Discrete Proton Structure (82+7)

Discrete Proton Structure (82+7): A continuous circle cannot be 3-divided cleanly — the leftover picks the single nozzle. Exponent = gauge lemma, prefactor = sector count; the +1 in 82 is the nozzle. This section explains why the canonical shell structure has exactly one inlet rather than several. Grade [F] forced.

A continuous circle cannot be 3-divided cleanly — the leftover picks the single nozzle. Exponent = gauge lemma, prefactor = sector count; the +1 in 82 is the nozzle.

8.0 Inflow geometry: single inlet from C₃ symmetry breaking

This section explains why the canonical shell structure has exactly one inlet rather than several. The "7-out-of-8 closed, 1 open" pattern is a derived consequence of C₃ symmetry breaking, not an assertion; the derivation follows.

8.0.1 The continuous-circle vs lattice-3-fold mismatch

Rotational quanta naturally live on the continuous rotation group SO(2) (one full circle). The cubic lattice's discrete rotational symmetry around any body diagonal is only C₃ (3-fold). A continuous group cannot be cleanly 3-divided into a discrete one. The residual orientational mismatch concentrates onto a single distinguished axis: the body diagonal that aligns with the rotational quantum's net orientation. This is the geometric origin of the single inlet.

8.0.2 C₃ decomposition of the 8 body diagonals

The eight body diagonals of the boundary cube R²≤6, that is the points (±1,±1,±1), split under the C₃ rotation cycling x→y→z into two classes:

Class	Points	Behavior under C₃
Unique-axis pair (2 points)	(1,1,1) and (-1,-1,-1)	Fixed (the axis itself)
Cyclic triple (6 points = 3 antipodal pairs)	(1,1,-1), (1,-1,1), (-1,1,1) and their negatives	Permuted cyclically by C₃

Thus 8 = 2 + 6: 2 axis-pair points (one body diagonal) plus 6 points forming 3 cyclic antipodal pairs (the 3-sector ring).

8.0.3 The 3-sector vector sum vanishes

For each of the three cyclic pairs, the two endpoints are antipodal (e.g., (1,1,-1) and (-1,-1,1)), so each pair sums to (0,0,0). Summing all three pairs:

\sum_{\text{3-sector 6 points}} \mathbf{r} = (0,0,0).

The three sectors form a balanced ring; they cannot produce a net inflow or outflow direction. Whatever inflow exists must be carried by the remaining 2 points (the fixed axis pair).

8.0.4 The single nozzle is forced

Because the 3-sector contribution to net inflow is identically zero (§8.0.3), any net direction along which the core exchanges material with its surroundings must lie along the unique fixed axis (§8.0.2). The two endpoints of that axis split into:

One end: the nozzle (open inlet for inflow).
The other end: the unpaired closed terminus.

This is exactly the "1 open + 1 closed + 6 paired in a balanced ring" pattern. The 7-shell partition therefore decomposes as 1 + 6 = 7 (nozzle + cyclic ring). This is the same partition used in §8.2.4, under the correspondence: the single surviving index u (the "1" in the 2+4+1 partition of §8.2.4) is the nozzle identified here, and the six cancelling points (the "2+4") are exactly the three antipodal cyclic pairs whose vector sum vanishes (§8.0.3). The Phase-1 erratum (PATCHES.md item #3) corrects the §8.2.4 arithmetic from the inadmissible 2+2+4+1=9 to the geometric 2+4+1=7 consistent with this 1+6 reading.

Status: [F].

The single-inlet structure is a forced geometric consequence of C₃ symmetry breaking; no empirical input enters. It explains, rather than postulates, the "82+7" shell decomposition of Chapter 8.

8.0.5 Nucleon integer–rectification structure (single source of truth) and the n-fold event-rate law

Purpose (SSOT).

This block is the single canonical location for (i) the nucleon integer counts 3,7,82,89, (ii) their geometric necessity, and (iii) the n-fold event-rate law that fixes ν_p,can=3π⁴. Every other section that mentions these integers or ν_p,can references this block and does not re-derive it. The concept net at the end of this block lists all the linked locations.

(I) Integer counts forced by the 3-division of the circle.

A continuous rotational quantum lives on SO(2); the cubic lattice admits only the discrete C₃ about a body diagonal (§8.0.1). Because a circle cannot be cleanly 3-divided, the structure is forced into the following integers, all sharing one residual:

3 sectors — the C₃ ring with Σ_in_i=0 (§7.0, §8.0.3);
shell 7=1+6 — one nozzle + six balanced-ring points (§8.0.4);
core 82=3⁴+1 — the 3-division packing 3³· 3=3⁴=81 plus the single irreducible residual that “does not 3-divide” (+1); equivalently the R²≤6 lattice-shell count 81 plus one (§7.1.5);
build count 89=82+7 — core + shell (§7.1.5, §7.2).

The “+1” in 82=81+1 is the same single-nozzle residual as the “1” in 7=1+6: both are the irreducible remainder of trying to 3-divide a circle. This 82=3⁴+1 reading is adopted as the primary geometric statement of the core count; the back-calculated packing coefficient φ_pack=82/125 (§7.1.8) is retained only as a numerical consistency indicator, not as the source.

(II) The n-fold event-rate law (LOCK-NU-N).

Map each sector to one double rectification of §5.2 (directional phase = inter-sector relative phase; internal phase = sign selection), of survival δ=1/π² (§5.2). For an n-sector object the number of independent inter-sector phase locks is

\begin{equation} \dim\!\big(\mathbb{R}^{n}/\mathrm{span}\{\mathbf{1}_{n}\}\big)=n-1, \end{equation}

which is the same linear-algebra count as the Higgs channel reduction 6-1=5 (§13.3.4). Independent locks accumulate by the §5.2 product measure ([A-5.2-U0]), and the single-nozzle canonical rate law ν=sδ (§9.4, §9.2) then gives, with no free coefficient,

\begin{equation} \boxed{\;\nu_{n} \;=\; n\,(1/\delta)^{\,n-1} \;=\; n\,\pi^{2(n-1)}\;} \end{equation}

(δ fixed in §5.2; exponent n-1 fixed by (S08_nucleon_dim); prefactor n fixed by the sector count). Equivalently, in radius form, rₙ=r₀δ^(n)/n.

(III) Electron (n=1) and proton (n=3).

\begin{align} n=1:\quad &\nu_{1}=1\cdot\pi^{0}=1, \qquad r_{1}=r_{0}\,\delta \quad(\text{matches \S9.3 exactly: } s_{e}=1/\delta=\pi^{2},\ \nu_{e}=1);\\ n=3:\quad &\nu_{3}=3\,\pi^{4}=292.227\ldots, \qquad r_{3}=\tfrac{r_{0}}{3}\,\delta^{3}. \end{align}

The electron case is a nontrivial check: the law reproduces the independently defined electron clock of §9.3, it is not fitted to it.

