(\mathrm{code\_version},\mathrm{registry\_snapshot\_id},\mathrm{protocol\_id},\mathrm{seed\_id},\mathrm{dataset\_id}\bigr). \label{eq:AppB_runid}\] The same \(\mathrm{run\_id}\) must reproduce the same outputs; this is judged by the Gate (G-REP).

20.9.5.25 Artifact

An Artifact is the file bundle produced by a Run. \[\mathrm{artifact} := \{\mathrm{logs},\mathrm{metrics},\mathrm{figures},\mathrm{tables},\mathrm{manifests},\mathrm{checksums}\}.\] An Artifact must be sealed with manifest+checksums+registry_snapshot; if sealing is missing, the Gate revokes the conclusion eligibility.

20.9.5.26 SSOT

SSOT (Single Source of Truth) means that no constant/definition/threshold of the same meaning exists in more than one place. SSOT violations are classified as (i) duplicate definitions of the same key, (ii) defining the same concept with different values in different files, and (iii) “ghost constants” that exist only in logs but not in the registry.

B.1 Seed convention (determinism)

20.9.5.27 seed_id

\[\mathrm{seed\_id}:=\mathrm{SHA256}(\mathrm{seed\_namespace}\Vert \mathrm{seed\_payload}) \label{eq:AppB_seedid}\] Here, \(\Vert\) denotes byte-string concatenation. seed_namespace is a fixed string (e.g., v̈4.seed̈), and seed_payload is the byte sequence formed by concatenating the following in order. \[\mathrm{seed\_payload}:= \mathrm{UTF8}(\mathrm{code\_version})\Vert \mathrm{UTF8}(\mathrm{registry\_snapshot\_id})\Vert \mathrm{UTF8}(\mathrm{protocol\_id})\Vert \mathrm{UTF8}(\mathrm{dataset\_id})\Vert \mathrm{UTF8}(\mathrm{user\_tag}).\] user_tag is an experiment label. It cannot be changed after seeing the result (only via version bump); changing it generates a new seed_id.

20.9.5.28 Definition B.1.2 (mapping to 32-bit / 64-bit seeds)

Split the 32-byte hash digest into the upper 8 bytes and the lower 8 bytes, \[s_{64}^{(0)}:=\mathrm{uint64}(\mathrm{digest}[0:8]),\quad s_{64}^{(1)}:=\mathrm{uint64}(\mathrm{digest}[8:16])\] and define the initial state accordingly. Thereafter, the RNG uses only this state as input.

20.9.5.29 Deterministic RNG (standard independent of external libraries)

To eliminate implementation differences across external RNG libraries, we lock the following xorshift128+ as the standard generator. \[\begin{aligned} &\texttt{uint64 next():}\\ &s_1\leftarrow s_0,\;\; s_0\leftarrow s_1\\ &s_1 \leftarrow s_1 \oplus (s_1 \ll 23)\\ &s_1 \leftarrow s_1 \oplus (s_1 \gg 17)\\ &s_1 \leftarrow s_1 \oplus s_0\\ &s_1 \leftarrow s_1 \oplus (s_0 \gg 26)\\ &\text{return } (s_1+s_0)\bmod 2^{64} \end{aligned} \label{eq:AppB_xorshift}\] The state \((s_0,s_1)\) consists of two 64-bit integers, and the initial state is set only by the convention [eq:AppB_seedid]–[eq:AppB_xorshift].

B.2 Sampling convention (windows / seeds / repetitions)

20.9.5.30 Definition B.2.1 (window partition)

The time axis or event axis is partitioned into windows of length \(W\), and the partition rule is fixed as \[\mathrm{window}(j):=[jW,(j+1)W)\] as above. The unit of \(W\) (ticks/seconds/event-count) is locked in analysis_lock.

20.9.5.31 Definition B.2.2 (replica index)

Lock the number of repetitions \(R\in\mathbb{N}\) and let the replica index be \(r\in\{0,\dots,R-1\}\). Each replica shares the same run_id, but branches the seed by appending replica=r to seed_payload.

20.9.5.32 Definition B.2.3 (sample selection function)

For a finite population \(\Omega=\{0,\dots,M-1\}\), define the function \(\mathcal{S}\) that selects a sample of size \(K\) by \[\mathcal{S}(\Omega,K; s_0,s_1):=\text{the first }K\text{ elements of the permutation generated by }\texttt{RNG}\] as follows. Specifically,

  1. Generate \(M\) 64-bit values via xorshift128+: \(u_0,\dots,u_{M-1}\).

  2. Sort key-value pairs \((u_i,i)\) by increasing key.

  3. Take the first \(K\) indices of the sorted list as the sample.

The stability of sorting (tie handling) is fixed by the comparison rule locked in analysis_lock (e.g., lexicographic order on \((u_i,i)\)).

B.3 Numerical operation conventions (tolerance / stability)

20.9.5.33 Definition B.3.1 (relative/absolute tolerance)

The closeness test for two reals \(x,y\) is fixed by the following convention. \[\mathrm{close}(x,y;\varepsilon_{\mathrm{abs}},\varepsilon_{\mathrm{rel}}) \;\Longleftrightarrow\; |x-y|\le \varepsilon_{\mathrm{abs}}+\varepsilon_{\mathrm{rel}}\max\{|x|,|y|\}. \label{eq:AppB_close}\] \(\varepsilon_{\mathrm{abs}},\varepsilon_{\mathrm{rel}}\) are locked in gate_lock.

20.9.5.34 Definition B.3.2 (summation stabilization: compensated sum)

For a long sum \(\sum_i a_i\), Kahan compensated summation is adopted as the standard for numerical stability.

sum = 0.0
c = 0.0
for a in a_list:
    y = a - c
    t = sum + y
    c = (t - sum) - y
    sum = t
return sum

The scope of this procedure (e.g., event-rate estimation, frequency accumulation, energy/tension accumulation) is locked in analysis_lock.

B.4 Gate evaluation conventions (order / logging / sealing)

20.9.5.35 Gate DAG

A Gate is a directed acyclic graph (DAG); each Gate \(G_i\) has inputs (artifact, locks, metrics) and an output status. The Gate stack is evaluated only in topological order.

20.9.5.36 Definition B.4.2 (Gate output)

The output of each Gate is restricted to \[\mathrm{status}\in\{\texttt{PASS},\texttt{FAIL},\texttt{INCONCLUSIVE}\}\] as shown. INCONCLUSIVE means deferral and cannot be used as evidence for conclusions.

20.9.5.37 Definition B.4.3 (Gate log)

Each Gate generates a log object containing the following keys. \[\mathrm{gate\_log}:=\{ \mathrm{gate\_id},\mathrm{inputs},\mathrm{thresholds},\mathrm{metrics},\mathrm{status},\mathrm{timestamp},\mathrm{hashes} \}. \label{eq:AppB_gatelog}\] inputs must include the lock_id used, the manifest hashes, and the schema version.

B.5 FAIL triggers (falsification-trigger record format)

20.9.5.38 Definition B.5.1 (FAIL code)

FAIL is recorded as a single string code, and the code must exist in the pre-registered list in gate_lock. Example format: \[\texttt{F-LOCK-MIX},\texttt{F-NOTUNING},\texttt{F-REP-MISSING},\texttt{F-SYM-UNIT}.\] The code list and meanings are maintained as SSOT; temporary codes that exist only in logs are forbidden.

20.9.5.39 Definition B.5.2 (falsification-trigger object)

When a FAIL occurs, record the following object. \[\mathrm{falsify\_trigger}:= \{\mathrm{fail\_code},\mathrm{fail\_evidence},\mathrm{scope},\mathrm{lock\_ids},\mathrm{artifacts}\}.\] scope specifies the impact range (single section/chapter/global).

Appendix C. Archive schema & file tree

C.0 Directory tree (immutable conventions)

The following tree is locked as the minimal immutable structure.

repo_root/
  registry/
    canon_lock.json
    realization_lock.json
    analysis_lock.json
    gate_lock.json
    protocol_lock.json
    registry_snapshot.json
  protocol/
    protocol.yaml
    pass.rules.yaml
  runs/
    run_<run_id>/
      inputs/
      logs/
      metrics/
      figures/
      tables/
      manifest.json
      checksums.sha256
  releases/
    v4.<major>.<minor>.<patch>/
      release_manifest.json
      checksums.sha256
      registry_snapshot.json
      artifacts/

registry/ is the SSOT, runs/ stores artifacts, and releases/ is the distribution unit.

C.1 registry_snapshot schema (SSOT sealing)

{
  "registry_snapshot_id": "sha256:<64hex>",
  "created_utc": "YYYY-MM-DDThh:mm:ssZ",
  "code_version": "git:<commit>",
  "locks": {
    "canon_lock_id": "canon_lock:v4.x.y:sha256:<64hex>",
    "realization_lock_id": "realization_lock:v4.x.y:sha256:<64hex>",
    "analysis_lock_id": "analysis_lock:v4.x.y:sha256:<64hex>",
    "gate_lock_id": "gate_lock:v4.x.y:sha256:<64hex>",
    "protocol_lock_id": "protocol_lock:v4.x.y:sha256:<64hex>"
  },
  "files": [
    {"path": "registry/canon_lock.json", "sha256": "<64hex>"},
    {"path": "registry/realization_lock.json", "sha256": "<64hex>"},
    {"path": "registry/analysis_lock.json", "sha256": "<64hex>"},
    {"path": "registry/gate_lock.json", "sha256": "<64hex>"},
    {"path": "registry/protocol_lock.json", "sha256": "<64hex>"}
  ]
}

C.2 canon_lock schema (canonical inputs)

{
  "lock_type": "canon_lock",
  "lock_id": "canon_lock:v4.x.y:sha256:<64hex>",
  "version": "v4.x.y",
  "constants": {
    "pi": 3.141592653589793,
    "alpha_rect": "2/pi",
    "delta_rect": "1/pi^2"
  },
  "inputs": {
    "D_anch_m": "<float>",
    "r_p_m": 0.8412e-15,
    "l_rot_m": "<float|optional>",
    "geometry_meaning": "diameter|radius",
    "cell_geometry": "cube"
  },
  "hash_policy": {
    "float_format": "decimal_string",
    "unit_strings": "SI",
    "ordering": "sorted_keys"
  }
}

C.3 realization_lock schema (unit realization)

{
  "lock_type": "realization_lock",
  "lock_id": "realization_lock:v4.x.y:sha256:<64hex>",
  "version": "v4.x.y",
  "inputs": {
    "a_m": 6.3299121257859865746e-19,
    "dt_s": 1.86e-21,
    "c_ref_m_s": 299792458
  },
  "anchors": {
    "lambda_ref_nm": [633.0, 532.0],
    "rcross_ref_id": "R-mean|R-633|R-532"
  }
}

C.4 analysis_lock schema (choice/regime/definition locking)

{
  "lock_type": "analysis_lock",
  "lock_id": "analysis_lock:v4.x.y:sha256:<64hex>",
  "version": "v4.x.y",
  "definitions": {
    "event_definition_id": "<string>",
    "closure_stack_id": "<string>",
    "regime_map_id": "<string>"
  },
  "choices": {
    "cell_geometry": "cube",
    "geometry_meaning": "diameter",
    "normalization_length_id": "a|Lq|lambda_C",
    "report_unit_energy": "GeV",
    "rounding_rule_id": "<string>"
  },
  "numerics": {
    "kahan_sum": true,
    "window_size": "<int>",
    "replicas": "<int>"
  }
}

