Yang–Mills existence and mass gap from the pentachoric density matrix

Minor update. Compilation align problems

v2: Inter-paper coherence and critical review. (i) Derived N=3 from P2 causal partition (3 future vertices → Herm₃; correspondence flagged as [T3]). (ii) Relabelled α* from T2 to T1† throughout (derived in P1 Prop.V.2, not external input). (iii) Added bridge remarks connecting P5 to P2 (gauge algebra, confinement) and P3 (scalar limit); explicit vertex-labelling bijection P₁↔v₀, P₂↔v₁, F₃↔v₂, F₁↔v₃, F₂↔v₄ added to Remark III.3. (iv) Added open problems O6 (EW sector) and O7 (fermion masses). (v) Fixed two broken LaTeX cross-references. (vi) Corrected inter-paper references: P2 Corollary III.4 → III.3 (confinement), P2 Theorem II.2 → II.1 (spectral filtration), removed phantom reference P2 Prop.3.4, fixed self-reference "companion paper P5" → "P1". (vii) Abstract rewritten: PA vs OS comparison now explicit on all seven directions (4 stronger, 1 equal, 2 weaker with mitigation). (viii) Confinement theorem tier refined: area law [T1], numerical σ = 3.73 [T2]. (ix) Strengthened two-level defence in proof of main theorem (point D) with explicit T2 flag on Yang–Mills identification. (x) Fixed six column overflows in twocolumn layout: notation table, HS-PD equation, factorisation equation (widetext), network gap equation, Clay checklist table, constants table. (xi) Nuanced conclusion: "physical reading" qualifier on master thesis, lattice-specific claim on final italic sentence. No quantitative results changed. Companion script: 389 tests, all PASS (unchanged), updated cross references with papers.

