Publication history: 2026-07-13

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• v478_entropic_continuum_legs.py (GRAV.ENTROPIC.LEGS.01). Leg A (AP2 continuum, exhibited): on the exactly solvable critical chain (infinite-volume sinc kernel, 50-digit precision) the compressed interval state has S(L){=}(c/3)\ln L{+}k with c_{\mathrm{est}}{=}1.00089/1.00038/1.00021 (monotone \to the Dirac c{=}1{=}2{\times}\tfrac12, the v450 reading; gapped control \sim6{\times}10^{-8}, area law); the Peschel modular Hamiltonian K{=}\ln((1{-}C_A)/C_A) organises on the CHM weight \beta(x){=}x(L{-}x)/L (corr 0.87{\to}0.99, shrinking plateau drift; constants deliberately not asserted), the even-range bands vanish identically (particle–hole, \sim10^{-36}) and the odd tail is 1.8\% with parabolic envelope — the state-side \Delta_\Sigma (the reading forced by v476) flows to the CHM/BW geometric form, exactly the object TFPT's Einstein derivation consumes (v323/v358): the bridge's modular side and the gravity side meet in the continuum limit; a lattice exhibit, NOT a continuum proof — AP2 stays [O]. Leg B (the measure): [E]single-scale corollary — d\mu{=}\delta(t{-}t_0)dt gives \mu_2/\mu_1^2{=}e^{t_0} exactly, so the v477 one-moment condition forces t_0{=}\ln\tfrac{72}{17}{+}9\ln(8\pi){=}30.461 (one dimensionless KMS time, closed form); the near-miss to h(E_8){=}30 (difference 0.461) is explicitly DECLINED per the v354/v355 no-free-pattern rule — deriving the measure from the entropic structure stays OPEN. Keybox extended in tfpt_research_contracts; high-precision numerical module, Python-only by nature (Wolfram count unchanged, 378/378; flagged in the Wolfram README); ledger row GRAV.ENTROPIC.LEGS.01; both legs stay [O], no marker moves.

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• v477_entropic_scaleflow.py (GRAV.ENTROPIC.SCALEFLOW.01). [E]The Frullani identity upgrades to a scale-flow representation: -\ln a=\int\tfrac{dt}t(e^{-ta}{-}e^{-t}) =2\int\tfrac{d\chi}\chi(e^{-a/\chi^2}{-}e^{-1/\chi^2}) (symbolic + 10^{-20} quadrature) — Bianconi's log action IS the flat scale-integral of relative heat-kernel actions, while TFPT's S_{\mathrm{rel},\chi} evaluates one KMS scale with a weight: the bridge between the two action principles is a choice of SCALE MEASURE. [E]Moment dictionary: a weight w(t)\,dt/t gives 3\beta R\,\mu_1+\tfrac{17}{24}\beta^2R^2\mu_2; the flat weight has \mu_1{=}\mu_2{=}1 and reproduces the v475 raw coefficients identically. [E]The ONE moment condition: demanding m^2{=}\cthree^7\Mbar^2 forces exactly \mu_2/\mu_1^2{=}\tfrac{72}{17}(8\pi)^9 — the v475 kill-test number IS a moment ratio — and the closure identity \tfrac{4608\pi^2/17}{(72/17)(8\pi)^9} {=}\cthree^7 holds identically: the “13 orders” are a scale-measure datum, not a bridge mismatch. [E]v36 consistency: f_0{=}6(4\pi)^2/\cthree^7\Leftrightarrow M^2/\Mbar^2{=}\cthree^7 — one KMS moment, two parametrisations, zero new dials. [C]A consistency statement, not a derivation (the KMS weight is TFPT input; the entropic action supplies the Einstein{}+\Lambda level and the scale-flow form, TFPT supplies the measure). Negative side-record: an exact lattice-BW commutator attempt for AP2 (Peschel–Eisler simple forms) did NOT hold on the half-filled open chain — kept out of the suite, logged in the research notes. Keybox extended in tfpt_research_contracts; Wolfram mirrors both exact blocks (378/378); ledger row GRAV.ENTROPIC.SCALEFLOW.01; the R^2 gate and the v359 typing stay [O], no marker moves.

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• v476_entropic_compression_toy.py (GRAV.ENTROPIC.COMPRESS.01). The last open work package of the v473 round, exercised on a gapped staggered-mass chain (N{=}16). The seam situation is reproduced (pure bulk: the ground-state covariance is a projection to 10^{-15}; thermal boundary: \operatorname{spec}(PCP)\subset(0,1), Peschel K{=}\ln((1{-}C_A)/C_A) finite, S{>}0 — v155 in the bridge context). The reading is decided: on the pure bulk f(x){=}x/(1{-}x) is singular on \operatorname{spec}(C){=}\{0,1\}, so the literal operator-side reading P f(C) P of the v473 conjecture is ill-posed while the state-side f(PCP) is finite — the compression identity can only mean “build \Delta_\Sigma from the compressed relative metric”, which matches Bianconi's own local-metric construction (AP2 retyped, not weakened). [E]Block algebra (symbolic): [C^2]_{AA}{-}(C_{AA})^2{=}BB^t, [C^3]_{AA}{-}(C_{AA})^3{=}ABB^t{+} BB^tA{+}BDB^t — first order in the cross block vanishes identically, so for any analytic f the two readings agree to second order in the cross-cut correlations: the modular (entanglement) content lives exactly in that quadratic correction. Gap control at the bounded (covariance) level: the Bianconi-shaped bulk identity f(C_T){=}e^{-h/T} holds to 10^{-11} relative; the mismatch obeys the quadratic law d\le x^2 (d/x^2\in[0.4,1.0]) and falls with the gap in the gapped regime, the decoupled cut agrees exactly — the two readings converge in the gap-dominated regime, where TFPT operates (\Delta_{\mathrm{eff}}{>}0, v76/v337). Keybox extended in tfpt_research_contracts; numerical module, Python-only by nature (Wolfram count unchanged, 376/376; the block identity flagged in the Wolfram README); ledger row GRAV.ENTROPIC.COMPRESS.01; AP2 (the continuum seam statement) stays [O], no marker moves.

