Three constants. One lattice. Zero free parameters.

The fine-structure constant α, the Higgs mass mH, and the proton mass × radius product mp·rp — three quantities the Standard Model treats as inputs or measurements — are derived in closed algebraic form from a single VP (Volume Particle) jamming lattice. Each derivation is reproducible in 4–5 steps with deterministic verification scripts.

Live derivations

The three derivations below use only integers (from lattice geometry), π (from rotational symmetry, §5), and external SI anchors (h, cref, λHeNe, rp). No fitting parameters at any step.

Whitepaper §14.5

Fine-structure constant αem−1

Standard Model: free input, no closed-form derivation. Measured to 10 digits.

Step 1 — Lattice integers:
Nshell = 7 (3D minimal cancellation: 2-pair + 4-tetrahedral + 1 survivor, §8.2)
Nsec = 3 (topological enclosure, §7.1)
Nspin = 1 (axial)

Step 2 — Permutation/projection ratio:
β = (7! / 3!2!2!) / (3 · 2⁶) = 210/192 = 35/32

Step 3 — Projection coefficient:
δproj = (35/32) · (2/π²) · (3/7) ≈ 0.09499

Step 4 — Closed form:
⇒ αem−1 = 4π · (7 + 3 + 1 − δproj)
         = 4π × 10.9050
         = 137.0364
137.0364 vs measured 137.035999 Match: 99.9997%  ·  Δ = 0.0003%
Free parameters: 0 Inputs: integers + π
Whitepaper §13.3

Higgs boson mass mH

Standard Model: free input. Measured at LHC (2012, refined 2024).

Step 1 — Anchor cell geometry:
Anchor cell is a cube → 6 faces

Step 2 — Gauge quotient:
Global reference shift u ~ u + c·1₆
dim(R⁶ / span{1₆}) = 6 − 1 = 5

Step 3 — Channel area (inscribed disk):
Each face: σ = π(a/2)²
Dimensionless: σ₀ = 4σ/a² = π

Step 4 — Effective cross-section:
σeff(H) = Σk σ₀ = 5 × π = 5π

Step 5 — Mass:
⇒ mH = Ulat / (5π)
     = 1958.7 GeV / 15.708
     = 124.69 GeV
124.69 GeV vs measured 125.25 GeV (LHC) Match: 99.55%  ·  Δ = 0.45%
Free parameters: 0 Inputs: cube geometry + π
Whitepaper §6.2 + §13.4

Proton invariant mp · rp

Standard Model: mp and rp measured separately, no algebraic relation.

Step 1 — Stability scaling laws (§6.2):
Π4(R) ∝ (1/α) · Lq⁴/R⁴   (collapse term)
Π5(R) ∝   Lq⁵/R⁵   (rigidity term)

Step 2 — Balance condition:
Π4(Rp) = Π5(Rp)
⇒ Rp/Lq = α = 2/π

Step 3 — Mass formula (§13.4):
mp = Ulat / Sp,   Sp = λC/a
λC = (π/2) · rp   (from Rp/Lq = 2/π)

Step 4 — a cancels algebraically:
mp = hc/λC = (2/π) · hc / rp

Step 5 — Anchor-free invariant:
⇒ mp · rp / (hc) = 2/π
         = 0.63662
0.63662 vs measured 0.63672 Match: 99.985%  ·  Δ = 0.015%
Free parameters: 0 Pure dimensionless

Why this is significant

"It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it… we don't know what kind of dance to do on the computer to make this number come out — without putting it in secretly."
— Richard Feynman, on α (QED, 1985)

Other closed-form derivations

The same backbone produces additional dimensionless invariants (full whitepaper for derivation chains):

WP § Dimensionless invariant Predicted Measured Match
§13.5 mp / me = 2π · νp,can 1836.78 1836.15 99.97%
§14.2 Coulomb KC(VP) ∝ (Ne/Np,core) · A−1/2 · hc 2.297 × 10⁻²⁸ N·m² 2.307 × 10⁻²⁸ 99.57%
§14.3 Elementary charge: emap = √(KC(VP)/ke) 1.5987 × 10⁻¹⁹ C 1.6022 × 10⁻¹⁹ C 99.78%
§15.5 Planck blackbody form ν³/(exp(hν/kT) − 1) structural derivation from rigidity-shell filtering exact form
§14.4 Casimir 1/d⁴ pressure law structural derivation from boundary screening exact form
§18.3 Chemical bond rupture at √2 ratio (cube diagonal) N₂, CO₂, H₂O experimental data confirms geometric