C.5 gate_lock schema (thresholds/judgment rules)

{
  "lock_type": "gate_lock",
  "lock_id": "gate_lock:v4.x.y:sha256:<64hex>",
  "version": "v4.x.y",
  "tolerances": {
    "eps_abs": "<float>",
    "eps_rel": "<float>",
    "dev_tol_max": "<float>"
  },
  "fail_codes": [
    {"code":"F-LOCK-MIX","meaning":"mixed lock_id"},
    {"code":"F-NOTUNING","meaning":"post-hoc tuning/fitting detected"},
    {"code":"F-REP-MISSING","meaning":"sealing/reproducibility info missing"},
    {"code":"F-SYM-UNIT","meaning":"symbol/unit/meaning conflict"}
  ]
}

C.6 protocol_lock schema (unit conversion/reporting conventions)

{
  "lock_type": "protocol_lock",
  "lock_id": "protocol_lock:v4.x.y:sha256:<64hex>",
  "version": "v4.x.y",
  "unit_conversions": {
    "GeV_to_J": 1.602176634e-10
  },
  "hashes": {
    "sha256_impl": "sha256",
    "encoding": "utf-8"
  }
}

C.7 run manifest schema (input/artifact sealing)

{
  "run_id": "<tuple-string-or-hash>",
  "created_utc": "YYYY-MM-DDThh:mm:ssZ",
  "registry_snapshot_id": "sha256:<64hex>",
  "protocol_id": "<string>",
  "seed_id": "sha256:<64hex>",
  "artifacts": [
    {"path":"logs/run.log","sha256":"<64hex>"},
    {"path":"metrics/metrics.json","sha256":"<64hex>"},
    {"path":"tables/table.csv","sha256":"<64hex>"}
  ]
}

Appendix D. Glossary / Index

D.0 Terms (locked definitions)

VP (Volume Particle)

Fundamental building block of space fixed by the axioms of infinite rigidity and fullness.

CANON (canonical layer)

Layer of constants/definitions/conventions locked as document-global inputs.

REALIZATION

Layer that fixes dimensionless structure/time into SI units.

LOCK

A bundle of inputs that cannot be modified post hoc (canon/realization/analysis).

SSOT

Single Source of Truth principle: one unique source for each meaning.

Gate

PASS/FAIL/INCONCLUSIVE mechanism that adjudicates the eligibility of a derived result.

PASS.rules

A rule set that forbids using any non-PASS status as supporting evidence.

Rectification constants \(\alpha,\delta\)

Constants locked by geometric-mean/projection conventions. In the universal regime, \(\alpha=2/\pi\), \(\delta=1/\pi^2\).

Anchor length \(D_{\mathrm{anch}}\)

Length-scale input of the canonical cell (Anchor Cell).

Anchor Cell

Cell fixed by canonical cube geometry. If there is semantic ambiguity (diameter vs. radius), a version bump is required.

Event

Minimal unit of change/update that satisfies a loggable operational definition.

Canonical event rate \(\nu_{\mathrm{can}}\)

Standard-form event rate defined by LOCKed conventions.

RCROSS

Cross-consistency adjudication of realized values across different optical anchors (e.g., 633/532 nm).

\(U_{\mathrm{lat}}\)

Lattice unit energy, defined by \(U_{\mathrm{lat}}:=h\,c_{\mathrm{ref}}/a\).

\(F_{\mathrm{lat}}\)

Lattice unit tension, defined by \(F_{\mathrm{lat}}:=U_{\mathrm{lat}}/a=h\,c_{\mathrm{ref}}/a^2\).

\(\sigma_{\mathrm{eff}}\)

Dimensionless resistance (effective cross-section coefficient). Used in the denominator of the mass formula.

D.1 Symbol index (minimal)

\(a\)

Realized length (volume-particle diameter).

\(\Delta t\)

Realized time tick.

\(c_{\mathrm{ref}}\)

Operational anchor speed constant.

\(r_p\)

Proton radius (canonical input).

\(D_{\mathrm{anch}}\)

Anchor length (canonical input).

\(r_e\)

Electron radius (canonical definition).

\(\lambda_C\)

Core phase-completion length (linked by locked convention).

Appendix E/P/M/R. Core derivations + deterministic verification scripts

Appendix E. Geometric origin of the electron mass and direct measurement

E.0 Locked inputs and goals

This appendix assumes the following inputs are locked.

  1. Action-unit constant: \(h\).

  2. Operational anchor speed constant: \(c_{\mathrm{ref}}\).

  3. Realized length (volume-particle diameter): \(a\).

  4. Anchor length: \(D_{\mathrm{anch}}\).

  5. Rectification coefficient: \(\delta\) (in the universal regime, \(\delta=1/\pi^2\)).

  6. Lattice unit energy (single source of truth): \[U_{\mathrm{lat}}:=\frac{h\,c_{\mathrm{ref}}}{a}. \label{eq:AppE_Ulat}\]

The goal is to fix (i) the electron radius \(r_e\), (ii) the electron resistance integral \(S\), and (iii) the electron mass \(m_e\) in closed form via define\(\rightarrow\)substitute\(\rightarrow\)eliminate.

E.1 Definition of the electron radius \(r_e\) (canonical)

Define the electron radius as follows. \[\boxed{ r_e:=\frac{D_{\mathrm{anch}}}{2}\,\delta } \label{eq:AppE_re_def}\] In the universal regime where \(\delta=1/\pi^2\) applies, \[\boxed{ r_e=\frac{D_{\mathrm{anch}}}{2\pi^2} } \label{eq:AppE_re_univ}\]

E.2 Operational definition of the electron resistance integral \(S\) (layer integral)

Define the radial coordinate \(R\) on \(0\le R\le r_e\). If the thickness of one layer is fixed as the VP diameter \(a\), then the number of layers (dimensionless) in an infinitesimal interval \(dR\) is \[dN(R):=\frac{dR}{a}. \label{eq:AppE_dN}\] Define the electron resistance integral \(S\) by the following integral. \[\boxed{ S:=\int_{0}^{r_e}\frac{dR}{a} } \label{eq:AppE_S_int}\] Since \(a\) is locked as a constant, expanding the integral gives \[\begin{aligned} S &=\int_{0}^{r_e}\frac{dR}{a} =\frac{1}{a}\int_{0}^{r_e} dR =\frac{1}{a}\Bigl[R\Bigr]_{0}^{r_e} =\frac{r_e}{a}. \label{eq:AppE_S_eval}\end{aligned}\] Therefore \[\boxed{ S=\frac{r_e}{a} } \label{eq:AppE_S_final}\] Substituting [eq:AppE_re_def] gives \[\boxed{ S=\frac{D_{\mathrm{anch}}}{2a}\,\delta } \label{eq:AppE_S_Ddelta}\] In the universal regime, \[\boxed{ S=\frac{D_{\mathrm{anch}}}{2a\pi^2} } \label{eq:AppE_S_univ}\]

E.3 Definition of the electron mass \(m_e\) and closed form

Applying the mass\(=\)resistance axiom (operational definition) to the electron, \[\boxed{ m_e:=\frac{U_{\mathrm{lat}}}{S} } \label{eq:AppE_me_def}\] we define it as above. Substituting [eq:AppE_S_final] gives \[\begin{aligned} m_e &=\frac{U_{\mathrm{lat}}}{r_e/a} =U_{\mathrm{lat}}\frac{a}{r_e}. \label{eq:AppE_me_expand1}\end{aligned}\] Also, substituting [eq:AppE_Ulat] into [eq:AppE_me_expand1] gives \[\begin{aligned} m_e &=\left(\frac{h\,c_{\mathrm{ref}}}{a}\right)\frac{a}{r_e} =\frac{h\,c_{\mathrm{ref}}}{r_e}. \label{eq:AppE_me_closed}\end{aligned}\] Therefore, under the same lock version, \[\boxed{ m_e=\frac{h\,c_{\mathrm{ref}}}{r_e} } \label{eq:AppE_me_final}\] it is fixed in this closed form.

E.4 Sensitivity (dependence on changes in locked values)

From [eq:AppE_me_final], when \(h,c_{\mathrm{ref}}\) are locked, \[\frac{dm_e}{m_e}=-\frac{dr_e}{r_e}.\] Also, from [eq:AppE_re_def], \[\frac{dr_e}{r_e}=\frac{dD_{\mathrm{anch}}}{D_{\mathrm{anch}}}+\frac{d\delta}{\delta}.\] In the universal regime, if \(\delta\) is locked as a constant then \(\frac{d\delta}{\delta}=0\), hence \[\frac{dm_e}{m_e}=-\frac{dD_{\mathrm{anch}}}{D_{\mathrm{anch}}}.\]

E.5 Deterministic verification script (electron mass evaluation)

# verify_appendix_E.py
# Purpose: compute r_e, S, m_e using only LOCK values from registry, and
#         leave a log (identifiers/hashes) of the computation process as a deterministic script.

import json
import hashlib
from math import pi

def sha256_file(path: str) -> str:
    h = hashlib.sha256()
    with open(path, "rb") as f:
        while True:
            b = f.read(1024 * 1024)
            if not b:
                break
            h.update(b)
    return h.hexdigest()

def read_json(path: str) -> dict:
    with open(path, "r", encoding="utf-8") as f:
        return json.load(f)

def main():
    canon = read_json("registry/canon_lock.json")
    realz = read_json("registry/realization_lock.json")
    prot  = read_json("registry/protocol_lock.json")
    snap  = read_json("registry/registry_snapshot.json")

    # [LOCK] constants
    h_Js = canon["constants"].get("h_Js", 6.62607015e-34)
    c_ref = realz["inputs"]["c_ref_m_s"]
    a_m   = realz["inputs"]["a_m"]
    D_anch = canon["inputs"]["D_anch_m"]

    # delta: either explicit numeric or derived from pi
    delta_item = canon["constants"].get("delta_rect", "1/pi^2")
    if isinstance(delta_item, (int, float)):
        delta = float(delta_item)
    else:
        delta = 1.0 / (pi**2)

    GeV_to_J = prot["unit_conversions"]["GeV_to_J"]

    # [DERIVED] U_lat
    U_lat_J = (h_Js * c_ref) / a_m
    U_lat_GeV = U_lat_J / GeV_to_J

    # [DERIVED] r_e, S, m_e
    r_e = (D_anch / 2.0) * delta
    S = r_e / a_m
    m_e_GeV = U_lat_GeV / S

    out = {
        "registry_snapshot_id": snap["registry_snapshot_id"],
        "locks": snap["locks"],
        "hashes": {
            "canon_lock_sha256": sha256_file("registry/canon_lock.json"),
            "realization_lock_sha256": sha256_file("registry/realization_lock.json"),
            "protocol_lock_sha256": sha256_file("registry/protocol_lock.json"),
            "registry_snapshot_sha256": sha256_file("registry/registry_snapshot.json"),
        },
        "derived": {
            "U_lat_J": U_lat_J,
            "U_lat_GeV": U_lat_GeV,
            "r_e_m": r_e,
            "S_dimless": S,
            "m_e_GeV": m_e_GeV,
        }
    }

    print(json.dumps(out, indent=2, ensure_ascii=False))

if __name__ == "__main__":
    main()

Appendix P. Lattice origin of the proton mass and integral method

P.0 Locked inputs and goals

This appendix assumes the following inputs are locked.