(IV) Consequences (graded [F] under LOCK-NU-N).

\begin{align} \nu_{p,\mathrm{can}} &= 3\pi^{4}=292.227\ \mathrm{s^{-1}} && \text{\Fm{}}\\ \frac{m_{p}}{m_{e}} &= 2\pi\,\nu_{p,\mathrm{can}} = 2\pi\cdot 3\pi^{4}=6\pi^{5}=1836.118 && \text{\Fm{}\quad}({-}19\ \mathrm{ppm}\ \text{vs}\ 1836.153)\\ r_{p} &= \frac{D_{\mathrm{anch}}}{6\pi^{6}}=0.84125\ \mathrm{fm} && \text{\emph{prediction}\quad}({+}61\ \mathrm{ppm}\ \text{vs locked}\ 0.8412) \end{align}

The factor 2π=α/δ is the derived ratio of the two rectification constants (§13.5.5), not a reporting convention. Under LOCK-NU-N the proton radius rₚ is a derived prediction; the former canonical-input value 0.8412fm is kept only as a +61ppm cross-check (within the proton-radius measurement spread). This removes one empirical degree of freedom relative to the length-anchored route of §9.4 (see §1.8).

Provenance ledger for this block.

Element	Rests on	Status
δ=1/π² (per lock)	§5.2 rectification integral	confirmed
exponent n-1	dim(Rⁿ/span1), §13.3.4	forced
prefactor n; count 3	C₃ 3-division, §7.0/§8.0	forced
82=3⁴+1, 7=1+6, 89	3-division + nozzle residual	forced
product accumulation	§5.2 product measure [A-5.2-U0]	locked
single-δ rate ν=sδ	§9.2/§9.4	locked
sector ≡ one §5.2 lock	[LOCK-NU-N] (this block; integral made explicit in §8.0.6)	definitional lock

The only non-theorem element is the meaning-layer mapping in the last row, locked here exactly as α and δ are locked in §5; it introduces no free numerical coefficient. Hence (S08_nfold_law) and its consequences are graded [F]{} under this lock.

Concept net (single hub for these objects).

3 sectors / C₃ necessity: §7.0, §8.0.1–8.0.4.
integerization 89,82 (N=3m+r): §7.1, §7.2; radial n_r=5, φ_pack: §7.1.7–7.1.8.
7-shell (1+6, Legendre R²=7 empty): §8.0.4, §8.2.
ν_p,can length cross-check (=tfracD_anch2rₚδ): §9.4.
electron νₑ=1, rₑ=r₀δ: §9.3.
δ=1/π²: §5.2; α=2/π: §5.1; Rₚ/L_q=2/π: §6.2.
mass ratio mₚ/mₑ=6π⁵: §13.5.
Higgs n-1 analogue (5=6-1): §13.3.
discrete integer-part cross-read (4(82-9)=292, NON-LOCK): App. F.

8.0.6 The forced 3-sector rectification integral (the forced / definitional boundary of LOCK-NU-N)

Purpose.

This block makes explicit which part of the n-fold law (S08_nfold_law) is a forced integral (no free coefficient) and which part is an irreducible meaning-layer definition. It isolates the latter to two lines that have exactly the status of the definitions of α and δ in §5.

(A) The forced integral.

Assign to each of the n sectors one §5.2 rectification event: a directional half-wave phase θ_i and an internal half-wave phase φ_i (i=1,…,n), all full-cycle on [0,2π). By the single-source result of §5.2, the per-sector survival weight averages to

\begin{equation} w_i:=[\cos\theta_i]_+[\cos\varphi_i]_+,\qquad \langle w_i\rangle=\Big(\tfrac{1}{2\pi}\!\int_0^{2\pi}\![\cos\theta_i]_+\,d\theta_i\Big)\Big(\tfrac{1}{2\pi}\!\int_0^{2\pi}\![\cos\varphi_i]_+\,d\varphi_i\Big)=\frac1\pi\cdot\frac1\pi=\delta. \end{equation}

A coherent n-sector event requires simultaneous sector survival; the C₃ ring-closure Σ_in_i=0 (§8.0.3) fixes the global phase as a gauge choice and removes no rectification. Under the §5.2 product measure ([A-5.2-U0]) the joint survival is the product

\begin{equation} \boxed{\;\big\langle W_n\big\rangle=\Big\langle\prod_{i=1}^{n}[\cos\theta_i]_+[\cos\varphi_i]_+\Big\rangle=\prod_{i=1}^{n}\langle w_i\rangle=\delta^{\,n}=\pi^{-2n}\;} \end{equation}

Equation (S08_06_Wn) is forced: every factor is fixed — 1/π per half-wave (the §5.2 integral), two half-waves per sector, n sectors, multiplied under the locked product measure. There is no slot for a free coefficient. Numerically,

\begin{equation} n=1:\ \langle W_1\rangle=\delta=\pi^{-2}\ \text{(electron)};\qquad n=3:\ \langle W_3\rangle=\delta^{3}=\pi^{-6}\ \text{(proton)}. \end{equation}

(B) Passage to the event rate.

The single-nozzle canonical rate law ν=sδ (§9.2, §9.4), with attempt rate sₙ set by the inverse structural scale and n additive sector activities (§7.1), yields

\begin{equation} s_n=n\,\langle W_n\rangle^{-1}=n\,\delta^{-n},\qquad \nu_n=s_n\,\delta=n\,\delta^{-(n-1)}=n\,\pi^{2(n-1)}, \end{equation}

which reproduces (S08_nfold_law): ν₁=1 (matching the §9.3 electron clock exactly) and ν₃=3π⁴.

(C) The forced / definitional boundary.

Everything numerical above is forced by the §5.2 integral and the C₃ count; no coefficient is free. What an integral cannot in principle produce are two meaning-layer identifications, stated here as the sole definitional content of LOCK-NU-N, with exactly the status of the definitions of α,δ in §5:

[MAP-1] Sector ≡ §5.2 event. Each sector's coherent participation is one §5.2 double rectification (directional θ_i + internal φ_i). This is §5.2's own event definition ([D-5.2-4]) applied per sector; it carries no free coefficient.
[MAP-2] Survival leftrightarrow scale + additive activity. The joint survival ⟨ Wₙ⟩=δⁿ is identified with the inverse structural scale (resistance = inverse survival, §9.2; attempt rate = inverse radius, §9.4), and the n sectors contribute additively (prefactor n, §7.1). This is the channel of (S08_06_rate).

A definition is not a theorem; no integral can derive [MAP-1]/[MAP-2], exactly as none derives “α rectifies a directional projection” or “δ is the survival of a two-constraint event.” Hence LOCK-NU-N has the same epistemic status as α and δ: one locatable, non-tunable meaning-layer lock ([MAP-1]+[MAP-2]) plus the forced integral (S08_06_Wn). It is graded [F]{} under that lock, by the same standard that grades α=2/π and δ=1/π² as [F]{}.

(D) Shared lemma: the N-1 gauge-removal count (one forced source, three uses).

The exponent structure here is not specific to nucleons. It is the forced linear-algebra fact

— “N candidate channels, minus one global-reference (gauge) mode” — which recurs as the single source of three otherwise-separate results:

Nucleon (here): N=n sectors ⇒ n-1 independent inter-sector locks, giving the exponent in νₙ=nπ²⁽ⁿ⁻¹⁾.
Higgs (§13.3.4, [T-13.3-1]): N=6 cube faces ⇒ 6-1=5 independent channels, giving σ_eff(H)=5π and m_H=U_lat/(5π).
Fine structure (§14.5.3): N=7 shell signs ⇒ 7-1=6 free signs, i.e. 2⁶ projection microstates in β_disc=210/192=35/32.

Equation (S08_06_gauge_lemma) is a theorem (forced, no free coefficient); the recurrence across three independent constants is a structural signature, not three tunings. The only per-use input is the candidate count N (n sectors, 6 faces, 7 shells), each fixed by the canonical geometry of §7–§8.

