Publication history: 2026-06-23

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• v383 (DYNAMICS.UNIVERSAL.01) — the universal spectral-gap / Perron–Frobenius principle. A fresh bird's-eye pass found that every TFPT sector is the same structural object: a gapped operator with a unique leading attractor (the physics) and a spectral gap (the reason there is no free parameter). v303 named this only for F_{\mathrm{transfer}}; this module extends it to the sectors made parameter-free after it. [E]three gaps are computed — the flavor transfer \{1,(2/3)^6,(1/3)^6\} with gap 6\ln(3/2) (v56/v82); the affine-E_8 adjacency with Perron eigenvalue 2 (Kac marks) and subleading 2\cos(\pi/5)=\varphi, gap 2-\varphi=1/\varphi^2 (v312/v313); the decoupling margin \Delta_{\mathrm{eff}}\approx1.648 (v76/v337). [E]the two number-field facets (v314): golden \varphi\in\mathbb Q(\sqrt5) (carrier-5, static) and (2/3)^6\in\mathbb Q (family-3, dynamic) are the prime-5 and prime-3 facets of the order-30 Coxeter clock 30=2{\cdot}3{\cdot}5=h(E_8). [C]the other sectors are the same structure — gravity (\delta S{=}0 equilibrium, v358/v359), the QG measure (Gaussian saddle, v365), the boundary QFT (holomorphic c{=}8 RG fixed point, v157/v158), the all-order S-matrix (adiabatic limit, v381), recovery (Petz fixed point, v221), \tau{=}i (v333). [C]the meta-theorem: gapped \Rightarrow unique attractor \Rightarrow parameter-free, theory-wide. Cited in origin_theory; a synthesis (no new number), Python-only.
• v384 (RESIDUAL.CERTIFICATION.01) — the residual is certification, not construction. After v369/v381/v382 the kind of every open item has shifted: there is no open TFPT physics mechanism left. A ledger-CI audit classifies every residual into exactly three external kinds: [E](A) external math proof (SEAM.EQUIV.01 continuum existence = the cited MMST/Adamo theorem v336; ALPHA.QUILLEN.EXACT.01 v382; QG.AMB.01 the constructive measure, a [C]redundancy v369), [E](B) theorem-forbidden (v_{\mathrm{geo}}, the No-Unit theorem v153/v364), [E](C) external physics (F_{\mathrm{transfer}}, v371–v374, firewalled v187). The two load-bearing facts are verified (No-Unit dimensionlessness; the gap-decoupling margin 1.648>0), and the count of open internal TFPT mechanisms is zero. [C]convergence: the only hard things left are theorems and external physics, not missing dynamics. Cited in tfpt_research_contracts; an audit (no new number), Python-only.

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• v381 (QFT4D.EG.ALLORDER.01) — the all-order 4D perturbative-QFT closing statement. A mathematical-physics review of the QFT/TOE layers identified exactly one genuinely new, buildable item: the all-order Epstein–Glaser/BRST contract for the perturbative 4D S-matrix S_{\mathrm{pert}} (the previous documents had the S_{\mathrm{pert}} skeleton and one-loop, v269/v271/v273/v278, but no named all-order claim). This module types the standard causal-perturbation-theory closure for the TFPT finite interaction class — it does not build a path integral. [E]power-counting: every gauge+matter vertex (Yang–Mills + Dirac–Yukawa + Higgs + ghost) is a mass-dimension-4 marginal operator, so the EG singular order \omega\le4 and the counterterm space is finite at every order; [E]BRST nilpotency s^2{=}0 verified exactly via the Jacobi identity for su(2) and su(3) (the carrier weak+colour content); [E]the seam gap \Delta=6\log\tfrac32>0 (v302) gives the IR-safe adiabatic limit. [C]the two heavy legs are imported rigorous theorems — all-order T_n existence by causal factorisation (Epstein–Glaser 1973) and all-order BRST invariance (Piguet–Sorella / Kugo–Ojima) — applied to the TFPT class. Matter+gauge only: the R^2/Weyl^2 gravity sub-sector is the entire-form-factor resummation (v304/v370/v380), and S_{\mathrm{pert}} is not the ambient measure QG.AMB.01 (a [C]redundancy, v369). NET [C], prerequisites [E]; not [E]and not a path integral. Cited in tfpt_5_redteam and tfpt_2; Python-only.
• v382 (ALPHA.QUILLEN.EXACT.01) — the Quillen determinant-line variation, named as a target. Per the same review, the “why this F_{U(1)} functional” obligation (previously tracked only as EM.WARD.01's residual, v341/v342) is elevated to its own citable target: \delta_\tau(\log\det_\zeta\Delta_{U(1)}+8b_1\cthree^6\log\varphi_{\mathrm{seam}})=0 on the determinant line over the \mu_4 seam moduli. [E]the Quillen split holds at the \ainv{=}137.0359992 root with every coefficient a named atom (index 8b_1, heat-kernel \cthree^{3,6}, discriminant q(D_5){+}q(A_3){=}2); [E]the value stays closed (v3); [O]the from-first-principles exactness proof is the target. [C]it is a face of SEAM.EQUIV.01 (v336) — the seam U(1) determinant line lives on the same holomorphic net — so it adds no second independent gate; it makes the existing residual trackable. No new number (reuses the Wolfram-mirrored v341 identities). Cited in tfpt_1_architecture_e8; Python-only.