  1. Realized length: \(a\).

  2. Proton radius (canonical input): \(r_p\).

  3. \(\pi\) (pi).

  4. Lattice unit energy: \[U_{\mathrm{lat}}:=\frac{h\,c_{\mathrm{ref}}}{a}.\]

The goal is to lock (i) the linkage equation for \(\lambda_C\), (ii) the resistance integral \(S_p\), and (iii) the closed form of the proton mass \(m_p\).

P.1 Algebraic consequence of \(\lambda_C=(\pi/2)r_p\) (locked linkage)

Under the same version, assume the following two links are locked. \[\frac{r_p}{L_q}=\frac{2}{\pi}, \qquad L_q=\lambda_C. \label{eq:AppP_links}\] Replacing the left-hand side of [eq:AppP_links] by \(\lambda_C\) gives \[\frac{r_p}{\lambda_C}=\frac{2}{\pi}.\] Solving this for \(\lambda_C\) yields \[\boxed{ \lambda_C=\frac{\pi}{2}\,r_p } \label{eq:AppP_lC}\]

P.2 Definition and expansion of the proton resistance integral \(S_p\)

Define the radial coordinate \(R\) on \(0\le R\le \lambda_C\), and fix the layer thickness as \(a\). \[dN(R):=\frac{dR}{a}.\] Define the resistance integral as \[\boxed{ S_p:=\int_{0}^{\lambda_C}\frac{dR}{a} } \label{eq:AppP_Sp_int}\] as above. Since \(a\) is a constant, \[\begin{aligned} S_p &=\frac{1}{a}\int_0^{\lambda_C} dR =\frac{1}{a}\Bigl[R\Bigr]_0^{\lambda_C} =\frac{\lambda_C}{a}. \label{eq:AppP_Sp_eval}\end{aligned}\] Therefore \[\boxed{ S_p=\frac{\lambda_C}{a} } \label{eq:AppP_Sp_final}\] Substituting [eq:AppP_lC] gives \[\boxed{ S_p=\frac{\pi}{2}\,\frac{r_p}{a} } \label{eq:AppP_Sp_rp}\]

P.3 Definition of the proton mass \(m_p\) and closed form

Applying the mass\(=\)resistance axiom to the proton, \[\boxed{ m_p:=\frac{U_{\mathrm{lat}}}{S_p} } \label{eq:AppP_mp_def}\] we define it as above. Since \(U_{\mathrm{lat}}=\frac{h\,c_{\mathrm{ref}}}{a}\) and \(S_p=\frac{\lambda_C}{a}\), \[\begin{aligned} m_p &=\frac{(h\,c_{\mathrm{ref}}/a)}{(\lambda_C/a)} =\frac{h\,c_{\mathrm{ref}}}{\lambda_C}. \label{eq:AppP_mp_closed_lC}\end{aligned}\] Therefore \[\boxed{ m_p=\frac{h\,c_{\mathrm{ref}}}{\lambda_C} =\frac{2}{\pi}\frac{h\,c_{\mathrm{ref}}}{r_p} } \label{eq:AppP_mp_final}\]

P.4 Deterministic verification script (proton mass evaluation)

# verify_appendix_P.py
# Purpose: deterministic script that computes lambda_C, S_p, m_p using only LOCK values from registry.

import json
import hashlib
from math import pi

def sha256_file(path: str) -> str:
    h = hashlib.sha256()
    with open(path, "rb") as f:
        while True:
            b = f.read(1024 * 1024)
            if not b:
                break
            h.update(b)
    return h.hexdigest()

def read_json(path: str) -> dict:
    with open(path, "r", encoding="utf-8") as f:
        return json.load(f)

def main():
    canon = read_json("registry/canon_lock.json")
    realz = read_json("registry/realization_lock.json")
    prot  = read_json("registry/protocol_lock.json")
    snap  = read_json("registry/registry_snapshot.json")

    h_Js = canon["constants"].get("h_Js", 6.62607015e-34)
    c_ref = realz["inputs"]["c_ref_m_s"]
    a_m   = realz["inputs"]["a_m"]
    r_p   = canon["inputs"]["r_p_m"]
    GeV_to_J = prot["unit_conversions"]["GeV_to_J"]

    # U_lat
    U_lat_J = (h_Js * c_ref) / a_m
    U_lat_GeV = U_lat_J / GeV_to_J

    # lambda_C, S_p, m_p
    lambda_C = (pi / 2.0) * r_p
    S_p = lambda_C / a_m
    m_p_GeV = U_lat_GeV / S_p

    out = {
        "registry_snapshot_id": snap["registry_snapshot_id"],
        "locks": snap["locks"],
        "hashes": {
            "canon_lock_sha256": sha256_file("registry/canon_lock.json"),
            "realization_lock_sha256": sha256_file("registry/realization_lock.json"),
            "protocol_lock_sha256": sha256_file("registry/protocol_lock.json"),
            "registry_snapshot_sha256": sha256_file("registry/registry_snapshot.json"),
        },
        "derived": {
            "U_lat_GeV": U_lat_GeV,
            "lambda_C_m": lambda_C,
            "S_p_dimless": S_p,
            "m_p_GeV": m_p_GeV
        }
    }

    print(json.dumps(out, indent=2, ensure_ascii=False))

if __name__ == "__main__":
    main()

Appendix M. Mass grand unification: geometric differentiation of lattice energy

M.0 Locked definitions (single source of truth)

\[U_{\mathrm{lat}}:=\frac{h\,c_{\mathrm{ref}}}{a}. \label{eq:AppM_Ulat}\] Lock the mass\(=\)resistance axiom as \[\boxed{ m(X):=\frac{U_{\mathrm{lat}}}{\sigma_{\mathrm{eff}}(X)} } \label{eq:AppM_mass_axiom}\] as shown. Here, \(\sigma_{\mathrm{eff}}(X)\) is the dimensionless resistance (effective cross-section coefficient) of object \(X\).

M.1 Resistance coefficients of three objects (unique definitions)

20.9.5.40 (1) H-mode

If the canonical cell (cube) locks the independent channel count to \(\kappa_H=5\) by removing one global reference degree of freedom from the 6 faces, and if the canonical cross-section of each channel is locked to \(\pi\), then \[\boxed{ \sigma_{\mathrm{eff}}(H):=5\pi } \label{eq:AppM_sigmaH}\]

20.9.5.41 (2) Proton

Define the proton resistance integral as \[S_p:=\int_0^{\lambda_C}\frac{dR}{a}=\frac{\lambda_C}{a}\] as above, and \[\boxed{ \sigma_{\mathrm{eff}}(p):=S_p } \label{eq:AppM_sigmap}\]

20.9.5.42 (3) Electron

Define the electron resistance integral as \[S:=\int_0^{r_e}\frac{dR}{a}=\frac{r_e}{a}\] as above, and \[\boxed{ \sigma_{\mathrm{eff}}(e):=S } \label{eq:AppM_sigmae}\]

M.2 Mass grand-unification theorem (same format)

20.9.5.43 Theorem M.2.1

Under the same lock version, if [eq:AppM_Ulat]–[eq:AppM_sigmae] hold, then \[\boxed{ m_H=\frac{U_{\mathrm{lat}}}{5\pi},\quad m_p=\frac{U_{\mathrm{lat}}}{S_p},\quad m_e=\frac{U_{\mathrm{lat}}}{S} } \label{eq:AppM_masses}\]

M.3 Invariants and dev definition (for cross-validation)

Define the invariants as follows. \[\begin{aligned} I_H &:= \frac{m_H(5\pi)}{U_{\mathrm{lat}}}, \label{eq:AppM_IH}\\ I_p &:= \frac{m_p S_p}{U_{\mathrm{lat}}}, \label{eq:AppM_Ip}\\ I_e &:= \frac{m_e S}{U_{\mathrm{lat}}}. \label{eq:AppM_Ie}\end{aligned}\] By definition, ideally \(I_H=I_p=I_e=1\) should hold (a channel that numerically reconfirms identical definitions).

Define the deviation of each invariant as \[\mathrm{dev}(I):=|I-1| \label{eq:AppM_dev}\] as above, and define the maximum deviation as \[\mathrm{dev}_{\max}:=\max\{\mathrm{dev}(I_H),\mathrm{dev}(I_p),\mathrm{dev}(I_e)\} \label{eq:AppM_devmax}\] as above. The tolerance threshold \(\mathrm{dev}_{\mathrm{tol}}\) for \(\mathrm{dev}_{\max}\) is pre-locked in gate_lock.

M.4 Gate decision format

Lock the following decision formula as the standard form. \[\boxed{ \texttt{PASS} \Longleftrightarrow \mathrm{dev}_{\max}\le \mathrm{dev}_{\mathrm{tol}} \quad\text{AND}\quad \text{(sealing/lock\_id/schema match)} } \label{eq:AppM_gate}\]

M.5 Deterministic verification script (mass-unification invariants)

# verify_appendix_M.py
# Purpose: compute U_lat, m_H, m_p, m_e and invariants I_H,I_p,I_e, dev_max, and generate Gate inputs.

import json
import hashlib
from math import pi

def sha256_file(path: str) -> str:
    h = hashlib.sha256()
    with open(path, "rb") as f:
        while True:
            b = f.read(1024 * 1024)
            if not b:
                break
            h.update(b)
    return h.hexdigest()

def read_json(path: str) -> dict:
    with open(path, "r", encoding="utf-8") as f:
        return json.load(f)

def main():
    canon = read_json("registry/canon_lock.json")
    realz = read_json("registry/realization_lock.json")
    prot  = read_json("registry/protocol_lock.json")
    gate  = read_json("registry/gate_lock.json")
    snap  = read_json("registry/registry_snapshot.json")

    h_Js = canon["constants"].get("h_Js", 6.62607015e-34)
    c_ref = realz["inputs"]["c_ref_m_s"]
    a_m   = realz["inputs"]["a_m"]
    r_p   = canon["inputs"]["r_p_m"]
    D_anch= canon["inputs"]["D_anch_m"]

    delta_item = canon["constants"].get("delta_rect", "1/pi^2")
    delta = float(delta_item) if isinstance(delta_item, (int, float)) else 1.0/(pi**2)

    GeV_to_J = prot["unit_conversions"]["GeV_to_J"]