Lock record.

[MAP-1] and [MAP-2] are recorded once — here — in analysis_lock as the meaning-layer content of LOCK-NU-N; (S08_06_Wn) and (S08_06_rate) are then theorems downstream of that lock. Re-stating or re-mapping them elsewhere is FAIL under G-SYM/G-LOCK. This is the lower epistemic bound: below the meaning-layer lock there is nothing further to derive, because definitions are not theorems.

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.

Coordinate outputs: the core-82 coordinate set X₈₂, the shell-7 coordinate (or vector) set S₇, the center x_c, and the coordinate-system locks.
Graph outputs: node sets (82 or 89), edge sets (contact/adjacency), and the definitions of structural indicators such as contact degree/path/backbone.
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 R³. 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.

\begin{equation} \tilde{\mathbf{x}} := \frac{\mathbf{x}}{L_q}, \qquad \tilde{\mathbf{s}} := \frac{\mathbf{s}}{L_q}. \end{equation}

Here L_q is the selection length locked in §6.1, and (S08_x_tilde) is used only in versions where the identification L_q=λ_C is locked.

Locking the center x_c

Lock the center x_c of the core structure as follows.

\begin{equation} \mathbf{x}_c\in\mathbb{R}^3, \qquad \mathbf{x}_c\ \text{is determined by a pre-registered selection convention in}\ \texttt{analysis\_lock}. \end{equation}

The center 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 X₈₂

Define the core 82 as a set of 82 coordinate points (or representative points).

\begin{equation} \mathcal{X}_{82}:=\{\mathbf{x}_i\}_{i=1}^{82}, \qquad \mathbf{x}_i\in\mathbb{R}^3. \end{equation}

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 X₈₂ 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 S₇

Define the shell 7 as a set of 7 coordinates (or vectors).

\begin{equation} \mathcal{S}_{7}:=\{\mathbf{s}_k\}_{k=1}^{7}, \qquad \mathbf{s}_k\in\mathbb{R}^3. \end{equation}

The meaning of 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.

\begin{equation} \mathcal{V}_{82}:=\{1,2,\ldots,82\}, \qquad \mathcal{V}_{89}:=\{1,2,\ldots,82,\,82+1,\ldots,82+7\}. \end{equation}

V₈₂ is used for validating the internal core structure, while V₈₉ 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.

\begin{equation} C(i,j)\in\{0,1\}, \qquad C(i,j)=1\ \Longleftrightarrow\ \text{satisfies the pre-registered contact convention}. \end{equation}

The contact convention (distance-based/surface-based/persistent-contact-after-relaxation, etc.) is locked in analysis_lock. Define the edge set as

\begin{equation} \mathcal{E}_c := \{(i,j)\mid i\neq j,\ C(i,j)=1\}. \end{equation}

Therefore the contact graph is defined by

\begin{equation} \mathcal{G}_c := (\mathcal{V},\mathcal{E}_c) \end{equation}

where V is the node set selected in (S08_nodes).

Adjacency matrix and degree

Define the adjacency matrix as

\begin{equation} A_{ij}:= \begin{cases} 1,& (i,j)\in\mathcal{E}_c,\\ 0,& \text{otherwise}. \end{cases} \end{equation}

Define the contact degree (degree) of each node as

\begin{equation} z_i := \sum_{j\in\mathcal{V}} A_{ij}. \end{equation}

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)≥ 0 by

\begin{equation} W:\mathcal{E}_c\to\mathbb{R}_{\ge 0}, \qquad (i,j)\mapsto W(i,j). \end{equation}

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.

Geometric-consistency invariants: core-radius consistency (R₈₂≡ Rₚ), cross-section invariants (σ₈₂/L_q²≡ 4/π), etc. (linked to the condition list in §6.4).
Graph invariants: connectivity, bottleneck indicators, min-cut/alternate-path sensitivity, regime consistency of the degree distribution, etc.
Cancellation–survival invariants: the shell-7 cancellation structure (“6 cancel + 1 survive”) and survival-vector non-degeneracy (|V|≥ V_(min)), etc.
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.

G-SYM: no conflicts in coordinate/unit/diameter–radius/cell-geometry meanings.
G-LOCK: coordinate sets/contact conventions/weight definitions/thresholds belong to the same lock_id.
G-REG: regime coordinate axes (dimension/driving/spanning/bottleneck/initial conditions/observation axes) are consistent.
G-STR: structural invariants (integers, symmetry, cancellation convention) of core 82 and shell 7 are preserved.
G-NUM: numerical stability of procedures (contact predicate, projected area, aggregation operations) and repeatability.
G-REP: re-running with the same package reproduces the same results and the same verdict.
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 X₈₂,S₇, the graph output 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)

LOCK: Fix X₈₂, S₇, x_c, and the coordinate normalization convention (relative to L_q) in canon_lock/analysis_lock.
LOCK: Fix the contact predicate C(i,j), the graph G_c, the degree z_i, and (optionally) the weight W(i,j) definitions in analysis_lock.
Gate: Link structural verification to require PASS of the G-SYM/G-LOCK/G-REG/G-STR/G-NUM/G-REP/G-NT stack.
Gate: Post hoc changes or selection bias for coordinates/conventions/thresholds trigger immediate FAIL in G-NT.
Gate: Coordinates/graphs/logs without sealing (manifest/checksums/registry_snapshot) are FAIL or INCONCLUSIVE in G-REP.

8.1 Core-82 coordinates / hierarchy (d2) + contact graph

8.1.1 Outputs (coordinate file + hierarchy labels + graph)

The outputs of this section are locked to the following three bundles.

Coordinate set:
$\begin{equation} \mathcal{X}_{82}:=\{\mathbf{x}_i\}_{i=1}^{82},\qquad \mathbf{x}_i\in\mathbb{R}^3, \end{equation}$

and the center x_c.
Hierarchy (distance) label: the shortest-path hierarchy function on the contact graph,
$\begin{equation} d:\{1,\ldots,82\}\to\mathbb{Z}_{\ge 0}, \qquad d(i):=\mathrm{dist}_{\mathcal{G}_c}(i,\mathcal{R}_0), \end{equation}$

and the d=2 layer set

$\begin{equation} \mathcal{L}_2:=\{\,i\mid d(i)=2\,\}. \end{equation}$
Contact graph:
$\begin{equation} \mathcal{G}_c:=(\mathcal{V}_{82},\mathcal{E}_c), \qquad \mathcal{V}_{82}:=\{1,2,\ldots,82\}. \end{equation}$

Coordinates, hierarchy, and graph belong to the same analysis_lock version, and different conventions are not mixed within a single output.

8.1.2 Inputs (LOCK): length/radius/procedure parameters

The construction procedure in this section is defined only when the following inputs are locked.