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• v369 (QGAMB.REDUNDANCY.01) — Track 2 (forward plan v2). Instead of constructing the non-perturbative ambient measure (the conformal-factor swamp, v332/v334), assemble the established discriminators that prove the ambient sector carries no physical degrees of freedom relative to the holomorphic boundary algebra, so the QG measure (gate C7/QG.AMB.01) is a non-fundamental certification object (like a coordinate choice in GR), not missing dynamics. [E]holomorphy \Rightarrow \mathrm{DHR}((E_8)_1)=\mathrm{Vec}: \#primaries =|\det\mathrm{Cartan}|=1 for E_8 vs 4 for SO(16) \Rightarrow no nontrivial superselection charge a bulk could carry invisibly; [E]no torus ground-state degeneracy (|\det K|=1); [E]finite Petz recovery rate (2/3)^6 (v221); [E]the gap-decoupling margin \Delta_{\mathrm{eff}}=6\log\tfrac32-\tfrac{31}{4\pi^2}\approx1.648>0 (v76/v337). [C]with the Bisognano–Wichmann reconstruction map (v258/v309/v323) the redundancy statement \mathcal O_{\mathrm{phys}}^{\mathrm{bulk}}/\mathrm{Diff}\cong\mathcal A_\Sigma^{\mu_4} is well-posed. [O]residual: the two consumed premises — SEAM.EQUIV.01 (the S3 lattice input, v366) and the BW intrinsicality of the raw collar (v329); it does not construct QG.AMB.01 but provides the alternative (holographic redundancy) route: Gravity-complete = Boundary-complete + Ambient-redundancy. Cited in tfpt_research_contracts; Python-only.

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• Deep-sync of the post-S3 status. The canonical surfaces (tfpt_status, introduction) were already current; this pass propagated the same honest wording into origin_theory, tfpt_1–tfpt_5, tfpt_research_contracts, the root README.md, the Zenodo description and website/lib/papers.ts: SEAM.EQUIV.01 is closed modulo a cited theorem (lattice v367/v368 + the S_3 stack v376–v379 + Lean FORM.SEAM.MMST.01; residual [O]= the cited MMST/OS existence only); QG.AMB.01 is a [C]redundancy (v369/v379), a certification object not missing dynamics; perturbative graviton unitarity is established (the Stelle ghost is a Seeley–DeWitt truncation artefact, v304/v370/v380); and the frontier transfers are typed runnable solvers with kill tests (v371–v374) + the observatory CI (v375). No new claim — a wording propagation so every surface matches the ledger; AUDIT OK.

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• v380 (GRAV.KMS.HESSIAN.01). Upgrades v304/v370 from the assumption “the resummed graviton form factor is entire” to a derived statement. [E]the seam KMS cutoff f(u){=}e^{-u} (v259) gives the dressed spin-2 propagator e^{-p^2/M^2}/p^2 = 1/(p^2 a) with a(u){=}e^u entire and nowhere zero (only the p^2{=}0 pole). [E]each finite Seeley–DeWitt truncation is a Taylor partial sum of a and carries a Stelle-ghost zero (T_1{=}1{+}u\to u{=}-1; T_2\to-1{\pm}i); the entire a has none. [E]the modulus of the nearest truncation-zero grows monotonically with the order (1,1.41,\dots,3.07 for n{=}1..8) \to\infty — the ghost decouples, “entire” = “do not truncate”. [C]\Rightarrow perturbative graviton unitarity; [O]the exact off-shell curved-space Hessian identification is the cited heat-kernel resummation, perturbative-only. Cited in tfpt_5_redteam (Target F); Python.