    # U_lat
    U_lat_J = (h_Js * c_ref) / a_m
    U_lat_GeV = U_lat_J / GeV_to_J

    # m_H
    m_H = U_lat_GeV / (5.0 * pi)

    # proton: lambda_C, S_p, m_p
    lambda_C = (pi/2.0) * r_p
    S_p = lambda_C / a_m
    m_p = U_lat_GeV / S_p

    # electron: r_e, S, m_e
    r_e = (D_anch/2.0) * delta
    S = r_e / a_m
    m_e = U_lat_GeV / S

    # invariants
    I_H = (m_H * (5.0*pi)) / U_lat_GeV
    I_p = (m_p * S_p) / U_lat_GeV
    I_e = (m_e * S) / U_lat_GeV

    dev_H = abs(I_H - 1.0)
    dev_p = abs(I_p - 1.0)
    dev_e = abs(I_e - 1.0)
    dev_max = max(dev_H, dev_p, dev_e)

    dev_tol = gate["tolerances"]["dev_tol_max"]

    status = "PASS" if dev_max <= dev_tol else "FAIL"

    out = {
        "registry_snapshot_id": snap["registry_snapshot_id"],
        "locks": snap["locks"],
        "hashes": {
            "canon_lock_sha256": sha256_file("registry/canon_lock.json"),
            "realization_lock_sha256": sha256_file("registry/realization_lock.json"),
            "protocol_lock_sha256": sha256_file("registry/protocol_lock.json"),
            "gate_lock_sha256": sha256_file("registry/gate_lock.json"),
            "registry_snapshot_sha256": sha256_file("registry/registry_snapshot.json"),
        },
        "derived": {
            "U_lat_GeV": U_lat_GeV,
            "m_H_GeV": m_H,
            "m_p_GeV": m_p,
            "m_e_GeV": m_e,
            "lambda_C_m": lambda_C,
            "S_p_dimless": S_p,
            "r_e_m": r_e,
            "S_dimless": S,
            "I_H": I_H,
            "I_p": I_p,
            "I_e": I_e,
            "dev_max": dev_max,
            "dev_tol_max": dev_tol
        },
        "gate_eval": {
            "gate_id": "G-RATIO-MASS-UNIFICATION",
            "status": status
        }
    }

    print(json.dumps(out, indent=2, ensure_ascii=False))

if __name__ == "__main__":
    main()

Appendix R. Lattice tension theory: deriving the absolute scale of electromagnetic force

R.0 Locked inputs and goals

This appendix assumes the following inputs are locked.

  1. \(h\), \(c_{\mathrm{ref}}\), \(a\), \(\pi\).

  2. \(U_{\mathrm{lat}}:=h\,c_{\mathrm{ref}}/a\).

  3. Proton radius \(r_p\) and \(\lambda_C=(\pi/2)r_p\).

  4. Proton resistance integral \(S_p=\lambda_C/a\).

The goal is to compute (i) the lattice tension \(F_{\mathrm{lat}}\), and (ii) as a product of geometric dilution/coupling/accumulation coefficients, the force at the reference distance \(r_p\), \(F_{\mathrm{VP}}(r_p)\). In the comparison (target-text) section, the standard Coulomb law is used only as a verification metric.

R.1 Definition of the lattice unit tension \(F_{\mathrm{lat}}\)

Define the lattice unit tension as follows. \[\boxed{ F_{\mathrm{lat}}:=\frac{U_{\mathrm{lat}}}{a} =\frac{h\,c_{\mathrm{ref}}}{a^2} } \label{eq:AppR_Flat}\]

R.2 Definition of geometric dilution/coupling/accumulation coefficients

Fix the following three coefficients.

20.9.5.44 (1) Mass dilution factor

Define the first-order dilution into proton structure as \[\boxed{ D_{\mathrm{mass}}:=\frac{1}{S_p} } \label{eq:AppR_Dmass}\] as defined.

20.9.5.45 (2) Spatial dilution factor (at reference distance \(r_p\))

Define the scale expansion from lattice length \(a\) to core radius \(r_p\) as \[\boxed{ D_{\mathrm{space}}:=\frac{a}{r_p} } \label{eq:AppR_Dspace}\] as defined.

20.9.5.46 (3) Geometric coupling factor (projection\(\times\)accumulation\(\times\)spherical correction)

Set the static projection coupling as \(\eta_{\mathrm{static}}:=\delta^2\), and in the universal regime where \(\delta=1/\pi^2\), \[\eta_{\mathrm{static}}=\frac{1}{\pi^4}.\] Define the dynamic accumulation factor as \[\boxed{ \gamma_{\mathrm{acc}}:=\sqrt{2} } \label{eq:AppR_gamma}\] as defined, and define the effective coupling as \[\eta_{\mathrm{eff}}:=\eta_{\mathrm{static}}\gamma_{\mathrm{acc}} =\frac{\sqrt{2}}{\pi^4}\] as defined. In addition, combining the spherical geometric correction factor \(1/4\), fix the final coupling factor as \[\boxed{ D_{\mathrm{cpl}}:=\frac{\eta_{\mathrm{eff}}}{4} =\frac{\sqrt{2}}{4\pi^4} } \label{eq:AppR_Dcpl}\] as a locked definition.

R.3 Deriving the absolute force at reference distance \(r_p\)

Combining [eq:AppR_Flat]–[eq:AppR_Dcpl] yields \[\boxed{ F_{\mathrm{VP}}(r_p) := F_{\mathrm{lat}}\cdot D_{\mathrm{mass}}\cdot D_{\mathrm{space}}\cdot D_{\mathrm{cpl}} } \label{eq:AppR_Fvp_def}\] Define it as above. Expanding, \[\begin{aligned} F_{\mathrm{VP}}(r_p) &= \left(\frac{h\,c_{\mathrm{ref}}}{a^2}\right) \left(\frac{1}{S_p}\right) \left(\frac{a}{r_p}\right) \left(\frac{\sqrt{2}}{4\pi^4}\right). \label{eq:AppR_Fvp_expand}\end{aligned}\] Here, since \(S_p=\lambda_C/a\), we have \(1/S_p=a/\lambda_C\), and therefore \[\begin{aligned} F_{\mathrm{VP}}(r_p) &= \left(\frac{h\,c_{\mathrm{ref}}}{a^2}\right) \left(\frac{a}{\lambda_C}\right) \left(\frac{a}{r_p}\right) \left(\frac{\sqrt{2}}{4\pi^4}\right) = \frac{h\,c_{\mathrm{ref}}}{\lambda_C r_p}\cdot \frac{\sqrt{2}}{4\pi^4}. \label{eq:AppR_Fvp_cancel}\end{aligned}\] Also, substituting \(\lambda_C=(\pi/2)r_p\) yields \[\begin{aligned} F_{\mathrm{VP}}(r_p) &= \frac{h\,c_{\mathrm{ref}}}{(\pi/2)r_p^2}\cdot\frac{\sqrt{2}}{4\pi^4} = \frac{h\,c_{\mathrm{ref}}}{r_p^2}\cdot\frac{\sqrt{2}}{2\pi^5}. \label{eq:AppR_Fvp_final}\end{aligned}\]

R.4 Deterministic verification script (geometric derivation + comparison output)

# verify_appendix_R.py
# Purpose:
#  (1) compute F_lat, S_p, F_VP(r_p) using only LOCK values
#  (2) use the standard Coulomb law (target text) only in the comparison section to print numerical comparisons

import json
from math import pi, sqrt

def read_json(path: str) -> dict:
    with open(path, "r", encoding="utf-8") as f:
        return json.load(f)

def main():
    canon = read_json("registry/canon_lock.json")
    realz = read_json("registry/realization_lock.json")

    # [LOCK] constants
    h_Js = canon["constants"].get("h_Js", 6.62607015e-34)
    c_ref = realz["inputs"]["c_ref_m_s"]
    a_m   = realz["inputs"]["a_m"]
    r_p   = canon["inputs"]["r_p_m"]

    # [DERIVED] lattice tension
    F_lat = (h_Js * c_ref) / (a_m**2)

    # [DERIVED] lambda_C, S_p
    lambda_C = (pi/2.0) * r_p
    S_p = lambda_C / a_m

    # [DERIVED] coupling
    eta_static = 1.0 / (pi**4)      # delta^2 with delta=1/pi^2
    gamma_acc = sqrt(2.0)           # accumulation
    eta_eff = eta_static * gamma_acc
    Dcpl = eta_eff / 4.0            # spherical correction

    # [DERIVED] absolute force at r_p
    F_vp = F_lat * (1.0/S_p) * (a_m/r_p) * Dcpl

    out = {
        "derived": {
            "F_lat_N": F_lat,
            "lambda_C_m": lambda_C,
            "S_p_dimless": S_p,
            "eta_static": eta_static,
            "gamma_acc": gamma_acc,
            "eta_eff": eta_eff,
            "D_cpl": Dcpl,
            "F_VP_rp_N": F_vp
        }
    }

    # [COMPARE] target-text computation (verification only)
    # Standard Coulomb force between +/-e at separation r_p:
    # F = (1/(4*pi*epsilon0)) * e^2 / r_p^2
    e_C = 1.602176634e-19
    eps0 = 8.8541878128e-12
    F_std = (1.0/(4.0*pi*eps0)) * (e_C*e_C) / (r_p*r_p)

    out["compare_target_text"] = {
        "F_std_Coulomb_N": F_std,
        "ratio_FVP_over_Fstd": (F_vp / F_std),
        "abs_rel_diff": abs(1.0 - (F_vp / F_std))
    }

    print(json.dumps(out, indent=2, ensure_ascii=False))

if __name__ == "__main__":
    main()

Appendix F. Discrete decomposition hypothesis of event rate: integer part (active sites) and residual (oscillation)

F.0 Purpose and scope (NON-LOCK)

Section 9.4 derived \(\nu_{p,\mathrm{can}}\approx 292.339978\,\mathrm{s^{-1}}\) from only LOCK inputs (\(D_{\mathrm{anch}}, r_p, \delta\)) via the closed form [eq:S09_04_closed_form]. This appendix decomposes this value into integer part + residual, and records a post-hoc hypothesis interpreting it from a discrete-structure (integer lattice/channel) viewpoint. Therefore, the contents of this appendix do not modify the main-text (LOCK/Gate) conclusions, and are not included in the PASS/FAIL decision set of the current Gate version.

F.1 Definition: integer activity and residual

Define the following decomposition. \[\nu_{p,\mathrm{can}} = N_{\mathrm{act}} + \varepsilon, \qquad N_{\mathrm{act}} := \lfloor \nu_{p,\mathrm{can}} \rfloor, \qquad \varepsilon := \nu_{p,\mathrm{can}} - N_{\mathrm{act}}. \label{eq:AppF_integer_decomp}\] Using the LOCK value of 9.4 ([eq:S09_04_nup_numeric]), \[N_{\mathrm{act}} = 292, \qquad \varepsilon = 0.3399781225\ldots \label{eq:AppF_nact_eps_numeric}\] we obtain.

20.9.5.47 Interpretation (post-hoc hypothesis).

\(N_{\mathrm{act}}\) can be interpreted as the discrete count of “effective events that maintain structure” inside the proton, i.e., the number of active event sites. In contrast, \(\varepsilon\) can be interpreted as a residual oscillation/fluctuation component remaining due to the limits of rectification (\(\delta\)) and integer lattice matching. This residual may be conceptually similar to certain terms in standard physics (e.g., Zitterbewegung), but this whitepaper does not claim an equivalence at the current stage.

F.2 Candidate discrete mapping: \(82\times 4 - 9\times 4 = 292\)

The following is a candidate interpretation in which the integer part \(N_{\mathrm{act}}=292\) can be reproduced by a simple discrete-structure calculation.