Change log , no conclusion changed CHANGELOG — P5 Yang-Mills existence and mass gap Date: 2026-06-05 Author: JBB / Claude ================================================================ ADDITIONS (7 operations, all independent and commutative) ================================================================ 1. REMARK: Hodge structure dictionary (§III) Location: after Remark rem:P2_gauge CTRL+F anchor: "this paper constructs the connection and derives its dynamics on the Kuhn lattice.\end{remark}" Insert: AFTER Content: New Remark (rem:hodge) — table mapping p=0/1/2 to Delta_0/Delta_1/Delta_2, spectra {0^1,5^4}/{5^10}/{5^10}, physical roles (matter/gauge/curvature). Notes edge-face duality |E|=|F|=10 as Hodge duality Delta_1 = Delta_2. Forward reference to Theorem thm:topo_gap. Tier: T1 Size: ~12 lines 2. THEOREM + REMARK: Topological mass gap beta_1=0 (§IX) Location: after Theorem thm:network_gap CTRL+F anchor: "This bound is rigorous: it requires no causal hypothesis and holds for all $\rho_v$ with eigenvalues in~$[0,1]$.\end{theorem}" Insert: AFTER, before "We now extend from a single" Content: - Theorem thm:topo_gap [T1]: beta_1(Delta^4)=0, ker(Delta_1)=H^1=0, spec(Delta_1)={5^10}, no zero mode. Proof via contractibility. - Remark rem:dob_vs_topo: Dobrushin gives the VALUE (0.025), topology gives the REASON (no 1-cycle). Photon masslessness from acoustic branch closing in thermodynamic limit (U(1)), SU(3) gap stays open (confinement). Tier: T1 Size: ~25 lines Impact: MAJOR — answers the first question any algebraic topology reviewer will ask. 3. REMARK: Deaf mode and simplicial blindness (§Discussion) Location: after "minimum energy cost" paragraph CTRL+F anchor: "This exponential decay is the mass gap: the minimum energy cost of creating a non-trivial excitation above the vacuum." Insert: AFTER this sentence, BEFORE "The $20$ oriented edges" Content: New Remark (rem:deaf_mode) — - Delta_0 on K5: 4 modes at lambda=5 - Delta_0 on K4 (perceived): 3 modes at lambda=4 - Missing mode e_w = (1,1,1,1,-4)/sqrt(20) concentrated on viewpoint - Connection to Dobrushin kappa_v as restoring force - 16 invisible simplices (P22) include {w} = support of deaf mode - Parallel: blindness (Delta_1, p=1) -> spatial curvature = gravity deafness (Delta_0, p=0) -> scalar sector = time dilation - Links to P22 Corollary 4, P23 lapse N^2=5/8 Tier: T1 (algebra) / T2 (physical identification) Size: ~20 lines 4. OPEN PROBLEM O8: Navier-Stokes regularity (§Discussion) Location: after O7 CTRL+F anchor: "The relationship between this spectral structure and the Dobrushin mass gap $\Delta \ge 0.025$ established here (which bounds gauge excitations, not fermionic masses) is an open question." Insert: AFTER Content: Hodge decomposition C^1(K5) = im(d_0) + im(delta_2), dim(im(delta_2))=6 for incompressible subspace. NS on K5 = 6-dim polynomial ODE, global existence by Picard-Lindelof. No blow-up in finite dimensions. Continuum singularity is an artefact. Tier: T1 (Hodge dim + ODE existence) / T2 (interpretation) Size: ~8 lines 5. BIBLIOGRAPHY: 3 new entries Location: after \bibitem{Mehta} CTRL+F anchor: "\bibitem{Mehta}" Insert: AFTER Content: - \bibitem{P21} Electricity and conductivity from K5 - \bibitem{P22} Simplicial blindness - \bibitem{P23} Gravity as perspective Required by: Remark rem:hodge (P21), Remark rem:deaf_mode (P22, P23) 6. COUNTER UPDATE: 389 -> 393 tests CTRL+F targets (4 occurrences): - Abstract: "$389$~tests across $30$~blocks" -> "$393$~tests across $30$~blocks" - Remark rem:vulns: "$389$ computational tests" -> "$393$ computational tests" - Conclusion: "$389$ quantitative claims ($369$ at tier T1)" -> "$393$ quantitative claims ($373$ at tier T1)" - Companion section: "$389$~tests across $30$~blocks" -> "$393$~tests across $30$~blocks" 7. COMPANION BLOCK: B16 additions Location: before B29 entry in companion itemize CTRL+F anchor: "\item B29\vermark{B29}: Exact $S_{\mathrm{eff}}$ formula" Insert: BEFORE Content: Description of 4 new tests: - TOPO01: ker(Delta_1) = {0} on K5 via eigenvalues > 0 - TOPO02: beta_1(Delta^4) = 0 by rank of boundary maps - DEAF01: projection of e_w onto K4 proportional to constant vector - NS01: dim(im(delta_2)) = 6 on K5 All T1. ================================================================ COMPANION SCRIPT: 4 new tests to add ================================================================ Test ID Block Content Tier TOPO01 B16 eigenvalues(Delta_1) all > 0 on K5 T1 TOPO02 B16 beta_1 = 0 via rank(partial_1), rank(partial_2) T1 DEAF01 B16 proj(e_w, K4) proportional to (1,1,1,1) T1 NS01 B16 dim(im(delta_2)) = 6 on K5 T1 Total: 389 + 4 = 393 tests ================================================================ NO MODIFICATIONS to existing text (all insertions only) ================================================================ ================================================================ PRIORITY ================================================================ Non-negotiable: Insertion 2 (topological mass gap beta_1=0) Reason: a referee trained in algebraic topology will immediately ask why Dobrushin is used when Hodge gives the answer in one line. Correct answer: Dobrushin gives the VALUE, Hodge gives the REASON. Both must be present. High value: Insertion 3 (deaf mode / P22 connection) Reason: gives the mass gap a geometric interpretation via simplicial blindness. Unifies P5/P22/P23. Nice to have: Insertions 1, 4 (Hodge dictionary, Navier-Stokes) Reason: structural clarity and second Millennium connection. Mechanical: Insertions 5, 6, 7 (bibliography, counters, companion) Reason: required by the above additions.