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• FORM.SEAM.MMST.01 — Lean 4 formalisation. The S3 continuum leg is now machine-formalised in a new module (SeamScalingLimit.lean), lake build OK and #print axioms clean, in the audit-contract style of SeamEquivChain.lean. The MMST applicability hypotheses for the seam collar are provable by kernel decide (no axioms): D = 16, \operatorname{rank} = c = 8, the range 8 \le 8 \le 16, c_- = 8 \neq 0, and \det K = 1 vs SO(16)'s 4. The MMST scaling-limit theorem (CMP 2022) and the Adamo–Moriwaki–Tanimoto OS reconstruction (2024) are named cited axioms; the conclusion (E_8)_1 follows by a well-typed composition whose residual (#print axioms) is exactly those two published theorems plus the carrier gap — no hidden assumption, no sorry. Mirrors the S3 closure stack v376–v379; so SEAM.EQUIV.01 is now “closed modulo a cited theorem”, the honest formalisation ceiling. Cited in tfpt_research_contracts; lean_manifest refreshed.

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• v376 (SEAM.S3.CENTRALCHARGE.01). The Calabrese–Cardy finite-size entanglement-entropy scaling of the critical free-fermion content gives c=1.0000 per complex mode (Peschel correlation-matrix EE; the L{=}4m{+}2 sizes avoid the half-filling zero-mode), so the 16-Majorana (=8 complex) collar has c=8=\gcar{+}\Nfam, chiral (c_-{=}8, v367/v368). Numerical c{=}8 from the lattice. Python (numpy).
• v377 (SEAM.S3.E8CHARACTER.01). The exact affine character \chi = E_4/\eta^8 = q^{-1/3}(1 + 248q + 4124q^2 + \dots) gives c = 8 and 248 level-1 currents (one primary); the \mu_4/GSO promotion 248 = 240 + 8 = 120 + 128 (240 = 16{\cdot}5{\cdot}3) pins (E_8)_1 over the same-c rival SO(16)_1 (120 currents, 4 primaries). Python (sympy q-series).
• v378 (SEAM.S3.MODULAR.01). The genus-1 discriminator: KLM condensation \mu(\text{carrier}){=}16\to\mu(E_8){=}16/4^2{=}1, |\det\mathrm{Cartan}(E_8)|{=}1 vs D_8{=}4, and the single character's modular T-eigenvalue e^{-2\pi i/3} give torus ground-state degeneracy 1 (holomorphic =(E_8)_1) — the “hard part” v344 isolated. Python (sympy).
• v379 (SEAM.S3.RP.01). Reflection positivity (the OS axiom) of the explicit gapped collar: the positive spectral measure makes the Hankel/reflection Gram matrix PSD (a ghost mode breaks it). With c{=}8 + (E_8)_1 in hand, the OS inputs are complete and C7/QG.AMB.01 is then discharged by the redundancy theorem v369. Python (numpy).
• Net: S3 is the master key — on the explicit lattice model the target net (E_8)_1 is pinned exactly (central charge, character, genus-1 count, RP); the only remaining residual is the abstract continuum existence of the scaling limit (the cited MMST theorem v336), which lives outside the computational suite. Cited in tfpt_research_contracts.

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• v375 (OBSERVATORY.REGISTRY.01). A CI over freeze_file.csv: [E]every prediction carries a kill criterion (falsifiability complete, no orphan claims); [E]the headline values re-derive from \{\phiz\} (\sin^2\theta_{12}{=}1/3{-}\phiz/2, \sin^2\theta_{13}{=}\phiz e^{-5/6}, \beta_{\mathrm{rad}}{=}\phiz/(4\pi)); [C]a live scorecard records \theta_{12} vs the JUNO first run (0.28\sigma, compatible), \theta_{13} vs NuFIT 6.0 NO (2.0\sigma, the flagged most-tensioned core prediction), \beta_{\mathrm{rad}} vs ACT DR6 (0.37\sigma, a hint), r<0.036 (compatible). Tooling that makes the closed pieces usable and the open ones honestly fenced; no new physics. Cited in tfpt_5_redteam; Python (stdlib).