Then, if we set the total capacity points and static points as \[N_{\mathrm{cap}}:=4N_{\mathrm{core}}=328, \qquad N_{\mathrm{static}}:=4N_{\mathrm{static\_q}}=36 \label{eq:AppF_cap_static}\] as above, \[N_{\mathrm{cap}}-N_{\mathrm{static}}=292 \label{eq:AppF_328_36_292}\] we get [eq:AppF_nact_eps_numeric] as the same integer part.

20.9.5.48 Caution (definition locking required).

Equation [eq:AppF_328_36_292] is until the meaning of “point”, “static throat”, “4-point locking”, and “9-input” is fixed by LOCK, only a post-hoc interpretation. In the current version, [eq:AppF_328_36_292] is not incorporated into the (LOCK/Gate) conclusion, and it is recorded separately as NON-LOCK to avoid being misconstrued as a degree of freedom for numerical fitting.

F.3 Future Gate design (declaration)

To elevate this candidate interpretation to a main-text conclusion in the future, at minimum the following are required.

F.4 Deterministic computation snippet (verification)

The snippet below computes \(\nu_{p,\mathrm{can}}\) from the LOCK values of 9.4, and prints the integer part/residual together with the candidate calculation [eq:AppF_328_36_292].

# verify_appendix_F.py (demonstration; NON-LOCK interpretation)
from math import pi

D_anch = 4.854194962126561e-12   # m
r_p    = 0.8412e-15              # m
delta  = 1.0/(pi*pi)

s_p = D_anch/(2.0*r_p)
nu  = s_p*delta

N_act = int(nu)          # floor for positive nu
eps   = nu - N_act

N_core = 82
points_per = 4
N_static_q = 9
N_cap = N_core*points_per
N_static = N_static_q*points_per
N_active_points = N_cap - N_static

print("s_p=", s_p)
print("nu_p_can=", nu)
print("N_act=", N_act)
print("eps=", eps)
print("N_active_points=", N_active_points)

Appendix G. Geometric derivation of gravity: lattice yield limit and saturation theory

G.-1 Applicable regime (Gate): derivation only in the stiffness regime

All derivations in this appendix assume the stiffness regime defined in 3.2 (\(\chi_{\mathrm{ST}}=1\)). If, within the observation window, the fluidity index \(\phi(\mathcal{P};W)\) is meaningfully large (i.e., stiffness failure is frequent), then the step that treats the “lattice stiffness limit” as a fixed constant is not justified, so the conclusion for that time window/protocol is treated as INCONCLUSIVE. Therefore, all upper-bound/saturation conclusions in this appendix qualify as conclusions only in regimes with \(\phi\approx 0\).

G.0 Core claim: 9.8 is the yield strength of space (the lattice)

In this theory, gravity is not “a force by which mass attracts,” but is redefined as the restoring pressure of a jammed lattice attempting to fill the Void.

Because the lattice has finite material properties (rigidity / transmission limits), there exists an upper limit on the maximum acceleration that the lattice can transmit directly to matter under contact/rest conditions. Denote this by \(\,g_\star\,\) (lattice yield limit).

\[g_\star \;\equiv\; c^2\,\Psi_{\rm yield} \quad\text{(corresponding to the yield strength of space / a surface-tension analogue)}\]

Earth happens to provide an environment with \(\Psi_{\rm geom}\approx\Psi_{\rm yield}\), so the representative value observed at the surface \(g\approx 9.8\,{\rm m/s^2}\) is interpreted not as “a value that grows without bound as mass increases,” but as a value near the lattice yield limit.

G.1 Defense logic: gravity is observed as a \(2\)-channel quantity, not a single component

Reports of values that appear like \(g>9.8\) on Jupiter/the Sun (or values back-calculated from orbital data) are not an immediate contradiction. The reason is that the observation protocol (context) differs.

In this appendix, we decompose gravitational acceleration into two components (channels): \[g_{\rm (concept)} \;=\; g_{\rm geom} \;+\; g_{\rm restore}.\]

20.9.5.49 Note (operational definition): “sum” does not mean simple addition within the same protocol.

The above decomposition does not mean “they add simultaneously in the same experiment to produce a single scale reading.” In this theory, observables branch by protocol, and the actually comparable observables are defined as follows: \[g_{\rm obs} \;=\; \begin{cases} g_{\rm geom}\ (\approx g_{\rm pot}) & \text{Orbit / Free-fall mode (free-fall/orbit)}\\ g_{\rm restore}\ (\le g_\star) & \text{Contact / Surface mode (contact/static)} \end{cases}\] That is, the component read by orbital data (far-field motion) and the component read by a scale/normal force (contact rest) are in principle different channels.

20.9.5.50 (A) \(\;g_{\rm geom}\) : channel of curvature

It is the geometric curvature component created by mass deficit. It governs orbits (satellites/probes), tides, and far-field motion, and can continue to grow with mass/radius.

20.9.5.51 (B) \(\;g_{\rm restore}\) : channel of restoring pressure

It is the restoring acceleration, under contact/rest conditions, by which the lattice pushes in directly to block the Void. This component is limited by the lattice rigidity (\(c^2\)) and the yield curvature demand (\(\Psi_{\rm yield}\)), and only this component saturates. \[0 \le g_{\rm restore} \le g_\star.\]

G.2 Geometric part: identity between deficit–curvature demand–potential acceleration

G.2.1 Lattice rigidity

\[K \equiv c^2 \quad [{\rm m^2/s^2}]\]

G.2.2 Deficit radius (geometric equivalent quantity)

Mass \(M\) corresponds to a “deficit (Void)” in the lattice; define the equivalent deficit size as \[R_s \equiv \frac{2GM}{c^2}\] and denote it so (a geometric length scale).

G.2.3 Geometric curvature demand (unit \(m^{-1}\))

Define the curvature demand at distance \(R\) by \[\Psi_{\rm geom}(R) \;\equiv\; \frac{R_s}{2R^2} \;=\; \frac{GM}{c^2R^2} \quad [m^{-1}]\] and denote it so. Here, \(\Psi\) is not the dimensionless strain of standard elasticity, but a curvature demand (unit \(m^{-1}\)).

G.2.4 Potential curvature acceleration (Newton-equivalent)

If rigidity \(c^2\) converts curvature demand into acceleration, \[g_{\rm pot}(R)\;\equiv\; c^2\,\Psi_{\rm geom}(R) \;=\; \frac{GM}{R^2}.\] that is, \(\;g_{\rm pot}\) takes the same form as the conventional Newtonian curvature acceleration.

G.3 constitutive law

The key is the yield axiom: \(\Psi\) does not diverge without bound.

G.3.1 Yield curvature demand

\[\Psi_{\rm eff} \le \Psi_{\rm yield}.\] Therefore, the curvature demand that the lattice actually “accepts” is \[\Psi_{\rm eff} \;=\; \Psi_{\rm yield}\,\Phi\!\left(\frac{\Psi_{\rm geom}}{\Psi_{\rm yield}}\right)\] and can be written as. Here \(\Phi(x)\) is a saturation function with \(0\le\Phi\le 1\).

G.3.2 Hard yield (the simplest saturation)

\[\Phi(x)=\min(x,1) \quad\Rightarrow\quad \Psi_{\rm eff}=\min(\Psi_{\rm geom},\Psi_{\rm yield}).\] Then the lattice restoring acceleration is \[g_{\rm restore} \equiv c^2\Psi_{\rm eff} = c^2\min(\Psi_{\rm geom},\Psi_{\rm yield}) = \min(g_{\rm pot},g_\star).\]

G.3.3 Hourglass/Janssen-type “gentle saturation” (continuous-transition model)

In jamming/granular media (hourglass/silo), vertical stress saturates with depth; this is analogous to Janssen-type shielding. Accordingly, we allow a gentle saturation curve of the following kind.

Normalized variable: \[x \equiv \frac{g_{\rm pot}}{g_\star} = \frac{\Psi_{\rm geom}}{\Psi_{\rm yield}}.\]

Representative kernel: \[\Phi_{\rm J}(x)=1-e^{-x} \quad\Rightarrow\quad g_{\rm restore}=g_\star(1-e^{-x}).\]

Or (a “velocity-like” form inspired by the master-curve of hourglass discharge rate): \[\Phi_{\rm H}(x)=\sqrt{1-e^{-x}}.\] (However, this form tends to break the small-\(x\) linearity (\(\Phi\sim x\)); when applying directly to acceleration, a correction (e.g., linear-near matching) is required.)

G.4 Handling counterexamples (Jupiter/Sun/neutron star, etc.)

In this theory, \(g_\star\) does not mean “the gravitational field is fixed to 9.8 everywhere in the universe,” but the maximum acceleration under contact/rest conditions by which the lattice can directly support matter.

20.9.5.52 (i) Moon/Mars/Earth:

If a solid surface (contact) exists and \(g_{\rm pot}<g_\star\), \[g_{\rm restore}=g_{\rm pot},\] then surface mechanics appears identical to conventional predictions.

20.9.5.53 (ii) Jupiter/Sun:

There is no solid surface (continuous fluid/plasma), so even if \(\,g_{\rm pot}\gg g_\star\) is possible, a “solid-contact-based surface reference” with \(\,g_{\rm restore}\le g_\star\) does not form. Therefore, it is difficult to define “\(g\) measured by a scale at the surface” itself. Far-field motion/orbits are explained by \(g_{\rm geom}\approx g_{\rm pot}\).

20.9.5.54 (iii) Near neutron stars/black holes:

\(\,g_{\rm pot}\) becomes extremely large, but a “solid-contact-based static condition” typically collapses/transitions to fluidization/nonlinear states, and the lattice restoring-pressure component saturates at \(g_\star\). The excess is dispersed into other constraints (fluid pressure/electromagnetism/rotation/compaction stress, etc.).