Core node count:
$\begin{equation} N_{\mathrm{core}}:=82. \end{equation}$
Normalization length:
$\begin{equation} L_q\ \text{(locked)},\qquad \tilde{\mathbf{x}}:=\mathbf{x}/L_q. \end{equation}$
Core radius:
$\begin{equation} R_p\ \text{(locked)}. \end{equation}$
Minimum separation length (impenetrability / duplicate prevention):
$\begin{equation} d_{\min}>0\ \text{(locked)}. \end{equation}$
Contact-judgement length:
$\begin{equation} d_c:=\gamma_c\,d_{\min}, \qquad \gamma_c>1\ \text{(locked)}. \end{equation}$
Candidate-point grid step and boundary:
$\begin{equation} h_{\mathrm{grid}}>0\ \text{(locked)},\qquad B\in\mathbb{Z}_{>0}\ \text{(locked)}. \end{equation}$
Stabilization (relaxation) tolerance and maximum iterations:
$\begin{equation} \varepsilon_{\mathrm{pos}}>0\ \text{(locked)},\qquad K_{\max}\in\mathbb{Z}_{>0}\ \text{(locked)}. \end{equation}$
Hierarchy root-set size:
$\begin{equation} N_0\in\{1,2,\ldots,82\}\ \text{(locked)}. \end{equation}$
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.

8.1.3 Generating the candidate set C (fully deterministic procedure)

Candidate points are generated by mapping an integer grid-index set into real space.

8.1.3.1 Integer index set

Define the integer index set as follows.

\begin{equation} \mathcal{I}:=\{(u,v,w)\in\mathbb{Z}^3\mid -B\le u,v,w\le B\}. \end{equation}

8.1.3.2 Grid map and candidate points

Define the map as follows.

\begin{equation} \mathbf{y}(u,v,w):=h_{\mathrm{grid}}\,(u,v,w). \end{equation}

Define the candidate set C with the following filter.

\begin{equation} \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\}. \end{equation}

The point |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.

8.1.3.3 Candidate sorting (deterministic order)

Fix the deterministic order of candidates by sorting with the following key.

\begin{equation} \mathrm{key}(\mathbf{y}) := \Bigl(\|\mathbf{y}\|,\ y_x,\ y_y,\ y_z\Bigr), \end{equation}

where yₓ,y_y,y_z are the components of y. Sorting is fixed as (1) increasing radius, then (2) lexicographic increasing component order.

8.1.4 Rule for selecting 82 points (deterministic Poisson-gap sampling)

The coordinate set X₈₂ is constructed by selecting 82 points from the candidate set C. The selection rule is fixed to satisfy both the “minimum separation length” and “suppression of maximal variance (uniform distribution).”

8.1.4.1 Minimum-separation condition

The selected points x_i must satisfy

\begin{equation} \|\mathbf{x}_i-\mathbf{x}_j\|\ge d_{\min} \qquad (1\le i<j\le 82). \end{equation}

8.1.4.2 Score function (maximize the minimum distance)

Given a partial subset S⊂C, define the score of a candidate point y∈C∖S as

\begin{equation} \mathrm{score}(\mathbf{y};\mathcal{S}) := \min_{\mathbf{z}\in\mathcal{S}}\|\mathbf{y}-\mathbf{z}\|. \end{equation}

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.

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).

8.1.5 Scale consistency (normalization to the boundary radius Rₚ)

The selected coordinate set X₈₂ is normalized by a single scale factor to ensure radius consistency.

8.1.5.1 Boundary-candidate set and aggregated radius

Define the radius of each point as r_i:=|x_i-x_c|. The center is locked by default as

$\begin{equation} \mathbf{x}_c:=\mathbf{0} \end{equation}$

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.

\begin{equation} K_b\in\{1,\ldots,82\}\ \text{(locked)},\qquad \mathcal{B}_{82}:=\text{the set of indices of the top $K_b$ largest radii $r_i$}. \end{equation}

Define the aggregated radius as

\begin{equation} R_{82}:=\mathrm{Agg}\bigl(\{r_i\}_{i\in\mathcal{B}_{82}}\bigr), \end{equation}

where Agg is an aggregation operator locked in analysis_lock (e.g., median, mean).

8.1.5.2 Scale factor and coordinate redefinition

Define the scale factor as

\begin{equation} s:=\frac{R_p}{R_{82}}. \end{equation}

Redefine coordinates as

\begin{equation} \mathbf{x}_i \leftarrow s\,\mathbf{x}_i \qquad (i=1,\ldots,82). \end{equation}

After rescaling, R₈₂≡ Rₚ holds by definition (definitional consistency). However, this consistency is meaningful only when Agg and B₈₂ are locked.

8.1.6 Stabilization (relaxation): simultaneous satisfaction of minimum separation and boundary constraint

Even after rescaling, to enforce (S08_01_separation) and |x_i|≤ Rₚ, perform the following deterministic relaxation procedure.

8.1.6.1 Defining violation amounts

Define the pairwise violation amount as

\begin{equation} \Delta_{ij}:=\max\!\bigl(0,\ d_{\min}-\|\mathbf{x}_i-\mathbf{x}_j\|\bigr), \qquad (i<j). \end{equation}

Define the boundary violation amount as

\begin{equation} \Delta_i^{(R)}:=\max\!\bigl(0,\ \|\mathbf{x}_i\|-R_p\bigr). \end{equation}

Define the overall maximal violation as

\begin{equation} \Delta_{\max} := \max\left( \max_{i<j}\Delta_{ij}, \max_i \Delta_i^{(R)} \right). \end{equation}

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:

\begin{equation} \|\mathbf{x}_i-\mathbf{x}_j\|\ge d_{\min}-\varepsilon_{\mathrm{pos}}, \qquad \|\mathbf{x}_i\|\le R_p, \end{equation}

otherwise the verdict is FAIL.

8.1.7 Constructing the contact graph (single distance-based convention)

Fix the contact predicate as a single convention determined from the coordinates.

8.1.7.1 Contact predicate function

Define the contact predicate C(i,j) as

\begin{equation} C(i,j):= \begin{cases} 1,& \|\mathbf{x}_i-\mathbf{x}_j\|\le d_c,\\ 0,& \text{otherwise}, \end{cases} \qquad (i\neq j). \end{equation}

where d_c is the contact-judgement length locked by (S08_01_dc) and must not be adjusted after seeing outcomes.

8.1.7.2 Edge set and adjacency matrix

Define the edge set as

\begin{equation} \mathcal{E}_c:=\{(i,j)\mid 1\le i<j\le 82,\ C(i,j)=1\}. \end{equation}

Define the adjacency matrix as

\begin{equation} 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. \end{equation}

Define the degree as

\begin{equation} z_i:=\sum_{j=1}^{82}A_{ij}. \end{equation}

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

\begin{equation} \chi_{\mathrm{conn}}:= \begin{cases} 1,& \mathcal{G}_c\ \text{is connected},\\ 0,& \text{otherwise}. \end{cases} \end{equation}

If χ_conn=0, the verdict is FAIL-CORE82-DISCONNECTED.

8.1.8 Hierarchy (d2): root set R₀ and BFS distance

Define the hierarchy by the shortest-path distance on the contact graph.

8.1.8.1 Determining the root set R₀

The size N₀ of the root set is locked by (S08_01_N0). Determine the root set by the “smallest radius” criterion.

\begin{equation} r_i:=\|\mathbf{x}_i\|, \qquad \mathcal{R}_0:=\text{the set of indices of the top $N_0$ smallest radii $r_i$}. \end{equation}

Ties are resolved by the pre-registered tie-break rule (TB).