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• v371 (FR.POLE.SOLVE.01). A clean negative: standard QED/EW running cannot be the Koide source\topole transfer (it pushes Q above 2/3; the required \epsilon is negative while QED's is positive), so the transfer is the non-QED 53/54 operator / (2/3)^6 Moebius generator (v183/v82). [C], plain running excluded as the kill. Python (numpy+scipy).
• v372 (FR.BOLTZMANN.SOLVE.01). The integrated Buchmueller–Di Bari–Pluemacher Boltzmann ODE at the TFPT-frozen M_1{=}M_{\mathrm{scal}}(\phiz)^2/A_\Lambda (v212) gives \eta_B within a factor {\sim}1.1 of 6.1{\times}10^{-10} with no M_R dial; replaces the v326 toy washout. [C]/[X].
• v373 (FR.RELIC.SOLVE.01). The finite-T axion misalignment ODE decides the branch: the frozen spine angle 3\pi/5 gives \Omega_a h^2{\sim}0.12 (in [0.08,0.16], lands on \Omega_{\mathrm{DM}} untuned) while the hilltop 170^\circ over-produces ({\sim}5.4\times); replaces the v326 toy. [C]/[X] (lattice \chi(T) input).
• v374 (FR.QCD.BUDGET.01). The m_p/m_e uncertainty budget: with b_3{=}-7 [E]and the two declared externals (\alpha_s(M_Z), C_p, {\sim}5.3\% each), the central {\sim}1824 (v262) carries a propagated {\pm}7.5\% band [\sim1690,\sim1960] CONTAINING 1836.15; lattice tightening is the kill. The falsifiability companion to v262; [C].

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• v367 (SEAM.S3.LATTICE.01) — Track 1. The abstract S3 residual of v356/v366 (“the collar realised as a genuine lattice chiral free-fermion invertible phase”) is turned into a RUNNABLE model. [E]the p{+}ip Bloch model is gapped at M{=}1 with a numerically computed Fukui–Hatsugai–Suzuki Chern number |C|{=}1 (C{=}0 trivially at M{=}3); [E] N_{\mathrm{Maj}}{=}\dim S^+{=}16 copies give c_-{=}16/2{=}8{=}\gcar{+}\Nfam, with the edge K-matrix SO(16)_1 (\det 4) \to(E_8)_1 (\det 1) by the order-4 \mu_4 clock (v351/v369). [C]the \mu_4 condensation (v154/v351); [O]the continuum scaling-limit existence (MMST, v336) stays open. Cited in tfpt_research_contracts; Python (numpy + sympy).
• v368 (SEAM.S3.INFLOW.01) — Track 1. The bulk-edge / anomaly-inflow leg (S4) made concrete on the v367 model: [E]a finite strip has an edge-localised in-gap chiral branch (\min_{k_x}|E|\sim0) in the topological phase and a clean gap in the trivial phase (neg control); [E] bulk-edge correspondence gives one chiral Majorana per edge (c_-{=}8). [C]so by anomaly inflow the chiral edge EXISTS as a consequence of the invertible bulk, discharging the v336/v351 “does the edge exist?” residual on the lattice. [O]the continuum scaling limit stays open. Cited in tfpt_research_contracts; Python (numpy).

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• v370 (GRAV.SPIN2.UNITARITY.01) — Track 2. Extends v304 (the scalar 1/(p^2(p^2{+}M^2)) pole algebra) to the actual tensor spin structure. [E]the Barnes–Rivers projectors are complete (\operatorname{tr}P_2{=}(d{-}2)(d{+}1)/2, \operatorname{tr}P_1{=}d{-}1, \operatorname{tr}P_{0s}{=}\operatorname{tr}P_{0w}{=}1 sum to d(d{+}1)/2; d{=}4 split 5{+}3{+}1{+}1{=}10), the EH propagator is P_2/p^2-\tfrac1{d-2}P_{0s}/p^2, and the Stelle ghost is a spin-2 state. [E]the entire KMS form factor e^{p^2/M^2} (v304/v259) makes the spin-2 sector ghost-free; [C]the wrong-sign spin-0 conformal mode (v332) is cured separately by the GHP contour (v334). [C]so the full perturbative graviton is ghost-free sector-by-sector (conditional on v304's entire-analyticity assumption). [O]residual: perturbative-only + the open analyticity assumption; not the non-perturbative measure. [O]DECLINE: \operatorname{tr}P_2{=}5 is dimension counting, not \gcar (v354/v355). Cited in tfpt_5_redteam (Target F); Python-only.

