The Pentachoron and the Fundamental Discretisation of Spacetime

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This version supersedes v4. All changes are responses to an adversarial self-review conducted prior to journal submission. No result is retracted; several proofs are strengthened and epistemic labels are made more precise. ══════════════════════════════════════════════════════ A. CORRECTIONS LOGIQUES — PREUVES ══════════════════════════════════════════════════════ A1. Lemma 2 (Rigidity of simplices) — proof restructured. The previous proof of the necessity direction incorrectly invoked Connelly (1977), a result applicable to non-convex polyhedra. The lemma is now stated as sufficiency only: a d-simplex satisfies (RC). The necessity direction is not required: Theorem 5 is restructured so that (MC) uniquely fixes n = 5, and five affinely independent points in R^4 are a 4-simplex by definition. Proof path changed from RC → simplex → MC → n=5 to MC → n=5 → simplex → RC. Reference to Connelly [10] retained in context (§9.3) with correct scope. Reference Blumenthal [18] added for the Cayley–Menger injectivity argument. A2. Theorem 5 (Uniqueness of the pentachoron) — proof restructured accordingly (Steps 1–3 reordered). Case (p,q)=(1,4) for the causal constraint (CC) restored explicitly as Step 4a (was omitted in v4 restructuring). A3. Theorem 4 — circularity eliminated. Previous statement derived c = ℓ_P/τ_0 from its own definition. Theorem is now stated as: in the ergodic limit, the effective propagation speed is path-independent and the O(α^1) correction vanishes identically by stationarity. A new cancellation lemma is included in the proof. Section title updated: "Path-independence and universality of causal propagation speed". A4. Lemma 1 (Multiplicative structure) — monotonicity hypothesis added explicitly. The Cauchy functional equation h(x+y) = g(x)·g(y) has pathological non-measurable solutions without a regularity assumption on g. Monotonicity of g is now stated as an explicit hypothesis. Reference Aczél [19] added. Duplicate paragraph removed. A5. Theorem 3 (Non-integrability of proper time) — causal 2-connectivity added as an explicit hypothesis. The proof requires two chains with disjoint interior vertex sets; this fails on tree-like graphs. A remark confirms that 2-connectivity holds generically for the BCC-Delaunay realisation (degree 14 at all interior vertices). ══════════════════════════════════════════════════════ B. CORRECTIONS ÉPISTÉMIQUES — STATUTS T1/T2 ══════════════════════════════════════════════════════ B1. Theorem 5, statement — "macroscopic arrow of time" replaced by "past/future asymmetry of the elementary cell". The propagation to macroscopic thermodynamic irreversibility is labelled [T2] in both the statement and the abstract, consistent with Step 5 of the proof. B2. Theorem 7 (Regge convergence) — the unimodular identification Eq.(17) is now labelled [T2]. The variational limit of the standard Regge action gives the Einstein equations, not the traceless equations, without the additional identification that ρ fixes det(g). This identification is stated as a T2 conjecture pending a companion derivation. Table 2 updated. B3. Theorem 8 (Geometric origin of spin) — relabelled [T1]|[T2]. Step A (assignment of excitation spaces E_d) constitutes a physical identification hypothesis [T2]: it requires a 3+1 decomposition motivated by Axiom 2.2 but not derived from it. The spin assignments conditional on Step A are [T1]. An epistemic note is inserted between Steps A–B and the case-by-case derivation. B4. Theorem 10 — split into Theorem 10 + Corollary 6. Theorem 10 [T1]: v_γ/c = 1 + O(α²Var(ρ̃)), proved by ergodic averaging with explicit O(α^1) cancellation. Corollary 6 [T1†]: numerical bound 10^{-76}, conditional on the identification α²Var(ρ̃) ~ (ℓ_P/λ)² via the pentachoric dispersion relation (verified numerically [C2], not yet derived analytically). Remark on epistemic status added. Table 2 updated. B5. Corollary 4 (Electronic shells) — relabelled [T1]|[T2]. The combinatorial fact dim H_l(S³) = (l+1)² is [T1]. The connection to the hydrogen spectrum via Fock's SO(4) symmetry is [T2]: Fock's result operates in momentum space via the Runge–Lenz vector; the connection to the pentachoric S³ is a structural analogy, not a first-principles derivation. A remark clarifies this explicitly. B6. Theorem 9 (Wave-particle duality) — hypothesis ρ̃_e = 0 for edge excitations stated explicitly in the theorem. The condition ρ̃_v = ρ̃_w = 0 on incident vertices is the discrete translation of zero rest mass; its derivation from graph dynamics is labelled [T2]. B7. "Zero free parameters" — all occurrences replaced by "no continuous free parameter". Structural choices (conformal Ansatz, fermionic fibre, graph-entropy correspondence) are identified as theoretical framework, not numerical parameters. One canonical formulation retained in §5.3; all others reformulated consistently. B8. §1.4 (Epistemic conventions) — explicit clarification added: verification keys [Vxx]/[Cxx] denote reproducibility, not proof. A result is T1 by virtue of its proof, not its verification key. B9. Theorem 2 (Metric censorship) — remark added clarifying that the bound ℓ_e ≥ ℓ_P e^{-α} follows from elementary monotonicity on a compact interval. The conceptual value (structural vs. dynamical