Why π emerges from the lattice (§5.1, §5.2.5)

A common question: aren't π, 2/π, 1/π² simply external mathematical constants? In this framework they are derived from the lattice itself, not imposed:

  1. 2π emerges from rotational closure of phase variables — full-cycle angular states return to the same configuration. Lattice-symmetry property.
  2. Uniform measure 1/(2π) emerges from maximum entropy under "no a priori bias" (§5.2.5) — an information-theoretic principle, not an assumption.
  3. α = ⟨|cos θ|⟩ = 2/π from full-cycle absolute rectification.
  4. δ = ⟨[cos θ]+ · [cos φ]+⟩ = 1/π² from two independent half-wave constraints.
  5. 4π in shell counting emerges from 3D isotropic averaging (§14.3); anisotropic regimes give different values.

So π and its powers are outputs of rotational symmetry plus maximum-entropy uniform distribution — in the same operational sense that Kolmogorov scales emerge from turbulence dynamics rather than being imposed.

Gravity: a new lens (Appendix G)

Gravity is reframed as the restoring pressure of a jammed lattice attempting to fill the mass-equivalent void. The lattice has finite yield curvature demand Ψyield, so contact-mode acceleration cannot exceed a saturation limit:

2-channel decomposition (G.1):
  gobs = ggeom(R) = GM/R²    orbit / free-fall mode
  gobs = grestore ≤ g★    contact / static mode
Yield limit: g★ ≡ c² · Ψyield

Falsifiable predictions distinguishable from Newton

Methodology: LOCK → Derive → Gate

Every numerical statement in the whitepaper carries three traceable identifiers: LOCK provenance, derivation chain, Gate verdict. Sentences without all three are not promoted to conclusions.

LOCK

Inputs frozen at a single source (SSOT). Changes exist only as new versions.

Derive

Only LOCK items + permitted transformations. SSOT enforced.

Gate

PASS / FAIL / INCONCLUSIVE. Required stack: G-SYM, G-LOCK, G-REG, G-RECT, G-STR, G-RCROSS, G-REP.

No-Tuning

Post-hoc adjustment forbidden within a version. Selection bias requires pre-registered rules.

Scope and acknowledged limitations (§17.3)

The whitepaper enumerates limitations rather than claiming completeness. Items below are recorded as hypotheses [H], not conclusions [F]:

  • Claim tier: primarily Level-C0 (internal consistency) + Level-C1 (scale reproduction). Level-C2+ (quantitative experimental claims) requires attached Gate reports verified as PASS.
  • me direct form (§13.5): closed-form re = (Danch/2)·δ off by π² from measurement; reporting convention mp/me = 2π·νp,can agrees. Documented, not hidden.
  • Ψyield (Appendix G): currently a LOCK input. Earth's 9.8 m/s² is a consistency anchor, not a prediction; the falsifiable content is the saturation behavior across regimes.
  • Ncore = 82 (§7.1.8.2): φpack = 82/125 explicitly flagged as back-calculated consistency indicator, not a prediction.
  • Danch, A precision: deterministic simulation outputs (§6.1, §10.3); reproducing them requires running bundled scripts.
  • Identifiability: §17.3 notes different closures may reproduce the same observations — agreement does not prove uniqueness.
  • Numerical agreement is recorded as Gate evidence, not used as theory justification (§0.1.5).

Extended application modules

The following are conditional extensions at claim-type [H] (hypothesis with declared regime). Conclusion status is independent of the core derivations above; each module requires its own Gate stack to PASS.

Hypothesis [H]

Lattice cosmology & optics

Light propagation, lensing, expansion constraints. Tired-light-style redshift requires separate Gates.

 Hypothesis [H]

Emergent fluidics

Continuity and Navier–Stokes emergence from VP jamming stability.

 Hypothesis [H]

Deterministic DNA pipeline

Rule-based decoding for Mouse/mm39 genome. Independent verification required.

 Hypothesis [H]

Bio-physical applications

Modular bio-physical applications with reproducibility bundles.

 Hypothesis [H]

Geodynamics (Atlantic opening)

Pressure-deficit "void suction" mechanism. Pre-registered PASS/HOLD/FAIL tests.

 Hypothesis [H]

Energy & catalyst engineering

ESS → power → desalination. Pt catalyst / H₂ pathways under energy-ledger accounting.

DOI-anchored whitepapers

All frameworks pre-registered on Zenodo with deterministic verification scripts and machine-readable LOCK files.

Core Whitepaper · 10.5281/zenodo.17932567 Cosmology & Optics Fluid Dynamics DNA Geodynamics Applications