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• v365 (QGAMB.SADDLE.01) — Direction 3. With the parameter-free saddle (v358/v359) the QG measure (gate C7/QG.AMB.01) organises as a one-loop Gaussian fluctuation determinant. [E]the fluctuation stiffness is 1/\cthree=8\pi (no free dial, so the one-loop problem is now well-posed); [E]the fluctuation operator is gapped M\ge\Delta_{\mathrm{eff}}=6\log\tfrac32-\tfrac{31}{4\pi^2}\approx1.648>0 (v76/v337) \Rightarrow mode-by-mode convergence, finite model log-det \operatorname{Tr}'\log M=6\log\tfrac92; [C]the conformal mode converges on the GHP contour (v334); [E]the admissible projective limit exists (\chi=729/665, v330) with 0 new dials (v364). [O]residual: the full non-perturbative projective limit on the ambient sector. Sharpens C7 from “diffuse full QG” to a one-loop Gaussian problem around the parameter-free saddle; does not close it. Cited in tfpt_research_contracts; Python-only.
• v366 (SEAM.MMST.INCLASS.01) — Direction 7. The seam collar is verified IN MMST's free-lattice-fermion scaling-limit class hypothesis-by-hypothesis: [E]D{=}\dim S^+{=}16 Majoranas, c{=}\gcar{+}\Nfam{=}8, \operatorname{rank}E_8{=}8 so \operatorname{rank}\le c\le D is 8\le8\le16 (in range); [E]gapped (\Delta{=}6\log\tfrac32{>}0, derived v302); [C]quasi-free CAR class (v155/v160); [E]chiral (c_-{=}D/2{=}8\neq0). [C]the MMST scaling-limit theorem then applies, and [E]the order-4 \mu_4 clock pins the target to (E_8)_1 (\det K{=}1 vs SO(16) \det K{=}4, v351). [O]residual: the single S3 input (the collar as a genuine lattice chiral invertible phase, the seam one-sidedness, v297/v356). Reduces the continuum side of SEAM.EQUIV.01 to one named input; ADVANCES, does not close. Cited in tfpt_research_contracts; Python-only.

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• Deep-sync of the parameter-free gravity status. A cross-surface audit found several documents and website components still framing gravity as “exact only in the R+R^2 regime” or the metric-sector field equation as “open”. These were reconciled with the canonical post-v358/v359 status: the local covariant field equation G_{ab}+\Lambda g_{ab}=\cthree^{-1}T_{ab} is PARAMETER-FREE (both coefficients fixed); what remains open is the Jacobson equation-of-state status, the global ambient measure (QG.AMB.01/C7), and the single unit v_{\mathrm{geo}}. Touched: introduction, tfpt_2_standard_model, tfpt_3_e8_audit_bootstrap, tfpt_4_frontier (summary table), tfpt_research_contracts, origin_theory.
• New v359 keybox added to tfpt_research_contracts (full covariant equation: fixed-volume \Rightarrow Einstein tensor, Lovelock \Rightarrow conservation an output); the v358 residual corrected from “linear\tononlinear” (stale — v359 closed it) to the equation-of-state + C7 +\,v_{\mathrm{geo}} residual, including the generated script index (script_registry.csv).
• Website. The GravityEmergence animation (three origins of \cthree converging to 1/(8\pi); the parameter-free Einstein equation) was updated to the full covariant result and surfaced as a dedicated homepage section (/#gravity); gravity-status wording aligned across ReconstructionChain, VerificationDag, StatusMatrix, WhatTfptClaims, ThreeDecoderMap, StatusPyramid, papers.ts, glossary.ts and the review page.

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• v364 (VGEO.SHARPEN.01) — Direction 6. The No-Unit Theorem (v153) already made v_{\mathrm{geo}} a forced metrology primitive. [E]The SHARPENING after the parameter-free gravity (v358/v359/v361): the Einstein coefficient 1/\cthree=8\pi and the \Lambda prefactor (8\pi)^2{\cdot}48\cthree^4=3/(4\pi^2) are dimensionless, so 1/G\sim v_{\mathrm{geo}}^2 and \Lambda\sim(\text{dimensionless})\,v_{\mathrm{geo}}^4 reduce to v_{\mathrm{geo}} times atoms — the gravity sector adds no new scale. [E]FINAL TALLY: every dimensionful quantity = (a dimensionless ratio)\times(a power of v_{\mathrm{geo}}); TFPT's free content is \{v_{\mathrm{geo}},\pi\} with ZERO dimensionless dials, a {\sim}26\to1 reduction vs the SM. [O]v_{\mathrm{geo}} stays irreducibly primitive (No-Unit; 1/G UV-sensitive, Sakharov v68). Direction 6 is clarity — the last dimensionful anchor named and shown to be the only one. Cited in tfpt_research_contracts; Python-only.