G.5 Final summary (summary equations)

\[\boxed{ g_{\rm pot}(R)=\frac{GM}{R^2},\quad x=\frac{g_{\rm pot}}{g_\star},\quad g_{\rm restore}=g_\star\,\Phi(x),\quad 0\le g_{\rm restore}\le g_\star }\]

G.6 Deterministic verification script (reproducible output)

The script below prints, in one run, the core identities and saturation kernels (hard/gentle) of Appendix G.

import numpy as np

# =============================================================================
# [APPENDIX G] GRAVITY (CONSOLIDATED)
# =============================================================================

def f_hard_clip(x: float) -> float:
    """Hard yield clamp: f(x)=min(x,1)."""
    if x <= 0.0:
        return 0.0
    return x if x < 1.0 else 1.0

def f_soft_clip(x: float, n: float = 64.0) -> float:
    """Smooth approximation of min(x,1): f = x/(1+x^n)^(1/n)."""
    if x <= 0.0:
        return 0.0
    if n <= 0.0:
        raise ValueError("n must be > 0")
    u = -n * np.log(x)
    log_1pexp_u = np.logaddexp(0.0, u)
    log_f = -(1.0 / n) * log_1pexp_u
    return float(np.exp(log_f))

def f_tanh(x: float) -> float:
    """Generic smooth saturation: f=tanh(x)."""
    if x <= 0.0:
        return 0.0
    return float(np.tanh(x))

def f_janssen(x: float) -> float:
    """Janssen-like saturation curve: f=1-exp(-x)."""
    if x <= 0.0:
        return 0.0
    return float(1.0 - np.exp(-x))

def f_sqrt_janssen(x: float) -> float:
    """Unified discharge master-curve shape: f=sqrt(1-exp(-x))."""
    if x <= 0.0:
        return 0.0
    return float(np.sqrt(1.0 - np.exp(-x)))

def classify_regime(x: float) -> str:
    if x < 0.1:
        return "Fluid (linear)"
    if x < 1.0:
        return "Transition (0.1<=x<1)"
    if x < 3.0:
        return "Near-yield (1<=x<3)"
    return "Saturated (x>=3)"

def g_potential_newton(G: float, M: float, R: float) -> float:
    return G * M / (R ** 2)

def schwartz_radius(G: float, M: float, c2: float) -> float:
    return (2.0 * G * M) / c2

def psi_from_rs(Rs: float, R: float) -> float:
    return Rs / (2.0 * (R ** 2))

def fmt_auto(x: float, width: int = 12, prec: int = 6) -> str:
    ax = abs(x)
    if ax == 0.0:
        return f"{0:{width}.{prec}f}"
    if ax >= 1e6 or ax < 1e-3:
        return f"{x:{width}.{prec}e}"
    return f"{x:{width}.{prec}f}"

def print_header(title: str) -> None:
    print("=" * 88)
    print(title)
    print("=" * 88)

def print_section(title: str) -> None:
    print("\n" + "[" + title + "]")

def evaluate_models_for_gpot(g_pot: float, g_limit: float, models: dict[str, callable]) -> dict[str, float]:
    x = g_pot / g_limit if g_limit > 0 else np.inf
    out: dict[str, float] = {}
    for name, f in models.items():
        frac = float(f(x))
        g_restore = g_limit * frac
        if g_restore > g_limit:
            g_restore = g_limit
        out[name] = g_restore
    return out

def main() -> None:
    c = 299_792_458.0
    c2 = c ** 2
    G = 6.67430e-11

    print_header("[APP G] GRAVITY: LATTICE STIFFNESS LIMIT & SATURATION + HOURGLASS(JANSSEN)")

    M_earth = 5.9722e24
    R_earth = 6.3710e6

    g_pot_earth = g_potential_newton(G, M_earth, R_earth)
    Rs_earth = schwartz_radius(G, M_earth, c2)
    psi_earth_from_g = g_pot_earth / c2

    psi_yield_geo = psi_earth_from_g
    g_limit_geo = c2 * psi_yield_geo

    g0_standard = 9.80665
    psi_yield_std = g0_standard / c2

    YIELD_MODE = "earth"
    if YIELD_MODE.lower().startswith("earth"):
        g_limit = g_limit_geo
        psi_yield = psi_yield_geo
    else:
        g_limit = g0_standard
        psi_yield = psi_yield_std

    n_soft = 64.0
    models: dict[str, callable] = {
        "hard": lambda x: f_hard_clip(x),
        f"soft_n{int(n_soft)}": (lambda x, n=n_soft: f_soft_clip(x, n=n)),
        "tanh": lambda x: f_tanh(x),
        "janssen": lambda x: f_janssen(x),
        "sqrt_janssen": lambda x: f_sqrt_janssen(x),
    }
    kernel_names = list(models.keys())

    bodies_geom = {
        "Moon": (7.342e22, 1.7374e6),
        "Mars": (6.4171e23, 3.3895e6),
        "Earth": (M_earth, R_earth),
        "Jupiter": (1.898e27, 7.1492e7),
        "Sun": (1.989e30, 6.963e8),
        "NeutronStar": (1.4 * 1.989e30, 1.0e4),
    }

    print_section("TABLE: (M,R)->g_pot then g_restore by kernels")
    header = ["OBJECT", "g_pot", "x"] + [f"g_{k}" for k in kernel_names] + ["REGIME"]
    print(" | ".join([f"{h:<12}" for h in header]))
    print("-" * (15 * len(header)))

    for name, (M, R) in bodies_geom.items():
        Rs = schwartz_radius(G, M, c2)
        psi_geom = psi_from_rs(Rs, R)
        g_pot = c2 * psi_geom
        x = g_pot / g_limit
        g_restore = evaluate_models_for_gpot(g_pot, g_limit, models)
        regime = classify_regime(x)

        row = [name, fmt_auto(g_pot), fmt_auto(x)]
        for k in kernel_names:
            row.append(fmt_auto(g_restore[k]))
        row.append(regime)
        print(" | ".join([f"{c:<12}" for c in row]))

    print("\nEND APP G")

if __name__ == "__main__":
    main()

21 Appendix H: Theory of Geometric Rigidity (v2.1)

This appendix records a compact, self-contained derivation that links an effective coordination number \(N\) (jamming state) to a rigidity measure \(\Psi\). It is intended as an extension note: the algebra is locked and reproduced by a script in the DOI bundle, while the physical interpretation can be debated and improved.

21.1 H.1 Geometric gap and barrier

Define the geometric gap \[\delta \equiv 4 - N.\] Assume a minimal inverse-gap barrier for rearrangement: \[U_{\mathrm{barrier}} = \frac{C_{\mathrm{geo}}}{4-N}.\]

21.2 H.2 Statistical stability and the rigidity form

With a Boltzmann-type escape probability \(P_{\mathrm{escape}} \propto \exp(-U_{\mathrm{barrier}}/E_{\mathrm{th}})\), define rigidity as inverse failure: \(\Psi \propto 1/P_{\mathrm{escape}}\). This yields an exponential rigidity law, normalized at \(N=3\): \[\Psi(N) = \Psi_{\mathrm{base}} \exp\Big[k \Big(\frac{1}{4-N} - 1\Big)\Big],\] where \(k \equiv C_{\mathrm{geo}}/E_{\mathrm{th}}\) is a dimensionless geometric coupling constant.

21.3 H.3 Diagnostic inversion: effective coordination from modulus ratios

If a material-specific mapping allows \(\Psi\) to be compared via a measurable rigidity proxy (e.g., a bulk-modulus ratio \(R \approx K_s/K_l\) across a transition), then the equation inverts to \[N_{\mathrm{eff}} = 4 - \frac{1}{1 + (1/k)\ln R}.\]

21.4 H.4 DOI reproducibility artifacts (LOCK)

The following files fully reproduce the numerical table used in this appendix:

22 Appendix I: DOI-anchored Citation Registry (No Missing DOI)

This appendix resolves the internal citation tokens used in the white paper, written in the form [cite: XX]. Each cite-ID maps to a DOI and a concrete bundle path.

ID What it anchors Gate DOI / bundle pointer
ID What it anchors Gate DOI / bundle pointer
20 Fe-Mo catalyst: asymmetric compression-field (design note) UNLOGGED 10.5281/zenodo.17932567;
21 N2 bond-break critical amplitude ( \(\pm 290\,\mathrm{fm}\) ) PASS 10.5281/zenodo.17932567;
26 N2 case internal simulation/report pointer (scalar traceability) PASS 10.5281/zenodo.17932567;
30 Asymmetric strike / alignment-interference protocol (design note) UNLOGGED 10.5281/zenodo.17932567;
47 N2 case secondary note pointer PASS 10.5281/zenodo.17932567;
61 CO2 case: high-temperature decomposition ( \(\pm 300\,\mathrm{fm}\) ) PASS 10.5281/zenodo.17932567;
65 CO2 case: break threshold + recombination/quenching note PASS* 10.5281/zenodo.17932567;
68 CO2 case: quenching + alignment\(<70\%\) recombination suppression UNLOGGED 10.5281/zenodo.17932567;
77 Seawater process: H2O/ion amplitude ranges PASS 10.5281/zenodo.17932567;
83 Seawater process: water operational-range record PASS 10.5281/zenodo.17932567;
85 Ion expulsion efficiency 97.3% (pending) UNLOGGED 10.5281/zenodo.17932567;
102 Water stiffness cross-check via \(P_{\mathrm{idx}}\) table PASS 10.5281/zenodo.17932567;

Note: PASS* indicates that the scalar threshold is reproduced in-bundle (PASS), while any additional mechanistic claim attached to the same sentence is treated as pending unless a primary run log is bundled.

23 Appendix J: Black-Coated Copper ESS — impedance-transform collection, granular-lattice storage, and black-copper transduction (engineering appendix; NON-LOCK)

23.1 J.0 Scope and status (NON-LOCK)

This appendix records an engineering blueprint motivated by VP axioms. It is not used as an input to any main-text LOCK chain. Therefore, any numerical value in this appendix is NON-LOCK unless it is explicitly tied to (i) a pre-registered measurement protocol and (ii) sealed artifacts in the Zenodo bundle (10.5281/zenodo.17932567).

23.1.0.1 Why a dedicated appendix?

The proposed “Black-Copper ESS” is treated here as a single, closed logic chain: collect (solar) \(\rightarrow\) inject (into Cu lattice) \(\rightarrow\) store (as granular-lattice vibration) \(\rightarrow\) extract (as electricity) \(\rightarrow\) optional feedback (electricity back into the cavity as heat). This is an engineering application layer; it must not retroactively tune VP constants.

23.2 J.1 Engineering definitions of lattices and interfaces

We define the module as four coupled subsystems.

23.2.1 J.1.1 Copper lattice \(\Lambda_{\mathrm{Cu}}\) (low-impedance lattice amplifier)

Within the VP interpretation, copper is treated as a low-impedance lattice that supports large flow-type amplitudes (current-like / velocity-like variables) for a given power throughput. For engineering calculations, we may reference typical material properties (not LOCKed here): mass density \(\rho\), elastic modulus \(\kappa\), and a small dissipation parameter \(\gamma\).

23.2.2 J.1.2 Black coating interface \(\Gamma_{\mathrm{BC}}\) (broadband absorber / impedance matcher)

“Black Copper” means copper with a nano-structured black coating used to (i) suppress reflection over the target spectrum and (ii) convert incident photon flux into lattice injection (phonon/electron excitation) as a boundary process. In practice, coating performance must be characterized by a measured reflectivity spectrum \(R(\omega)\) and a durability envelope (temperature/oxidation), both sealed as DOI artifacts.

23.2.3 J.1.3 Granular storage lattice \(\Lambda_{\mathrm{Sand}}\) (non-continuum phonon reservoir)

The storage medium is modeled as a disordered granular network (e.g., silica/alumina), treated as a discrete oscillator ensemble. At the module level, the leading-order capacity is still well-approximated by the bulk heat capacity model.

23.2.4 J.1.4 Glass cavity boundary \(\Gamma_{\mathrm{Glass}}\) (radiative trap + insulation interface)

The cavity boundary is engineered to (i) suppress conductive loss (outer insulation) and (ii) trap internal infrared exchange (inner liner), approaching an adiabatic boundary condition.