P5 v5 CHANGELOG ======================================== SINGLE ADDITION: Strong CP problem resolved (θ_QCD = 0). 1. Corollary IX.4 "Strong CP from contractibility" [T1] Added after Theorem IX.3 (Topological mass gap). Δ⁴ contractible → H_n = 0 for n≥1 → every G-bundle trivial → c₂ = 0 → θ F∧F has no support. Network recollement: 2-cell, chain, ring all β₁ = 0. Causal (2,1,2) prevents torus (T⁴ has β₁ = 4). 2. Companion: v4 → v5, 30 → 31 blocks, 393 → 409 tests 3. B31 bullet added to companion structure list After "B30: Structural completeness map": + B31: Strong CP (homology, Hodge, recollement, contrast) NO other changes. Body of proof untouched.

The Dobrushin gap proof of v1 is invalid (the Jaynes entropy is concave; companion B32) and is replaced by a Jentzsch/Perron-Frobenius argument requiring no convexity (Theorem thm:jentzsch, Δ > 0 for all α* > 0). All Dobrushin constants removed; numerical gap Δ = 0.934 ± 0.003 per cell (scalar sector, T2) with an explicit combinatorial oscillation bound and a Birkhoff floor. The π discussion is reformulated: all three irrationals (e, ln 2, π) are emergent through limits; the lattice performs counting and growth, never isotropy. Companion v6: 433 tests, 34 blocks, all PASS (Dobrushin tests retained as DEPRECATED). Erratum included. Topological gap theorem, Strong CP corollary, gauge construction, and Wilson sector unchanged.

v9 (July 2026) — changes from the published v6 (433 tests): Gap value: 0.934 +/- 0.003 (trapezoidal Richardson) replaced by 0.9369 +/- 0.0005 (Gauss-Legendre, converged). The trapezoidal family was biased low by the s*ln(s) endpoint kink. Entropy shift corrected from ~0.2% to +0.3%. New results: - Exact 1D reduction of the Dirichlet sector (Sec. IX.B): the nonzero spectrum equals that of an explicit 1D integral operator; the gap is created entirely by the Pauli walls. - Appendix A: rank-quantised density alphabet. Measure theorem (binomial C(3,k) derived from microstate counting). Exact finite-matrix spectra. Survival-ratio certificate R(r;q) = 0, degree 12, integer coefficients in q = exp(-1/12). Cyclotomic skeleton traced to K5 geometry. Tick rationality a* ln2 = 1/4. Structural improvements: - Mass gap on every finite volume (Thm IX.7, T1) promoted to main lattice theorem; causal transfer demoted to Corollary (T2). The gap no longer depends on the causal hypothesis. - W_f = I defence consolidated in a single Remark (XIII.4): gap independent of effective action; Pauli compactness distinguishes from Kazakov-Migdal; Yang-Mills identification is T2 by design. - Forgetting-rate interpretation of the mass gap (Discussion). - Subdominant mode characterised: lambda_2 > 0 (monotone decay), parity +1 (density mode), 96% pair-mean. - S_max by exhaustive enumeration on the rank-quantised alphabet (1024 configs, T1). Birkhoff floor promoted to T1 on the alphabet. - "No pi" thesis refined: pi enters through the erf of the 1D reduction, exits under rank quantisation. - Non-claim disclaimer added to Section XIV (Clay). Companion: P5_unified_companion_v9.py (498 tests, 43 blocks, T1 = 461, T2 = 37, T3 = 0). New blocks: B35-B38 (v7 audit), B39 (Gauss-Legendre + 1D reduction), B40 (rank-quantised enumeration), B42 (measure theorem), B43 (survival-ratio certificate). Minor bug fixes