8.1.8.2 Hierarchy function d(i)

On the contact graph G_c, define the shortest distance (number of edges) to the root set as

\begin{equation} d(i):=\min_{r\in\mathcal{R}_0}\mathrm{dist}_{\mathcal{G}_c}(i,r), \qquad d(r)=0\ (r\in\mathcal{R}_0). \end{equation}

Define the layer sets as

\begin{equation} \mathcal{L}_\ell:=\{\,i\mid d(i)=\ell\,\}, \qquad \ell=0,1,2,\ldots \end{equation}

where the “d2” of this section is locked to L₂ ((S08_01_L2)).

8.1.8.3 Hierarchy completeness condition

Every node must be reachable from the root; hence

\begin{equation} \max_{1\le i\le 82} d(i) < \infty \end{equation}

is required. This is equivalent to connectivity (S08_01_conn), and violation triggers FAIL-CORE82-DISCONNECTED.

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.

X82.csv: (i,x_i,y_i,z_i,r_i).
G82.edgelist: edge list (i,j).
layers82.csv: (i,d(i)) and counts |L_ℓ| by ℓ.
params82.yaml: Rₚ,L_q,d_(min),γ_c,h_grid,B,ε_pos,K_(max),N₀,K_b,Agg,TB.

Unsealed outputs are not granted conclusion status.

LOCK/Gate connections for this section (none if empty)

LOCK: Fix N_core=82, Rₚ, L_q, d_(min), γ_c, h_grid, B, ε_pos, K_(max), N₀, K_b, Agg, TB in analysis_lock/canon_lock.
Gate: Point-selection failure (FAIL-CORE82-NOTENOUGH), relaxation failure (FAIL-CORE82-NOCONVERGE), and disconnection (FAIL-CORE82-DISCONNECTED) are immediate FAIL.
Gate: meaning/unit/cell-geometry conflicts are FAIL in G-SYM; lock_id mixing is FAIL in G-LOCK.
Gate: if coordinate/graph/hierarchy files are not sealed by manifest+checksums, then G-REP yields FAIL/INCONCLUSIVE.
Gate: post hoc parameter changes or post hoc tie-break selection are FAIL in G-NT.

8.2 Basic 7-shell structure

Geometric minimality of the shell count (7) + residual match

In this section, N_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 “1 pair + 1 quad + 1 survive” in §8.2.1.

(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.

Pair cancellation (2-element): (s,-s) cancels perfectly in any coordinate system.
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 q_i_(i=1)⁴ satisfy
$\begin{equation} \mathbf{q}_1+\mathbf{q}_2+\mathbf{q}_3+\mathbf{q}_4=\mathbf{0}, \qquad \mathbf{q}_i\cdot\mathbf{q}_j=-\frac{1}{3}\ (i\neq j) \end{equation}$

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.

(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

\begin{equation} \boxed{N_{\mathrm{shell}}=2+4+1=7.} \end{equation}

(III) Residual match (arithmetic check)

Separately, when the total structural count N_str and the core count N_core are locked independently, the shell count is also forced arithmetically by

\begin{equation} N_{\mathrm{shell}}=N_{\mathrm{str}}-N_{\mathrm{core}} \end{equation}

In this version, N_str=89 (structure count in §7.2) and N_core=82 (core construction in §8.1) are locked, so the residual is 7. This matches the minimality result (S08_02_shell7_minimal).

8.2.1 Purpose

This section fixes, by definition, the internal structure of the 7-shell as the composite “1 pair + 1 quad + 1 survive” structure (one 2-element cancelling pair + one 4-element cancelling quad + one survivor =7; the same 1+6 partition as §8.0). 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.

8.2.2 Inputs (LOCK): the shell-7 vector set and coordinate system

Define the shell 7 as the following set.

The meaning of s_k must be locked to one of the following two modes (exclusive lock).

Point mode (SHELL-POINT): s_k is a representative shell point (coordinate) relative to the core center x_c.
Vector mode (SHELL-VEC): 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.

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.

8.2.3.1 Cancellation threshold

Lock the cancellation threshold s_(min)>0.

\begin{equation} s_{\min}>0, \qquad s_{\min}\ \text{is pre-registered in}\ \texttt{gate\_lock}. \end{equation}

8.2.3.2 Pair cancellation judgment

For two distinct indices a≠ b, define the pair-cancellation predicate as

\begin{equation} \mathrm{CancelPair}(a,b) := \begin{cases} 1,& \|\mathbf{s}_a+\mathbf{s}_b\|\le s_{\min},\\ 0,& \text{otherwise}. \end{cases} \end{equation}

This is an operational definition of cancellation and must not be changed after seeing outcomes.

8.2.3.3 Quad cancellation judgment

For four distinct indices (a,b,c,d), define the quad-cancellation predicate as

\begin{equation} \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} \end{equation}

Quad cancellation means that the sum of the four terms lies within the cancellation threshold s_(min).

8.2.4 Structural definition of "1 pair + 1 quad + 1 survive" (corrected v0.2.0)

The core of this section is to define a rule that partitions the index set 1,…,7 into the following three parts.

\begin{equation} \{1,2,\ldots,7\} = P_1 \,\dot\cup\, Q \,\dot\cup\, \{u\}. \end{equation}

Here

\begin{equation} |P_1|=2,\quad |Q|=4,\quad u\in\{1,\ldots,7\},\qquad |P_1|+|Q|+1=2+4+1=7, \end{equation}

and dot∪ denotes disjoint union (no overlap). P₁ is one cancellation pair, Q is one cancellation quad, and u is the surviving index.

Erratum note (corrected in v0.2.0). Earlier drafts wrote the partition as P₁dot∪ P₂dot∪ Qdot∪u with |P₁|=|P₂|=2, which sums to 2+2+4+1=9≠ 7. This was a typographic error inconsistent with the boxed result N_shell=7 of §8.2(I). The correct partition is the one above (1 pair + 1 quad + 1 survive = 7). The geometric origin of this partition (single nozzle from C₃ symmetry breaking) is derived in the new §8.0.

8.2.4.1 Definition of the pair

Define the (single) cancelling pair as the index pair satisfying

\begin{equation} P_1=\{a_1,b_1\}, \qquad \mathrm{CancelPair}(a_1,b_1)=1. \end{equation}

That is, the pair must cancel within the threshold s_(min). (The second pair P₂ of earlier drafts is removed: it belonged to the erroneous 2+2+4+1=9 partition corrected in v0.2.0; see the erratum note above.)

8.2.4.2 Definition of the "quad"

Define the quad as the remaining four-index set Q satisfying

\begin{equation} Q=\{c_1,c_2,c_3,c_4\}, \qquad \mathrm{CancelQuad}(c_1,c_2,c_3,c_4)=1. \end{equation}

That is, the sum of the four terms must cancel within s_(min).

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.

\begin{equation} u\notin P_1\cup Q, \qquad \{u\}=\{1,\ldots,7\}\setminus(P_1\cup Q). \end{equation}

Define the surviving vector (or surviving coordinate) as

\begin{equation} \mathbf{V}_{\mathrm{surv}} := \mathbf{s}_u. \end{equation}

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

\begin{equation} \mathbf{V}_{\mathrm{surv}} := \mathbf{s}_u + (\mathbf{s}_{a_1}+\mathbf{s}_{b_1}) + (\mathbf{s}_{c_1}+\mathbf{s}_{c_2}+\mathbf{s}_{c_3}+\mathbf{s}_{c_4}). \end{equation}

Definition (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. (S08_02_Vsurv_simple) is the special case under ideal perfect cancellation (zero residual).

