Publication history: 2026-08-30

Summary

The mandatory-dynamics leg of QFT4D.LATTICE.FUNDAMENTAL.01 is CLOSED at Hamiltonian-class level. Cited articles/ 2026-08-30/ tfpt_ thermodynamic_ dynamics_ en.tex/.pdf: uniform Lieb–Robinson bound J=12/5, R=2, z=4, \kappa=16043/450, v_{\mathrm{LR}}=32086e/225\sim387.64; norm-convergent \tau_t; gauge-invariant quasilocal \mathcal A^G (Gauss caveat resolved on the invariant subalgebra); existence of ground and \beta-KMS states. Numeric twin: measured v=2.0\ll proved bound; analytic finite-volume bound dominates nested differences at buffers 8/16/32; 1/r mutant ratio 4.84; Gauss-positive control commutes, violating embedding has commutator norm 1. Remaining: phase uniqueness, limit gap, IR universality. Decision typing UNCHANGED. (2) v1014_ bridge_ refinements.py (17/17): finite detline restriction isomorphism VERIFIED (c_{\mathrm{phase}} constancy 3.7{\times}10^{-16}, conjugation covariance exact, C=W_{\mathrm{bulk}}=W_{\mathrm{seam}}=+1 / mirror -1); orientation fixes only the conjugate branch (2\to1); the residual constant U(1) is the A_0 reference normalization — recorded on ALPHA.QUILLEN.EXACT.01. W-bridge: P-anch DERIVED (e_1=R^{-1}(1,1,2)^T via the winding lock); P-dem PARTIAL (the \mathbb Z_4 average annihilates the nontrivial character space); missing object = the character-blind determinant-response map — the same MMST/Quillen externalized leg as TEL-B/ALG-EXH/P1. AX.P2.01 stays an axiom. Convergence: every remaining axiom-core and continuum gap now runs through ONE externalized theorem. Surfaces: tfpt_research_contracts executed-update paragraphs; tfpt_4_frontier + introduction late-evening note; website papers.ts / VerificationDag.

Summary

T5 IR universality on the physical QWZ edge — c\to1 (limit 0.99993), clustering dichotomy \xi{\cdot}\mathrm{gap}\approx1.089 vs power-law exponent 2.034, Lieb–Robinson cone v_{\mathrm{LR}}=5/3\ge c with leakage 0.015, cubic curvature \to1/6 with volume; TYPED SPLIT: gauge holonomy + seam clock are not propagating modes, so the common-c test needs dynamical gauge excitations. T8 \rho_0 eight-candidate table ALL unique; hard sectors kill the SK response (\sim10^{-31}); every response-alive set is exactly \rho_{\mathrm{KMS}} even without \mu_4 — entropic-proximity selection is VACUOUS at finite level; verdict NO_SET_WORKS with the double diagnosis; binding requirement = a non-entropic principle (Euclidean-cap / orientation-branch). T6 \kappa six-fold NULL: six pre-declared D4-odd candidates, all parity-checked, all NULL (five at 10^{-14}; K_4=3.55{\times}10^{-3} is 18.8{\times} too large); the 1.885{\times}10^{-4} cancellation requires the true 4D functional — third and final scaffold-level nu-mechanism bound this week. T7 interacting TT: conserved interaction-complete lattice stress \sim10^{-15}, TT spectral positivity exact, sum rules \sim10^{-15}, dominant positive-Z pole persists under \phi^4 (gapped); Z trend k_1 suppressed 0.958 / k_2 enhanced 1.033; free analytic match. Missing: TFPT content, k\to0 volume scaling, Ward identities. Ledger notes only on TFPT.TOE.COMPLETE.01, FTRANSFER.SK.RHO0.01, FLAV.NU.TEXTURE.MECHANISM.01, GRAV.SPIN2.EMERGENCE.01 (all stay [O]). Surfaces: tfpt_research_contracts executed-update paragraphs; tfpt_4_frontier + introduction evening-wave note; website papers.ts / VerificationDag.