23.3 J.2 Impedance mismatch and amplitude transformation (no energy gain)

To avoid ambiguity, we use the standard effort/flow representation for waves. Let \(E\) denote an effort-type amplitude (voltage/force) and \(J\) a flow-type amplitude (current/velocity). Define the (effective) impedance by \[E = Z\,J, \qquad P = \frac{1}{2}\,EJ = \frac{1}{2}ZJ^2 = \frac{1}{2}\frac{E^2}{Z}. \label{eq:AppJ_power_effort_flow}\]

Assume the black-coating interface is engineered so that reflection is small in the operating band and the absorbed power is approximately transmitted into the copper lattice: \[P_{\mathrm{in}}\;\approx\;P_{\mathrm{Cu}}. \label{eq:AppJ_power_continuity}\]

Then the flow-type amplitude scales as \[\boxed{ \frac{J_{\mathrm{Cu}}}{J_{\mathrm{in}}} = \sqrt{\frac{Z_{\mathrm{in}}}{Z_{\mathrm{Cu,eff}}}} } \label{eq:AppJ_flow_gain}\] while the effort-type amplitude scales oppositely, \[\boxed{ \frac{E_{\mathrm{Cu}}}{E_{\mathrm{in}}} = \sqrt{\frac{Z_{\mathrm{Cu,eff}}}{Z_{\mathrm{in}}}} } \label{eq:AppJ_effort_gain}\] and the power remains the same (no energy creation).

23.3.0.1 Numerical illustration (NOT a universal claim).

If we take \(Z_{\mathrm{in}}\) as the free-space impedance \(Z_0\approx 376.73\,\Omega\) and (for illustration only) adopt an effective copper-lattice impedance \(Z_{\mathrm{Cu,eff}}\approx 0.01\,\Omega\), then \[G_J := \frac{J_{\mathrm{Cu}}}{J_{\mathrm{in}}} = \sqrt{\frac{376.73}{0.01}}\approx 194. \label{eq:AppJ_gain_example}\] In the LOCK\(\rightarrow\)Gate framework, \(Z_{\mathrm{Cu,eff}}\) must be treated as a measured-and-locked interface parameter (protocol + uncertainty), otherwise this gain is only a qualitative scaling statement.

23.3.1 J.2.1 VP interpretation: “a new electricity form” as a flow-amplitude variable

In conventional circuit language one typically emphasizes the voltage-like (effort) amplitude \(E\). In the present appendix we intentionally emphasize the flow-type amplitude \(J\) because a low-impedance lattice naturally supports large current/velocity response for a given absorbed power. In VP wording, this can be described as an “amplitude-current” state: not a new energy source, but a different representation of the same conserved power budget (high-\(J\)/low-\(E\) instead of low-\(J\)/high-\(E\)).

23.4 J.3 Geometry: “morning-glory” trumpet as an impedance transformer

The outer collector is specified to be a trumpet / horn / morning-glory geometry. This geometry is not an aesthetic choice: its purpose is to implement a distributed impedance transformation and suppress reflection by spreading the mismatch over a finite length.

23.4.1 J.3.1 Geometric concentration gain (area ratio)

Let the mouth (aperture) area be \(A_{\mathrm{m}}\) and the throat (coupling) area be \(A_{\mathrm{t}}\). For a given incident intensity \(I_{\odot}\) (W/m\(^2\)), the absorbed power is \[P_{\mathrm{in}}\approx \eta_{\mathrm{abs}}\, I_{\odot}\,A_{\mathrm{m}}, \label{eq:AppJ_Pin}\] where \(\eta_{\mathrm{abs}}\in[0,1]\) is the band-averaged absorption efficiency of \(\Gamma_{\mathrm{BC}}\) (must be measured). If the optical/thermal coupling transports that power into the throat area without major loss, the local intensity scale near the throat grows as \[I_{\mathrm{t}} \sim \frac{P_{\mathrm{in}}}{A_{\mathrm{t}}} \approx \eta_{\mathrm{abs}}\,I_{\odot}\,\frac{A_{\mathrm{m}}}{A_{\mathrm{t}}}. \label{eq:AppJ_Intensity_gain}\] Therefore, any field-like amplitude whose energy density scales as amplitude\(^2\) acquires a geometric gain \[\boxed{ G_{\mathrm{geo}} := \sqrt{\frac{A_{\mathrm{m}}}{A_{\mathrm{t}}}}. } \label{eq:AppJ_Ggeo}\] This part is purely geometry (no material constants).

23.4.2 J.3.2 Smooth-taper condition (reflection suppression)

Let \(z\in[0,L]\) be the horn axis and \(A(z)\) the cross-sectional area. In a slowly varying (adiabatic) transformer, the leading reflection scales with the gradient of the effective impedance. A standard engineering criterion is \[\bigl|\partial_z\ln Z_{\mathrm{eff}}(z)\bigr|\ll k \qquad (k:=2\pi/\lambda), \label{eq:AppJ_adiabatic}\] which says: make the taper long and smooth compared to the operating wavelength. The “morning-glory” flare is a geometric way to realize this condition without introducing discontinuities.

23.4.3 J.3.3 A simple closed profile (exponential horn)

As a concrete blueprint, an exponential flare can be used: \[r(z)=r_{\mathrm{t}}\,\exp\!\left(\frac{z}{L}\ln\frac{r_{\mathrm{m}}}{r_{\mathrm{t}}}\right), \qquad A(z)=\pi r(z)^2, \label{eq:AppJ_exponential_horn}\] with throat radius \(r_{\mathrm{t}}\) and mouth radius \(r_{\mathrm{m}}\). This profile has a constant relative expansion rate and is analytically simple to fabricate/parameterize.

23.5 J.4 Granular-lattice storage (sand battery) capacity

At leading order, the thermal storage capacity is \[E_{\mathrm{store}} \approx M\,c_p\,\Delta T. \label{eq:AppJ_Estore}\] For a \(1\,\mathrm{m}^3\) packed module with bulk density \(\rho_{\mathrm{bulk}}\approx 1600\,\mathrm{kg/m^3}\) and silica heat capacity \(c_p\approx 830\,\mathrm{J/(kg\cdot K)}\), and an operating swing \(\Delta T\approx 500\,\mathrm{K}\), one obtains \[\begin{aligned} M &\approx 1600\,\mathrm{kg},\\ E_{\mathrm{store}} &\approx 1600\times 830\times 500\,\mathrm{J}\approx 6.64\times 10^{8}\,\mathrm{J}\approx 184\,\mathrm{kWh}. \label{eq:AppJ_Estore_num}\end{aligned}\] This provides the correct order-of-magnitude for module sizing. Loss/retention must be evaluated from the actual boundary design (insulation, radiative trapping, leakage paths) and treated as a Gate-controlled performance claim.

23.6 J.5 Black-copper spring inside the cavity (dual use: heater + transducer)

The inner copper element is specified as a spring / helix geometry. At the blueprint level it has two roles.

23.6.1 J.5.1 Charge mode (electricity \(\rightarrow\) heat): injection into the cavity

When surplus electricity exists, it can be routed into the helix as a resistive/inductive drive. The black coating increases radiative coupling so that electrical energy is efficiently converted into cavity heat (lattice vibration), i.e., a controllable “power-to-heat” path.

23.6.2 J.5.2 Discharge mode (heat \(\rightarrow\) electricity): coupling to a generator stage

Extracting electricity from stored heat requires a heat-engine mechanism (temperature difference, non-equilibrium). In this blueprint the black-coated copper helix is treated as the thermal interface + electrical winding of a generator stage. The concrete conversion method (thermoelectric, thermomagnetic, Stirling/Brayton with generator, TPV, etc.) must be specified and validated separately; VP theory by itself does not remove the need for a temperature gradient and a closed energy budget.

23.6.3 J.5.3 Feedback loop (allowed interpretation)

The phrase “feed remaining electricity back into the ESS” is interpreted here as a bidirectional storage mode: electricity can be converted back into heat (charge mode) to avoid waste and maintain a target cavity state. This does not violate energy conservation; it is a control/storage feature.

23.6.4 J.5.4 Geometry lock: helix (spring) parameterization

Let the helix centerline be parameterized by radius \(R\), pitch \(p\), and turn count \(N_{\mathrm{turn}}\): \[\mathbf{x}(t)=\bigl(R\cos t,\;R\sin t,\; \tfrac{p}{2\pi}\,t\bigr), \qquad t\in[0,2\pi N_{\mathrm{turn}}]. \label{eq:AppJ_helix_param}\] This geometry is convenient because (i) it is fully determined by a small set of geometric DOFs \((R,p,N_{\mathrm{turn}})\), (ii) it provides a large coated surface area per unit volume for radiative/thermal coupling, and (iii) it maps cleanly to standard electrical parameters (inductance, resistance) once the wire diameter and material state are fixed.

To keep this appendix consistent with the main document style, any performance claim should be tied to the following minimum artifacts.

24 Appendix K: Methodological Validation via Classical Gas Acoustics (Mean-Free-Path Reconstruction)

24.1 K.0 Scope and intent (calibration module; NON-LOCK)

This appendix adds an external calibration for the VP “discrete-medium reconstruction” logic. It is not an analogy (“space is like air”) and it is not used to tune any VP LOCK constants. Instead, it is a black-box inversion test:

Given: macroscopic wave data and macroscopic transport data of a known discrete medium.
Recover: an accepted microscopic length scale of that medium.
Interpretation: if the same inversion structure works on a classical system with known micro-scale, then applying the same structure to the VP lattice problem is methodologically less arbitrary.

24.2 K.1 Why acoustics in air is a suitable calibration target

In a dilute gas, the speed of sound \(v_s\) is determined by equilibrium thermodynamics (mainly \(T\) and the adiabatic index \(\gamma\)), while the continuum validity of wave mechanics (and the frequency-dependent attenuation) is governed by collisional microphysics. The controlling micro-length is the mean free path \(\lambda_{\mathrm{mfp}}\).

A key point is that \(\lambda_{\mathrm{mfp}}\) cannot be inferred from \(v_s\) alone. One needs at least one macroscopic transport coefficient (e.g., viscosity) or attenuation data. This is exactly the same type of requirement we impose in VP theory: a wave speed plus a dissipation/transport gate is needed to infer a discrete scale.

24.3 K.2 Reconstruction of \(\lambda_{\mathrm{mfp}}\) from macroscopic data (one worked example)

We consider dry air near room temperature and \(1\,\mathrm{atm}\) as a calibration case. We use representative macroscopic inputs \[v_s \approx 343\,\mathrm{m/s},\quad \eta \approx 1.81\times 10^{-5}\,\mathrm{Pa\cdot s},\quad p \approx 1.01325\times 10^{5}\,\mathrm{Pa},\] together with \(\gamma\approx 1.4\) and molar mass \(M\approx 28.97\,\mathrm{g/mol}\).

24.3.1 K.2.1 Step 1: Temperature from sound speed

For an ideal gas, \[v_s=\sqrt{\gamma \frac{R T}{M}} \quad\Longrightarrow\quad T=\frac{v_s^2\,M}{\gamma R}.\] With the above inputs, \(T\approx 2.93\times 10^2\,\mathrm{K}\).