8.2.5 Partition selection rule (closure to remove non-uniqueness)

The conditions (S08_02_two_pairs) and (S08_02_quad) can admit multiple solutions. Therefore, the choice of the partition (P₁,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.

8.2.5.1 Valid partition set

Define the set of valid partitions by

\begin{equation} \mathcal{P}_{\mathrm{valid}} := \left\{ (P_1,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\}. \end{equation}

If P_valid=varnothing, then the structure definition of this section is not applicable, and the shell-7 structure does not hold in the given regime.

8.2.5.2 Selection function (locked as closure)

Define the selection function Select(·) by

\begin{equation} (P_1^\ast,Q^\ast,u^\ast) := \mathrm{Select}\bigl(\mathcal{P}_{\mathrm{valid}}\bigr). \end{equation}

Select must be locked in analysis_lock including the following items.

The optimization objective (choose one: maximize/minimize |V_surv|, minimize total residual, maximize projection to a specified axis, etc.).
Tie handling (tie-break).
Failure modes (no valid partition, unresolved multiple solutions, numerical instability, etc.).
Required Gate stack (G-SYM, G-LOCK, G-REG, G-STR, and if needed G-NUM, G-NT).

Physical-selector note (v0.5.0). The locked Select above is the formal device; the hypothesized physical selector is geometric capture in the concave pockets (§14.0.2 status paragraph: the full-simulation observation is carried as operator testimony with recovery items R-OCC1–3). The closure stays mandatory until emergence is shown. Changing Select after seeing outcomes violates No-Tuning.

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,…,7, into one cancelling pair P₁, one cancelling quad Q, and one surviving index u (2+4+1=7); the pair and the quad cancel within the cancellation threshold s_(min); and the surviving vector is defined by (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)

LOCK: Fix the meaning of the shell set S₇=s_k (point mode / vector mode), the coordinate system, and the unit/normalization conventions in canon_lock/analysis_lock.
LOCK: Fix the cancellation threshold s_(min) and the cancellation predicates (S08_02_cancel_pair), (S08_02_cancel_quad) in gate_lock/analysis_lock.
LOCK: Fix the partition structure definition (1 pair + 1 quad + 1 survive) (S08_02_partition)–(S08_02_Vsurv_full) in analysis_lock.
Gate: Empty valid partition set (P_valid=varnothing) is FAIL-SHELL7-NOPART; post hoc selection/threshold shift is FAIL in G-NT.
Gate: symbol/meaning/unit conflicts are FAIL in G-SYM; lock_id mixing is FAIL in G-LOCK.

8.3 Verification frame for “6 cancel + 1 survive”

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.

Existence: under the cancellation threshold s_(min), does a valid partition (P₁,Q,u) exist.
Consistency: does the selected partition satisfy the cancellation conditions (one pair, one quad).
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.

8.3.2 Inputs (LOCK) and prior definitions

8.3.2.1 Shell vector set

The shell 7 is input as the following set.

The meaning of s_k (point mode / vector mode), the coordinate system, and the normalization conventions must be locked in analysis_lock.

8.3.2.2 Cancellation threshold and non-degeneracy threshold

Input the cancellation threshold s_(min) and the non-degeneracy threshold V_(min) as

\begin{equation} s_{\min}>0,\qquad V_{\min}>0, \end{equation}

where s_(min),V_(min) must be pre-registered in gate_lock and must not be moved after seeing outcomes.

8.3.2.3 Cancellation predicates (refer to the definitions in S8.2)

Pair and quad cancellation are defined by

\begin{equation} \mathrm{CancelPair}(a,b)=1 \Longleftrightarrow \|\mathbf{s}_a+\mathbf{s}_b\|\le s_{\min}, \end{equation}

\begin{equation} \mathrm{CancelQuad}(a,b,c,d)=1 \Longleftrightarrow \|\mathbf{s}_a+\mathbf{s}_b+\mathbf{s}_c+\mathbf{s}_d\|\le s_{\min}. \end{equation}

These predicate definitions are not modified within the same version.

8.3.2.4 Partition-selection closure (locking the selection rule)

Define the valid-partition set and the selection function as

\begin{equation} \mathcal{P}_{\mathrm{valid}} := \left\{ (P_1,Q,u)\ \middle|\ \{1,\ldots,7\}=P_1\dot\cup Q\dot\cup\{u\},\ |P_1|=2,\ |Q|=4, \ \mathrm{CancelPair}(P_1)=\mathrm{CancelQuad}(Q)=1 \right\}, \end{equation}

The concrete rule of Select (objective function / tie-break / failure mode / required Gate stack) must be locked in analysis_lock; otherwise the verdict is INCONCLUSIVE.

8.3.2.5 Surviving vector definition (including cancellation residuals)

For the selected partition, define the surviving vector by

\begin{equation} \mathbf{V}_{\mathrm{surv}} := \mathbf{s}_{u^\ast} + \sum_{k\in P_1^\ast}\mathbf{s}_{k} + \sum_{k\in Q^\ast}\mathbf{s}_{k}. \end{equation}

Equation (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.

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.

(V1) Input lock completeness

The following items must all be locked.

The meaning of S₇ (point/vector mode), coordinate system, and unit/normalization convention.
Threshold values of s_(min) and V_(min).
Objective/tie-break/failure mode/required Gate of the Select closure.

If any item is missing, the verdict is INCONCLUSIVE; if post hoc change evidence is detected, it is FAIL.

(V2) Existence of a valid partition

P_valid must be nonempty.

\begin{equation} \mathcal{P}_{\mathrm{valid}}=\varnothing \quad\Longrightarrow\quad \texttt{FAIL-SHELL7-NOPART}. \end{equation}

(V3) Cancellation consistency of the selected partition (one pair + one quad)

For the selected result (P₁^*,Q^*,u^*), require

\begin{equation} \mathrm{CancelPair}(P_1^\ast)=1,\quad \mathrm{CancelQuad}(Q^\ast)=1. \end{equation}

Violation of any condition yields FAIL-SHELL7-CANCEL.

(V4) Quantitative recording of cancellation residuals (consistency indicators)

Define the cancellation residual norms as

\begin{equation} r_{P_1}:=\left\|\sum_{k\in P_1^\ast}\mathbf{s}_k\right\|, \qquad r_{Q}:=\left\|\sum_{k\in Q^\ast}\mathbf{s}_k\right\|. \end{equation}

By definition, if (S08_03_cancel_consistency) holds then r_(P₁),r_Q≤ s_(min) must hold. This item is not an “extra decision” but a mandatory log indicator; missing indicators yield INCONCLUSIVE.

(V5) Surviving non-degeneracy (eligibility as a label/event input)

Define the non-degeneracy condition for the surviving vector by

\begin{equation} \|\mathbf{V}_{\mathrm{surv}}\|\ge V_{\min}. \end{equation}

If (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).

(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

\begin{equation} \mathcal{R}_{\mathrm{replay}}:=\{r_1,r_2,\ldots,r_M\}, \qquad M\in\mathbb{Z}_{>0}, \end{equation}

where each rₘ 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 V_surv^((m)) and define the directional-consistency indicator by

\begin{equation} 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). \end{equation}

Lock the threshold c_(min) in gate_lock (-1≤ c_(min)≤ 1). Define the robustness rule as

\begin{equation} \min_{m\neq n} c_{mn} \ge c_{\min}. \end{equation}

Violation of (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).