Summary

Derivation matrix over four frozen seeds, unique remainder min S_4=2; anchor (1,1,2) is not among the three declared Milnor maps; \varphi_0 four novel candidates NULL. Bridge Q CANONICAL — the bit-selector ladder forces c=2 and family n=3 forces Q column sums (9,5,1); C's carrier-looking 5 is inherited from R column 3, not inserted (the v1005 circularity flag is corrected as forced provenance). Bridge W CHOICE-but-conditionally-unique: spectrum \{5/2,1,1\} alone has 1078 solutions; democratic-image + carrier-anchor kernel leave exactly the corpus shift (count 1); remaining obligation = seam-geometric derivation of those two directional premises. Milnor strong all-structure bridge NONEXISTENT at \mathbb F_2 (clock 2 vs 4, CP rank 2 vs 4) and at every upstairs home (\mathbb Z/4, Gaussian k=2,3, affine); only the coarse D-module+pairing bridge exists (witness constructed; \lvert\mathrm{Aut}(X)\rvert=36864, \lvert\mathrm{Aut}(Y)\rvert=8). Ring-internal identities of v1004 (rank clock, socle CP, self-clock) unaffected. SEAM.MILNOR.LOCALRING.01 stays [O](canonical status SHARPENED); AX.P2.01 stays an axiom (pencil rehabilitated, W premises remain). (2) v1011_ 3p1d_ ladder_ battery.py (32/32): B6–B9 compressed — minimal 3{+}1D viable (exact Gauss 32768, confinement W(A{=}1,g{=}2)\approx0.063 vs W(A{=}1,g{=}0.8)\approx0.995, exact in-face isotropy, g_\star\approx1.915); exact tree-fixing 524288\to4096 with tree-independence \sim10^{-14}; DET survives coupling; back-reaction; periodicity restores holonomy K; charged channels \sim2.014{\times} at weak g; charge-0 control matches B8 to 10^{-10}; K collapses above g\sim1.6. BINDING: the matching point of the coupling functional must sit in the deconfined/weak regime. QFT4D.LATTICE.FUNDAMENTAL.01 Decision typing UNCHANGED (Kronecker limitation removed); GAUGE.DETLINE.FIXPOINT.01 stays [O]with the matching-scale constraint recorded. Surfaces: tfpt_research_contracts executed-update paragraphs (MILNOR sharpening, P2 rehabilitation, QFT4D ladder, DETLINE matching-scale); tfpt_4_frontier + introduction afternoon note; website papers.ts / VerificationDag.

Summary

The MMST lemma battery, the DET16 stability battery, the 2{+}1D master-object assembly scaffold, and the \rho_0 constrained minimizer; suite 998 \to 1002, ledger 1173 no new rows, existing contract rows updated, NO status-marker upgrades; Python-only / Wolfram mirror deferred as for v974–v1005). (1) v1006_ mmst_ lemma_ battery.py (68/68): five lemmas in-house on SEAM.MMST.TYPEIII.CHARGED.01 [O]— L3 crossed-product; UGF K_G=\pi^2/4 (measured plateau \sim1.023, max measured K 1.029399 at N=16); TEL-a N^{-2} with exact isometries (A_{\mathrm{TEL}}=5120(1+\pi/2), measured envelope 3.420384<8); integer D5+A3 pairings; \mathbb Z_4 outerness in the limit + finite-inner correction. Residual boxed: TEL-B-EXTERNAL (Hankel C_R; HS remainder sourced from the paper max 2.105079358 / fit 2.107565497 / v998 bound 2.11, not recomputed) + ALG-EXH (two Buchholz–Verch estimates). Display stays [O]. Cited articles/2026-08-30/mmst_charged_scaling_limit_en.tex (21 pp) + externalization_mmst_handoff_v2_en.html. (2) v1007_ det16_ stability_ battery.py (67/67): T1/T2 proved (v1002); T3 Michalakis–Zwolak cited-verified (LTQO, t_*>0, gap \ge1/2). Counting line n=2,3; n=2 closing scan through t=4; dynamical \mathbb Z_2/\mathbb Z_4 uniquely gapped (full-probe min 0.728854416 pinned; compressed suite). SW 1-\alpha t^2+O(t^4); c_{\mathrm{geo}}/L bounded. 1/192 boxed (ordinary hopping not block-diagonal); BDL boxed (no volume-uniform gap remainder). CHIRAL4D.MIRROR.DET16.01 stays [C]; CHIRAL4D.NOMIRROR.01 unmoved. Cited articles/2026-08-30/det16_mirror_gap_theorem_en.tex. (3) v1008_ master_ assembly_ scaffold.py (22/22): 2{+}1D master-object scaffold-coherent — combinatorial K=0; frozen-link q^2\sim4, K(q=0)=0; unique-contractive \lvert R'\rvert<1; QWZ wall slope \approx-0.9989, DET gap \ge1.5 through t=0.6; assembly \mathbb Z_6 (1,2,3,0), clock [0,1,1,2]. L=3 spaces 262144 / 129024 out of suite. Decision QFT4D.LATTICE.FUNDAMENTAL.01 and GAUGE.DETLINE.FIXPOINT.01 [O]unmoved. (4) v1009_ rho0_ minimizer.py (13/13): finite uniqueness of the KMS compression (Klein); Hessian strictly positive; centre dim 1; identity \sim8{\times}10^{-17}. TYPED: V1000_CONSEQUENCE_MISMATCH (compression kills the SK response vs frozen contrast 3.550{\times}10^{-3}) + CENTER_FLATNESS_PREMISE_FALSE. Canonical formula needs a weaker admissible set. FTRANSFER.SK.RHO0.01 stays [O]. Surfaces: tfpt_research_contracts executed-update paragraphs; tfpt_4_frontier wave-7 paragraph; introduction status card; website papers.ts / VerificationDag.