24.3.2 K.2.2 Step 2: Effective collision diameter from viscosity (hard-sphere scaling)

A standard hard-sphere kinetic-theory scaling (Chapman–Enskog form) for dynamic viscosity is \[\eta \approx \frac{5}{16\,d^2}\sqrt{\frac{m\,k_B\,T}{\pi}},\] where \(d\) is an effective collision diameter and \(m=M/N_A\) is the molecular mass. Solving for \(d\) gives \[d \approx \sqrt{\frac{5}{16\,\eta}\sqrt{\frac{m\,k_B\,T}{\pi}}}.\] Numerically, this yields \(d\approx 3.7\times 10^{-10}\,\mathrm{m}\) (sub-nanometer scale, as expected).

24.3.3 K.2.3 Step 3: Mean free path

For a hard-sphere gas, \[\lambda_{\mathrm{mfp}}=\frac{k_B T}{\sqrt{2}\,\pi\,d^2\,p}.\] Using the reconstructed \(d\) and the macroscopic inputs above, we obtain \[\lambda_{\mathrm{mfp}}\approx 6.6\times 10^{-8}\,\mathrm{m}\;\approx\;66\,\mathrm{nm}.\] This is consistent with the commonly quoted order-of-magnitude for the mean free path of air near \(1\,\mathrm{atm}\) and room temperature (\(\sim 10^{2}\,\mathrm{nm}\)).

24.4 K.3 Regime gate: continuum acoustics vs. discrete-collision scale

A simple regime indicator is the Knudsen number for the wave, \[\mathrm{Kn}_\lambda := \frac{\lambda_{\mathrm{mfp}}}{\lambda_{\mathrm{wave}}} =\frac{\lambda_{\mathrm{mfp}}\,\nu}{v_s},\] where \(\lambda_{\mathrm{wave}}=v_s/\nu\) and \(\nu\) is the acoustic frequency. The continuum acoustic model is reliable for \(\mathrm{Kn}_\lambda\ll 1\), and breaks down (strong kinetic attenuation / non-continuum behavior) as \(\mathrm{Kn}_\lambda\to O(1)\). Thus the reconstruction is not only a number-fit; it also predicts the expected frequency gate where macroscopic wave theory ceases to be valid.

24.5 K.4 Isomorphic comparison table (acoustics vs. VP lattice reconstruction)

Isomorphic validation logic. The goal is not to equate “air” and “space”, but to show that inferring a discrete micro-scale from macroscopic wave data is a well-posed operation once an appropriate transport/damping gate is included.
Item Classical gas acoustics (air) VP lattice reconstruction (space)
Wave type Longitudinal pressure wave EM / lattice wave in VP medium
Macroscopic speed \(v_s\) (measured) \(c\) (defined/measured)
Micro length to infer \(\lambda_{\mathrm{mfp}}\) (collision scale) \(\ell_{\mathrm{VP}}\) (lattice scale)
Auxiliary macro input \(\eta\) (transport / damping gate) VP transport/damping gate (protocol-defined)
Regime gate \(\mathrm{Kn}_\lambda\ll 1\) (continuum valid) VP Gate stack (Jam/Unjam, protocol window)
Outcome \(\lambda_{\mathrm{mfp}}\sim 10^2\,\mathrm{nm}\) (known) \(\ell_{\mathrm{VP}}\) (predicted/anchored)

24.6 K.5 Claim level and limitation

This appendix supports a methodological claim: the inversion logic that maps (wave speed + transport gate) \(\rightarrow\) (micro-scale) is not unique to VP theory and succeeds on a classical benchmark. It does not by itself prove any specific VP lattice value; it only reduces arbitrariness by demonstrating an isomorphic reconstruction on a known system.

25 Appendix L: A 4-3-1 State Dictionary for Jamming-Regime Unification (Interpretive Module; NON-LOCK)

25.1 L.0 Scope (what this appendix is, and is not)

This appendix provides an interpretive dictionary that links familiar macroscopic phase language (solid–liquid–gas) to the discrete jamming language already defined in the main text (jammed lattice, failure-rate fluidity \(\phi\), and unjamming events). It is intended to improve conceptual closure and reader navigation; it does not introduce new axioms, and it is not used as an input to any locked numerical derivation.

25.2 L.1 The 4-3-1 dictionary (connectivity as a regime label)

To avoid symbol collision with electrical impedance \(Z\) used elsewhere (e.g., Appendix J), we denote the effective number of load-bearing bridges (an effective coordination count) by \(N_{\mathrm{bond}}\).

The 4-3-1 state dictionary. The state code is a compact label for a connectivity regime. It is a reader-facing dictionary that maps macroscopic intuition onto the already-defined VP jamming vocabulary.
State code \(\mathbf{N_{\mathrm{bond}}}\) Macroscopic analogy VP regime interpretation (this whitepaper)
4 \(\gtrsim 4\) Solid (e.g., ice): shape preserved, high stiffness Jammed lattice / space-like regime. \(\chi_{\mathrm{ST}}=1\), \(\phi \approx 0\); supports coherent wave propagation (“vacuum as a stiff lattice”).
3 \(\approx 3\) Liquid: persistent rearrangement / flow Near-jammed / flowing regime. \(\chi_{\mathrm{ST}}\) intermittently fails; \(\phi\) is nonzero due to repeated local unjamming events (structural failure-rate).
1 \(\lesssim 1\) Gas / chaotic dispersion Unjammed / released-carrier regime. Connectivity is insufficient to maintain a load-bearing enclosure; degrees of freedom behave as free carriers or high-entropy excitations.
2 \(=2\) (line-like) Geometrically unstable in 3D enclosure. A two-bridge configuration cannot enclose volume; it collapses toward State 1 or re-jams toward State 3/4 (see §L.3).

25.3 L.2 Energetics as “cost of unjamming” (a minimal staircase model)

Let \(\epsilon_{\mathrm{bond}}>0\) denote an effective binding (jamming) energy per load-bearing bridge. Then a minimal coarse-grained energy bookkeeping can be written as \[E_{\mathrm{tot}} \;\equiv\; E_{\mathrm{kin}} \;-\; N_{\mathrm{bond}}\,\epsilon_{\mathrm{bond}}, \label{eq:AppL_energy_staircase}\] where the potential well is represented by \(-N_{\mathrm{bond}}\epsilon_{\mathrm{bond}}\). In this language, “energy” acts as a key that pays the unjamming cost: \[\Delta E_{4\rightarrow 1} \;\approx\; \bigl(N_{\mathrm{bond}}^{(4)} - N_{\mathrm{bond}}^{(1)}\bigr)\epsilon_{\mathrm{bond}} \;+\; \Delta E_{\mathrm{kin}}. \label{eq:AppL_unjam_cost}\] Conversely, when the system re-jams (cooling / relaxation), the same amount of energy must be released as an effective latent heat of re-jamming (an energy budget associated with restoring connectivity).

25.4 L.3 Why “State 2” is forbidden (geometric enclosure failure)

A key reason the 4-3-1 labeling is useful is that it highlights a geometric constraint: a two-bridge configuration is line-like and cannot provide a closed enclosure in 3D. In the 2D cross-section language used in the main text, enclosing a core point requires at least three vectors (a minimum three-sector closure). Two vectors remain collinear and therefore cannot trap a region; four vectors are redundant. Hence \(N_{\mathrm{bond}}=2\) is not a stable regime, but a transient.

25.5 L.4 Relation to fluidity \(\phi\) (failure-rate view)

In §3.2.1, fluidity was defined as a failure rate (fraction of unjamming events in a window). The 4-3-1 dictionary can be read as a coarse mapping between connectivity and failure-rate: \[\text{State 4: }\phi\!\approx\!0,\quad \text{State 3: }0\!<\!\phi\!<\!1,\quad \text{State 1: }\phi\!\rightarrow\!1,\] with the understanding that \(\phi\) is protocol- and window-dependent, and must always be reported as \(\phi(\mathcal{P};W;\chi_{\mathrm{ST}})\) (see the main definition).

25.6 L.5 “Light is gas-like” without contradiction (medium vs excitation)

The statement “light is gas-like” can be made consistent if we distinguish the background regime from the excitation statistics. In VP language, the background vacuum is State 4 (a jammed stiff lattice) that supports coherent propagation. On top of that background, radiation can still be treated as a high-entropy excitation ensemble (photon-gas-like statistics), which is exactly what appears in the Planck form derived from rigidity-shell filtering (see the blackbody derivation).

Appendix K provides a classical calibration: the same “continuous wave \(\rightarrow\) discrete scale” logic applied to air acoustics reproduces a realistic mean-free-path scale. The purpose is methodological: it demonstrates that the reconstruction pipeline is not unique to VP, before applying it to the vacuum lattice scale.

25.8 L.7 Energy–volume exchange (cosmic “metabolism” as a gate-defined hypothesis)

If one models the release of a discrete volume quantum \(V_{\mathrm{VP}}\) as an unjamming work cost, the dimensionally consistent form is an effective pressure (energy density) scale: \[E_{\mathrm{unjam}} \;\equiv\; P_{\mathrm{jam}}\,V_{\mathrm{VP}}, \label{eq:AppL_energy_volume_pressure}\] where \(P_{\mathrm{jam}}\) is a jamming pressure scale. One may then express a gate-defined hypothesis: when an input energy packet exceeds a threshold (set by \(P_{\mathrm{jam}}V_{\mathrm{VP}}\) under the declared protocol), a local unjamming event occurs and an unjammed VP quantum is released. This statement is intentionally placed in a NON-LOCK appendix: it is interpretive until an explicit measurement or simulation gate is satisfied.

  1. S. K. Lamoreaux, “Demonstration of the Casimir Force in the 0.6 to 6 \(\mu\)m Range,” Phys. Rev. Lett. 78, 5–8 (1997). DOI: 10.1103/PhysRevLett.78.5.↩︎

  2. U. Mohideen and A. Roy, “Precision Measurement of the Casimir Force from 0.1 to 0.9 \(\mu\)m,” Phys. Rev. Lett. 81, 4549–4552 (1998). DOI: 10.1103/PhysRevLett.81.4549.↩︎

  3. Reference value (e.g., NIST/CODATA 2022): inverse fine-structure constant \(\alpha_{em}^{-1}=137.035\,999\,177(21)\).NIST database: physics.nist.gov (accessed 2025).The CODATA 2022 consolidated review was published as Rev. Mod. Phys. 97, 025002 (2025) with DOI: 10.1103/RevModPhys.97.025002.↩︎

  4. Planck’s original paper: M. Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Annalen der Physik 309(3), 553–563 (1901). DOI: 10.1002/andp.19013090310.↩︎

  5. For example, the Tolman surface-brightness test and tired-light constraints are discussed in L. M. Lubin & A. Sandage (2001), AJ 122, 1084–1103, DOI: 10.1086/322134; arXiv: astro-ph/0106566. Time-dilation discussions include G. Goldhaber et al. (2001), ApJ 558, 359–368, DOI: 10.1086/322460; arXiv: astro-ph/0104382.↩︎

  6. In this DOI bundle, \(\beta_{vac}=1.088\) and \(\beta_{bond}=1.000\) are fixed in data/chem/traceability_spec_v3.txt.↩︎

  7. The complete form including \(P_{avg}\) (\(P_{idx}=P_{avg}\Phi_{geo}(1-\Delta_{ov})\)) is summarized in docs/chem/DEFINITIVE_V3_CONSISTENCY.md within the same DOI bundle.↩︎