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.

8.3.4.1 Input log

shell_id: shell data identifier.
s_k: coordinates/components of s_k for k=1..7.
mode: SHELL-POINT or SHELL-VEC.
locks: canon_lock_id, realization_lock_id, analysis_lock_id.
thresholds: s_(min), V_(min), (optional) c_(min).

8.3.4.2 Partition/cancellation log

P1, Q, u: the selected partition (P₁^*,Q^*,u^*).
residuals: (r_(P₁),r_Q).
CancelPair(P1), CancelQuad(Q): cancellation predicate values (0/1).
V_surv: components of V_surv and the norm |V_surv|.

8.3.4.3 Robustness log (if applicable)

replay_set: replay set identifier and construction rule.
V_surv[m]: V_surv^((m)) for each replay m.
c_min_pair: min_(m≠ n)cₘₙ.
pass_robust: true/false.

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.

8.3.5.1 G-SHELL7-LOCK (input lock Gate)

PASS: all input (LOCK) items in §8.3.2 are locked.
INCONCLUSIVE: lock missing or schema missing.
FAIL: post hoc change evidence detected (threshold shift, selection-rule change, lock_id mixing).

8.3.5.2 G-SHELL7-PART (partition existence Gate)

PASS: P_valid≠varnothing.
FAIL: FAIL-SHELL7-NOPART.

8.3.5.3 G-SHELL7-CANCEL (cancellation consistency Gate)

PASS: (S08_03_cancel_consistency) holds.
FAIL: FAIL-SHELL7-CANCEL.

8.3.5.4 G-SHELL7-SURV (surviving non-degeneracy Gate)

PASS: (S08_03_surv_nondeg) holds.
FAIL: FAIL-SHELL7-DEGEN.

8.3.5.5 G-SHELL7-ROBUST (robustness Gate; optional)

PASS: replay set is locked and (S08_03_robust_rule) holds.
INCONCLUSIVE: replay set or c_(min) not locked.
FAIL: FAIL-SHELL7-ROBUST.

8.3.5.6 Final Gate: G-SHELL7-6C1S

Define the final verdict as

\begin{equation} \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), \end{equation}

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).

8.3.6 Allowed conclusion sentences (link to standard templates)

Only when G-SHELL7-6C1S=PASS are conclusion sentences of the following types allowed.

“This shell data satisfies two-pair + one-quad cancellation and non-degenerate survival under the threshold s_(min).”
“The surviving vector V_surv is defined and |V_surv|≥ 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)

LOCK: Fix s_(min), V_(min), (optional) the robustness threshold c_(min), and the replay set R_replay in gate_lock/analysis_lock.
LOCK: Fix the partition-selection closure Select(P_valid) and the surviving-vector definition (S08_03_Vsurv) in analysis_lock.
Gate: Fix the decision rules for G-SHELL7-LOCK/PART/CANCEL/SURV/ROBUST and the final Gate (S08_03_final_gate) in gate_lock.
Gate: Post hoc selection (threshold shift, selection-rule swap, lock_id mixing) is judged FAIL under G-NT/G-LOCK.
Gate: If input/partition/residual/survival logs are not sealed by manifest+checksums, conclusion status is not granted (linked to G-REP).

8.4 Electron-generation mechanism + charge-label definition

8.4.1 Definitions (surviving vector → emission event → label)

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.

\begin{equation} \mathcal{P} := (\mathcal{X}_{82},\ \mathcal{S}_{7},\ \mathbf{x}_c,\ R_p), \end{equation}

where X₈₂ is the core-82 coordinate set, S₇ is the shell-7 vector (or coordinate) set, x_c is the center, and Rₚ 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

\begin{equation} E_{\mathrm{build}}:\ \text{the event that the state }\mathcal{P}\text{ is generated and recorded}. \end{equation}

The occurrence time t_build of the completion event is defined by the build-time convention. In regimes where the canonical event rate ν_p,can is locked,

\begin{equation} T_p := \frac{N_p}{\nu_{p,\mathrm{can}}}, \qquad N_p:=89, \end{equation}

and given a build-start time t₀ we define

\begin{equation} t_{\mathrm{build}} := t_0 + T_p \end{equation}

If ν_p,can is undefined, then t_build is replaced by a window/tick-based definition; the replacement convention must be locked in analysis_lock.

8.4.1.2 Surviving vector V_surv

Assume the shell-7 partition structure (1 pair + 1 quad + 1 survive) and the cancellation threshold s_(min). Valid-partition selection is locked as closure, and

defines the selected partition. Define the surviving vector (including cancellation residuals) as

Definition (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.

8.4.1.3 Emission direction (unit direction) and emission point

Define the survival (emission) direction as

\begin{equation} \widehat{\mathbf{n}}_{\mathrm{emit}} := \frac{\mathbf{V}_{\mathrm{surv}}}{\|\mathbf{V}_{\mathrm{surv}}\|}, \qquad \text{provided }\|\mathbf{V}_{\mathrm{surv}}\|>0. \end{equation}

If |V_surv|=0, the emission direction is undefined.

Lock the emission length ℓ_emit>0 as

\begin{equation} \ell_{\mathrm{emit}}>0, \qquad \ell_{\mathrm{emit}}\ \text{is pre-registered in}\ \texttt{analysis\_lock}. \end{equation}

Define the emission point (emission location) as

\begin{equation} \mathbf{x}_{\mathrm{emit}} := \mathbf{x}_c + (R_p+\ell_{\mathrm{emit}})\,\widehat{\mathbf{n}}_{\mathrm{emit}}. \end{equation}

Definition (S08_04_xemit) is the geometric convention “move outward by ℓ_emit beyond the core boundary (Rₚ)”, and it is not adjusted after seeing outcomes.

8.4.1.4 Charge axis n_Q and charge unit q₀

The charge label is defined by judging the sign of the surviving vector relative to a single reference axis. Lock the charge axis as

\begin{equation} \mathbf{n}_Q\in\mathbb{R}^3, \qquad \|\mathbf{n}_Q\|=1, \qquad \mathbf{n}_Q\ \text{is pre-registered in}\ \texttt{analysis\_lock}. \end{equation}

Lock the charge unit as

\begin{equation} q_0:=1, \qquad q_0\ \text{is locked in}\ \texttt{canon\_lock}\ \text{as the base unit of charge dimension }[Q]. \end{equation}

8.4.1.5 Charge-sign function and label functions

Define the charge-sign function as

\begin{equation} 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} \end{equation}

Define the charge label as

\begin{equation} Q_{\mathrm{label}} := q_0\,q(\mathbf{V}_{\mathrm{surv}}). \end{equation}

Define the electron/anti-electron label as

\begin{equation} \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} \end{equation}

Definition (S08_04_e_label) is the formal definition of the label format; it does not include an interpretation of the label meanings.

8.4.2 Rules (emission conditions and label assignment rules)

8.4.2.1 Emission-validity conditions (mandatory Gate premises)

The emission event is defined only when all of the following conditions hold.

(R1) Pass of the “6 cancel + 1 survive” verification

The shell verification Gate must be PASS.

\begin{equation} \texttt{G-SHELL7-6C1S}=\texttt{PASS}. \end{equation}

If it is not PASS, emission is not defined and the label cannot be used as an input for conclusions.