Summary

The Milnor–Frobenius–Gray finite algebra, the circularity-flagged relative pencil, the local-ring identification contract, the Weinberg–Witten spin-2 firewall, and honest rest/SM/gravity wording; suite 996 \to 998, ledger 1171 \to 1173: two new rows, existing contract rows updated, NO status-marker upgrades; Python-only / Wolfram mirror deferred as for v974–v1003). (1) v1004_ milnor_ frobenius_ gray.py (40/40): CLASSICAL (cite, no claim) — Milnor algebra of x^2+y^3+z^5\cong\mathbb Q[y,z]/(y^2,z^4) dimension 8, spectral numbers = E_8 exponents \{1,7,11,13,17,19,23,29\}. NEW-IN-CORPUS EXACT [E]-finite: rank clock (2-a)(4-b) table-sum 30=h(E_8); S=pqr\cdot\mu/8; self-clock iff \mu=8; (2,3,5) unique among ordered spherical triples; Galois–Gray 11/7/-1 (binary Gray order 0\to1\to3\to2); CP = socle/Frobenius; integral dual-number bridge \mathbb Z[y,z]/(y^2,z^4)\otimes\mathbb F_2 free rank 4 over D=\mathbb F_2[\varepsilon]/(\varepsilon^2), basis-dependent. INTERPRETIVE [C]: PG(3,2)=AG(3,2)\sqcup PG(2,2). Mutants: (2,3,7) S=63\neq42; A_8 self-clock survives but exponents fail; wrong socle rank 4. NEW ledger row SEAM.MILNOR.LOCALRING.01 [O]: required canonical geometric/mechanical identification of the raw seam collar / E_8 transition bus with \mathbb F_2[y,z]/(y^2,z^4) preserving dual-number action, Frobenius pairing, Gray torsor, deck action and CP (rank clock and socle-CP as outputs); kill = second inequivalent identification or \mu_4-equivariance failure; currently [O]because Milnor algebra + exponents are classical and the corpus map is unproven. (2) v1005_ relative_ pencil_ bridges.py (43/43): EXACT [E]-finite Hankel p_n p_{n+2}-p_{n+1}^2=2^{n+1} + Jacobi spec \{1,2\} diag (4/3,5/3) (v106/v119) and winding spec \{5/2,1,1\} + fixed plane (v4). EXACT BUT CIRCULARITY-FLAGGED [C]: unique t=2, \lambda=7/3, factorisation (3\lambda-7)(\lambda^2-2\lambda+2), CM J_0^2=-I, dual number mod 2, index 25=g_{\mathrm{car}}^2 SNF (1,1,25) — BUT carrier 5 already sits in the column sums of Q_2 and C: NOT a P2 selector until Q, C, \varepsilon are derived from seam geometry independently; AX.P2.01 typing UNCHANGED. INTERPRETIVE [C]: Klein-four SNF (2,2) only after D=2d (side-blind nulls v521, not a theorem). Mutants: t=0 fails; perturbed C breaks factorisation; 2+3^n breaks Hankel. (3) registrations (no marker moves). NEW GRAV.WEINBERG.WITTEN.01 [O]: binding firewall BEFORE any spin-2 marker move — the microscopic Hamiltonian family must NOT possess a fundamental Lorentz-covariant local conserved T_{\mu\nu}; Lorentz, diffeomorphism Ward identities and the spin-2 mode must emerge in the SAME IR limit; acceptance = positive TT density, two helicities, massless pole, emergent diff Ward, universal soft coupling, named evaded Weinberg–Witten premise; kill = microscopic covariant local T_{\mu\nu} together with a massless composite spin-2 claim; child of T7, cross-link GRAV.SPIN2.EMERGENCE.01 (not back). (4) display corrections (honest wording, no claim changes). Status-at-a-glance: compiler rest v_{\mathrm{geo}}\oplus G_{\mathrm{net}}\oplus F_{\mathrm{transfer}} now sits NEXT TO the strict-physical-TOE rest (ten-component \mathrm{Rest}_{\mathrm{TOE}} / eight TFPT.TOE.COMPLETE.01 gates, one line). SM completeness retyped: the discrete Standard-Model structure (charges, families, flavor operators) is derived under the axioms; 4D existence, chiral measure, mirror gap, couplings and neutrino dynamics are open. Gravity display: “parameter-free” headlines replaced by “the classical covariant field equation carries no free dimensionless Newton coupling under the named QFT/entanglement premises [C]”. Surfaces: tfpt_research_contracts (new sections MILNOR.LOCALRING, WEINBERG.WITTEN; SPIN2 firewall note; T7 child; v358/v359 keybox titles); tfpt_4_frontier wave-6 paragraph; introduction status card + SM/gravity lines; tfpt_2 gravity heading + conclusion; website StatusMatrix / WhatTfptClaims / ClaimStack / GravityEmergence / OpenGates / VerificationDag.