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.