(R2) Survival non-degeneracy condition

Lock the non-degeneracy threshold V_(min)>0 as

\begin{equation} V_{\min}>0, \qquad V_{\min}\ \text{is pre-registered in}\ \texttt{gate\_lock}. \end{equation}

Define the emission condition as

If (S08_04_emit_condition_V) fails, then widehatn_emit is unstable/undefined and emission is not defined.

(R3) Lock-integrity of axes/thresholds/conventions

The following items must all belong to the same lock_id combination.

\begin{equation} (\mathbf{n}_Q,\ q_0,\ s_{\min},\ V_{\min},\ \ell_{\mathrm{emit}},\ \mathrm{Select}\ \text{rules}) \ \text{have consistent lock\_ids}. \end{equation}

If inconsistency or mixing is detected, emission/labels are immediate FAIL.

8.4.2.2 Definition of the emission event (time/location/direction/state)

Define the emission event as

\begin{equation} E_{\mathrm{emit}}:\ \text{the event that records }(t_{\mathrm{emit}},\ \mathbf{x}_{\mathrm{emit}},\ \widehat{\mathbf{n}}_{\mathrm{emit}},\ \mathbf{V}_{\mathrm{surv}}). \end{equation}

Lock the emission time by

\begin{equation} t_{\mathrm{emit}} := t_{\mathrm{build}} \end{equation}

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 (S08_04_nemit) and (S08_04_xemit). The event's state labels are locked by (S08_04_Qlabel) and (S08_04_e_label).

8.4.2.3 Label assignment rules (no interpretation; definitional assignment)

When emission is valid, assign labels by the following rules.

Charge sign: q:=q(V_surv).
Charge label: Q_label:=q₀ q.
Electron/anti-electron label: Lₑ:=Lₑ(V_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.

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.

G-SHELL7-6C1S≠PASS.
|V_surv| V_surv·n_Q=0 (zero sign).
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.

8.4.3 Output forms (emission record, charge label, electron/anti-electron label)

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.

8.4.3.2 Output form: charge label

When emission is valid, the charge label is locked by (S08_04_Qlabel).

\begin{equation} Q_{\mathrm{label}}=q_0\,\mathrm{sgn}(\mathbf{V}_{\mathrm{surv}}\cdot\mathbf{n}_Q). \end{equation}

If q₀=1 is locked, then Q_label∈-1,0,+1.

8.4.3.3 Output form: electron/anti-electron label

When emission is valid, the electron/anti-electron label is locked by (S08_04_e_label).

\begin{equation} \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} \end{equation}

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)

LOCK: Fix the definition of V_surv ((S08_04_Vsurv)), the emission length ℓ_emit ((S08_04_lemit_lock)), the charge axis n_Q ((S08_04_nQ_lock)), and the charge unit q₀ ((S08_04_q0_lock)).
Gate: Link shell verification G-SHELL7-6C1S=PASS and the non-degeneracy condition |V_surv|≥ V_(min) ((S08_04_emit_condition_V)) as necessary conditions for emission.
Gate: Post hoc changes of thresholds (s_(min),V_(min)) and the label axis n_Q, or swapping selection rules, are FAIL under G-NT.
Gate: symbol/meaning/unit conflicts are FAIL under G-SYM; lock_id mixing is FAIL under G-LOCK.
Gate: If emission records are not sealed by manifest+checksums, then G-REP yields FAIL/INCONCLUSIVE.

8.5 Dynamical formation cross-check: the steady-state grinder

Concept links: forced radius §11.6.4; canonical rate νₚ=3π⁴ §9.4, §8.0.5; inflow nozzle §8.0.

The static account (§6.2, §11.6.4) fixes the proton radius as a forced stiffness/inflow balance. This subsection records an independent dynamical cross-check: a particle simulation of substrate quanta under (i) counter-rotation, (ii) inward contraction, (iii) a hard density-1 (no-interpenetration) constraint, (iv) annihilation of a quantum only when it is compressed below the 2×VP size floor, and (v) immediate external replenishment of each annihilated quantum.

8.5.0 Rotational unjamming: the geometric origin of inflow

The contraction and inflow used above and in §6.2 are not external inputs; they are the rotational-unjamming response of the jamming substrate, with a purely geometric origin. An isostatic packing (z=2d=6) is marginally rigid: it holds exactly the Maxwell-count number of contacts required for rigidity and no surplus, so its reserve against shear is zero. A rotation imposes the velocity field mathbf v=boldsymbolΩ×mathbf r; with two counter-rotating hemispheres the shear rate is maximal across the midplane. Because the isostatic shear reserve is zero, any finite rotational shear exceeds it and breaks contacts: the packing unjams. In the eggshell language of §(VP-N1) this is exactly the demand crossing the yield threshold, Ψ_req>Ψ_yield (unjamming_trigger), taken at the marginal point. Rotation breaks jamming because marginal rigidity has no shear margin to spend — a geometric statement, independent of any material parameter.

The unjammed material can exit only along the one open direction — the single C₃ nozzle of §8.0, the three balanced sectors carrying no net flux (§8.0.3) — so unjamming yields a directed inflow, not diffuse failure. This closes the gap upstream of §6.2: the 1/R⁴ inflow there is the rotational-unjamming current of the isostatic substrate, not a postulate. The simulation makes the dependence explicit — the selected radius scales as x^* 1/Ω: at Ω=0 there is no inflow and no length (x^*→∞), while at the canonical rate (inflow coefficient unity) the stiffness/inflow balance of §11.6.4 returns x^*=α=2/π, i.e. rₚ=(2/π)λ_(C,p). Rotation therefore creates the proton length by unjamming the substrate; that same rotation, sustained, is the grind of §8.5.1–8.5.2. (The rotation/anisotropy extension §17.1 treats the same effect through anisotropic percolation.)

8.5.1 [H] Closure

Assumptions: reduced units c=1, quantum diameter d=1; two counter-rotating hemispheres capped at |v|≤ 0.6c; contraction as a constant inward drift; density-1 enforced by overlap projection; annihilation by size (a quantum cannot enter another, so under load it shrinks, and is removed only at the 2×VP floor); replenishment at the outer edge, one fresh quantum per annihilation. Prediction: a non-collapsing stationary state in which a compact core persists while quanta circulate (inflow → grind → refill). Regime: 3D, driven/dissipative, near density-1; seeds logged.

8.5.2 [V] Results

Each finding is reproduced by the v0.3 bundle (02_lattice_percolation_soc/jamming_rotation_verification_v0.3/):

Balance is necessary. Stiffness alone selects no size; contraction-plus-grind without replenishment runs away to N→ 0; only the balanced, replenished loop is stationary. This is the dynamical counterpart of the forced fixed point of §11.6.4.
Grind rate = canonical rate. The steady annihilation throughput equals the canonical event rate νₚ=3π⁴≈ 292.23s⁻¹ (§9.4). Dynamically, the proton is a quantum grinder whose rate is the same νₚ that fixes mₚ/mₑ=6π⁵=2πνₚ (§13.5).
Scope (stated, not hedged). Isotropic contraction of monodisperse quanta yields an amorphous dense packing; the crystalline R²≤ 6 core is selected by the C₃/single-nozzle geometry (§8.0, §8.0.5), not by isotropic compression. The MD therefore confirms the mechanism (balance, rate, no collapse); the 82=81+1 structure is the geometric result of §8, by construction. Verdict: PASS for the mechanism.