censorship) is distinguished from the technical content. ══════════════════════════════════════════════════════ C. CORRECTIONS FORMELLES ET BIBLIOGRAPHIQUES ══════════════════════════════════════════════════════ C1. \begin{figure}[H][t] → \begin{figure}[H] (two occurrences). The malformed optional argument produced literal "[t]" text visible in the compiled PDF above Figures 2 and 3. C2. Citation key Alexandrov2005 → Alexandrov1950 to match \bibitem{Alexandrov1950} in the reference list. C3. \eqref{eq:LIV_bound} → \eqref{eq:cgw_num} in Remark 14 (label orphaned after Theorem 10 restructuring). C4. \ref{thm:c} → \ref{thm:speed} in Companion Script item C3 (label updated following Theorem 4 renaming). C5. \label{lem:metric} and \label{lem:uniqueness} added to Lemma 3 (missing labels caused ?? in Theorem 5 proof). C6. §9.3 punctuation: [11] it → [11]; it (missing semicolon produced a run-on sentence). C7. Table 2 repositioned between §9.1 and §9.2 to prevent mid-paragraph insertion in §9.3. C8. Two bibliographic entries added: [18] Blumenthal, Theory and Applications of Distance Geometry, Oxford University Press (1953). [19] Aczél, Lectures on Functional Equations and Their Applications, Academic Press (1966). ══════════════════════════════════════════════════════ D. COMPANION SCRIPT — v4 → v5 ══════════════════════════════════════════════════════ D1. Script renamed: pentachoron_companion_v5.py. D2. Tests updated to reflect restructured Theorem 4 and Theorem 10 (Corollary 6 now tested separately). D3. Causal 2-connectivity check added (Theorem 3 hypothesis): BCC-Delaunay graph verified to be 2-connected at all resolutions n ∈ {5, 7, 8, 10, 12, 15}. D4. Cayley–Menger injectivity test added for the 4-simplex (Lemma 2, Blumenthal reference). ══════════════════════════════════════════════════════ RÉSULTATS INCHANGÉS ══════════════════════════════════════════════════════ All numerical predictions are unchanged: α* = 1/(4 ln 2) ≈ 0.3607 ρ* = 4 ln 2 ≈ 2.773 |c_GW/c − 1| ≤ 10^{-76} Electronic shell capacities: 2, 8, 18, 32 Spin spectrum: {0, 1/2, 1, 3/2, 2} The central result — algebraic uniqueness of the pentachoron from constraints (RC)+(MC)+(CC) — is unchanged. The proof path is more direct and fully rigorous.

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Major additions Exit multiplicity N(σ) — new primitive replacing 2s+1 New Appendix B (Section "Exit Multiplicity: Completing the Spin Framework"): introduces N(σ) = max(1, exit paths) as the primitive state count, from which the conventional spin label s = (N−1)/2 is derived. N reproduces the Standard Model polarisation spectrum {1, 2, 3, 5} (Higgs, fermions, massive/massless gauge bosons, graviton) with zero free parameters. Definition B.1: exit multiplicity for each sub-simplex of K₅ with causal partition (2,1,2). Exit multiplicity table (Appendix B): full table of exit paths, N values, particle identifications, and SM spin for all 5 object types. Three structural advantages over 2s+1 documented: (i) massless spin-1 (photon): N=2 correct, 2s+1=3 wrong; (ii) mass–polarisation unification: face saturation simultaneously provides mass (trapping) and the third polarisation state; (iii) spin spectrum derived from exit counting alone. Exit orientation: face circulation → spin direction via boundary operator ∂. Spin-statistics derived from Pauli bound (not independent postulate): fermions antisymmetric (different exit vertices), bosons symmetric ({a,b}={b,a}). Causal partition (2,1,2) derivation σ:E→F bijection from C(5,2) = C(5,3) = 10. K₃-sharing overlap analysis: overlap ∈ {1,2}; overlap=1 is the unique complete partition (3+3−1=5 vs 3+3−2=4<5). (2,1,2) = Past {P₁,P₂}, Present {F₃}, Future {F₁,F₂}. Tetrahedron reclassification (d=3) K₄ reclassified as inter-cell boundary, not propagating excitation. s=3/2 algebraically valid (Theorem VII.1(iv)) but physically inaccessible: no particle in the K₅ propagating spectrum carries N=4. Remark VII.5 renamed "Saturation hierarchy and boundary reclassification" with v2 update note. Gravitino exclusion (Corollary VII.8) strengthened: now argued from N=4 exit paths (boundary) rather than only from off-diagonal topological prohibition. Structural changes Abstract Rewritten point (4): exit multiplicity from a single geometric principle, including Hessian barrier (bosonic exits) and σ:E→F (fermionic exits). "131 tests" → "144 tests" (companion v5 → v6). Introduction (§I.B) "Four independent results" → "Five independent results". Added item (v): exit multiplicity N(σ) superseding 2s+1. Table 0 (page 1) Footnote ‡ rewritten: references N=4 exit paths, Section B, and Table B. Table I (§VII.3) Column N added: exit multiplicities for all 5 object types. Edge row shows "2/3" (free/trapped). Tetrahedron row shows "4‡" with "(boundary)". Caption rewritten: N is primitive, s derived. Footnote ‡ explains boundary reclassification. Text after table rewritten: propagating spectrum N ∈ {1,2,3,5}. Theorem VII.1 item (iv) — d=3 Reclassification note (v2) added: algebraic derivation valid, but exit-path analysis shows K₄ is boundary, not propagating. Conclusion (§X) Fifth-result paragraph rewritten: exit multiplicities N ∈ {1,2,3,5}, conventional spin derived, K₄ boundary, gravitino exclusion via N=4. Companion Verification Script section v5.py → v6.py. 131 claims → 144 claims; 126 PASS → 139 PASS.