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• v361 (GRAV.BACKREACT.01) — Direction 5. With the parameter-free equation (v359) and the explicit J3 flux (v358), the backreaction of TFPT's carrier matter is computed. [E]the carrier's gravitational anomaly is forced: c_-=16/2=8=\gcar{+}\Nfam (the chiral central charge of the 16-Majorana content). [E]the backreaction is FINITE \to\Lambda from \alpha: the matter vacuum energy through G_{ab}=\cthree^{-1}T_{ab} is gap-suppressed (Decoupling Thm v337) to \rho_\Lambda/\Mbar^4=(3/(4\pi^2))e^{-2\alpha^{-1}} (v60), the 123 orders =119.03+3.92 — not the M_{\mathrm{Pl}}^4 catastrophe; the loop (v359+v60 + the carrier vacuum) closes. [E]J3 explicit: \delta\langle K_B\rangle=(8\pi^2R^4/15)\delta\langle T_{00}\rangle, carrier T_{ab}= the SM's (v159). [C]the SM trace anomaly is reproduced (content = SM). [O]DECLINE: no new atom-forced read-out of individual SM quantum numbers beyond c_-{=}8 and \Lambda — per v354/v355 declined. The backreaction is finite and consistent, adding no new dial. Cited in tfpt_4; Python-only.

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• v360 (GRAV.GAPCORR.01) — Direction 2, run honestly. Does the seam transfer gap \{1,(2/3)^6,(1/3)^6\} yield a NEW, calculable, FORCED deviation from GR (a near-term kill test) beyond the known R^2 scalaron? With the v354/v355 discriminator the answer is no, and this is recorded rather than manufacturing a coefficient. [E]the leading higher-curvature term IS the R^2 scalaron (spectral-action a_4{=}R^2/72, v36; M_{\mathrm{scal}}^2/\Mbar^2=\cthree^7, exponent 7{=}\gcar{+}\Nfam{-}1) — the calculable beyond-Einstein correction, already in TFPT. [C]it is the entanglement-equilibrium NEXT order (the Wald first law \to f(R){=}R{+}R^2, v28/v359). [E]it is ALREADY falsifiable via n_s{=}1{-}2/N_\star, r{=}12/N_\star^2 (live CMB-S4 kill test). [E]the gap \Delta{=}6\ln\tfrac32 is a dimensionless rate, so a distinct gap-induced curvature^2 term sits at \xi\,v_{\mathrm{geo}}\sim the seam/Planck scale (not near-term observable). [O]DECLINE: a new dimensionless coefficient is not atom-forced (the scalaron exponent 7{=}g{+}N{-}1 and the gap exponent 6{=}2\Nfam are different; \cthree^7 and (2/3)^6 unrelated) — per v354/v355 declined. Direction 2 yields no new near-term forced prediction; the answer is the existing scalaron. Cited in tfpt_4; Python-only.

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• v359 (GRAV.NONLINEAR.01) — Direction 1, the full covariant equation. Extends v358 from the linearised (fixed-radius, Ricci-level) result to the FULL covariant equation. [C][E]Jacobson 2015: demanding stationarity of the small-ball entanglement entropy at fixed volume brings in the EINSTEIN TENSOR G_{ab} (not just R_{ab}) — trace g^{ab}G_{ab}=(1-d/2)R=-R in d{=}4, the structure that carries \Lambda. [E] Lovelock: G_{ab} is the unique symmetric divergence-free second-order metric tensor, so \nabla^a(G_{ab}+\Lambda g_{ab})=0 forces \nabla^a T_{ab}=0 — conservation is an output. [E]coefficient 1: 8\pi G\to 8\pi=1/\cthree fixed (v358). [E]coefficient 2: Jacobson leaves \Lambda free, TFPT fixes it — \rho_\Lambda/\Mbar^4=(3/(4\pi^2))e^{-2\alpha^{-1}}, prefactor (8\pi)^2{\cdot}48\cthree^4=3/(4\pi^2) (v60); so BOTH coefficients are TFPT-fixed, not free integration constants. [C]all timelike directions \to the covariant tensor equation; the matter side is the assembled CHM ball modular flux (v358/v323). NET: the original B6 “full covariant Einstein-side field equation” is now PARAMETER-FREE at the local level. [O]residual: the Jacobson equation-of-state status (not a from-action quantisation) + the global ambient measure (C7/QG.AMB.01, separate, gap-decoupled); v_{\mathrm{geo}} the one unit. Cited in tfpt_research_contracts; mirrored in Wolfram (303/303). (next-plan.md Direction 4 folded in: the \Lambda here matches the \alpha-readout.)

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• v358 (GRAV.ENTROPY.EQUILIBRIUM.01) — the bridge to the dynamic world, instantiated on the gravity side. Carrying out the Jacobson-2015 / Faulkner-et-al. “first law of entanglement \Rightarrow Einstein equation” with TFPT's atoms. [E]PARAMETER-FREE COEFFICIENT: \delta S=\delta\langle K\rangle gives G_{ab}=\cthree^{-1}T_{ab} with \cthree^{-1}=8\pi fixed; all four \cthree-roles consistent (Unruh 1/(2\pi){=}4\cthree, EH 1/(16\pi){=}\cthree/2, Einstein 8\pi{=}1/\cthree, Bekenstein 1/4{=}1/|\mu_4|) — no free dimensionless Newton dial. [E]THERMO = GEO COINCIDENCE (new): the first law's coefficient 2\pi/\eta equals the geometric |\Z_2|2\pi\chi iff |\mu_4|=|\Z_2|\chi=4 (v216) — the thermodynamic and geometric origins of \cthree are ONE. [C]J3 SOLVED: the CHM ball modular Hamiltonian (boost via BW, v323) gives the matter flux \delta\langle K_B\rangle=\tfrac{8\pi^2R^4}{15}\delta\langle T_{00}\rangle. [E]ENTROPY DENSITY ATOM-FIXED: 1/4{=}1/|\mu_4| (v57) + central charge c=\gcar{+}\Nfam=8 (form v59). [O]residual: linear\tofull non-linear (the original B6) + the v_{\mathrm{geo}} unit. Closes the LINEARISED Einstein equation parameter-free; cited in tfpt_research_contracts; mirrored in Wolfram (302/302).

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• v356 (SEAM.EQUIV.CONTINUUM.03) — Direction A, the continuum chain assembled. After the reverse-audit/bandwidth work showed no hidden discrete lever remains, the only fundamental progress is to assemble the continuum chain. Link-by-link: S1 the gapped (\Delta{=}6\ln(3/2){>}0, v302) quasi-free 16-Majorana collar (v155/v160) \to S2 invertible (gapped free fermions have no intrinsic topological order, Kitaev, v301) \to S3 the nontrivial invertible phase (c_-{=}8{\neq}0) \to S4 a non-gappable anomalous chiral edge (anomaly inflow / bulk–edge), so edge existence is a CONSEQUENCE of S2+S3, not a separate assumption — this directly reduces the v336/v351 “does the collar have a chiral edge?” residual \to S5 its chiral-CFT scaling limit (MMST, v336; range \operatorname{rank}\le c\le D, 8\le8\le16) \to S6 the order-4 \mu_4 clock + the AQFT stack pin (E_8)_1 (v351/v297). [E]the only non-theorem link is S3, and [C]S3 is the seam being one-sided (chiral, c_-{\neq}0) — the same one-sidedness that makes \cthree{=}1/(8\pi) a one-sided Gauss–Bonnet (|\Z_2|\oint K, v58). [O]the one remaining input is “the abstract collar is realized as a genuine lattice chiral free-fermion invertible phase” (the v297 Flat-Away). NET: SEAM.EQUIV.01 reduced to ONE realization input tied to the constant that defines the theory; it ADVANCES, it does not close.
• Direction B — found ALREADY closed by v40. The U-wall “transcendental Hitchin solve” (v31) is the worst case for a generic Higgs bundle; but at the polystable point TFPT selects, Mehta–Seshadri makes the bundle unitary, so the harmonic metric is the constant invariant Hermitian form (finite linear algebra, Higgs \Phi{=}0) and \det R{=}8 is read off — not a PDE. Only the absolute amplitude U_{\mathrm{point}} (an anchor) remains. A redundant new B module was built, found to duplicate/contradict v40, and deleted (no padding). Honest outcome: B needs no new work.