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• v475_entropic_scalaron_gate.py (GRAV.ENTROPIC.SCALARON.01). Work package 5 of the v473 bridge. [E]On a maximally symmetric background the relative curvature operator \widetilde{\mathcal R}\widetilde g^{-1} has eigenvalues exactly \{R;\,R/4{\times}4;\,R/6{\times}6\} (Bianconi's Appendix-B flattening verified symbolically, incl. the 6{\times}6 two-form block), giving the exact vacuum action \mathcal L_{\mathrm B}(R)=3\beta R+\tfrac{17}{24}\beta^2R^2+O(R^3) — the 3 re-derives her Eq. (45) (= the v473 pin), \tfrac{17}{24}{=}\tfrac12(1{+}\tfrac14{+}\tfrac16) is the exact tensorial R^2 factor. [E]With the pinned \beta'{=}\cthree/6 both mass routes (Starobinsky matching and f'/3f'') agree: the raw entropic scalaron is m_{\mathrm{raw}}^2{=}\tfrac{4608\pi^2}{17} \Mbar^2, i.e. m_{\mathrm{raw}}{\approx}51.7\,\Mbar{\approx}1.3{\times}10^{20} GeV — TRANS-PLANCKIAN, not light; v473's interpretation 2 (the TFPT scalaron as the light \mathcal G-trace mode) is dead in naive form; a viable mechanism must supply exactly \tfrac{72}{17}(8\pi)^9{\approx}1.69{\times}10^{13} in mass^2 (the kill-test number now WITH tensorial factors). [E]Domain: first log pole at R{=}1/\beta{=}384\pi^2\Mbar^2 (trans-Planckian) — the obstruction is the mass, not the domain. [E]Lorentzian-positivity witness: a timelike gradient gives the Gg^{-1} eigenvalue 1-\alpha v^2\le0 for v^2\ge1/\alpha (\ln undefined), spacelike/Euclidean stay positive — the v473 [C]caveat exhibited, the OS route sturdier. [C]Verdict: interpretation 1 survives (Einstein{}+\Lambda level), 2 is dead naive, 3 (KMS-spectral renormalisation) is the ONLY surviving R^2 route. Keybox extended in tfpt_research_contracts; Wolfram mirrors all three blocks (376/376); ledger row GRAV.ENTROPIC.SCALARON.01; no gate closes, no marker moves.

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• v474_entropic_hodge_carrier.py (GRAV.ENTROPIC.HODGE.01). Work packages 1 and 4-algebra of the v473 bridge, machine-checked on the finite carrier Fock space \Lambda^\bullet\mathbb C^5 (\dim 2^{\gcar}{=}32). [E]CAR{\Rightarrow}Clifford: the ten \Gamma's from (a_k^\dagger,a_k) satisfy \{\Gamma_a,\Gamma_b\}{=}2\delta_{ab} exactly (all 100 pairs, integer matrices); the 45 bilinears [\Gamma_a,\Gamma_b]/4 commute with \gamma{=}(-1)^N, span exactly 45{=}\dim\mathfrak{so}(10) and leak nothing from the 16-dim even subspace — the carrier half-spinor is the even exterior algebra with its spinor structure (the operator level of v2/v44/v197). [E]the Hodge–Dirac symbol: c(\xi){=}a^\dagger(\xi) {-}a(\xi) satisfies c(\xi)^2{=}-|\xi|^2\,\mathrm{Id} for symbolic \xi — Bianconi's D{=}d{+} \delta is Clifford multiplication at symbol level. [E]the fold is the 5{\to}\bar5 conjugation: the traceless \mathfrak u(5) weights of \Lambda^4 are the negatives of those of \Lambda^1, so her fiber 1{+}5{+}10 becomes the GUT half-spinor 16{=}1{+}\bar5{+}10 (the SM carrier decomposition). [E]honest obstruction: \Omega^{\le2} is not Clifford-invariant (\Lambda^2{\to}\Lambda^3 leak) — the bridge must go through the fold, never through degree truncation (a recorded constraint on the AP2 compression). [E]the Q-target enumeration (the v473 [C]conjecture decided at the algebraic level): \operatorname{Tr}Q^2{=}d\,k^2\cthree^4{=}32\cthree^4 with integer k has exactly the supports (d,k){=}(2,4),(8,2),(32,1) =\{|\Z_2|,\ \gcar{+}\Nfam{=}\operatorname{rank}E_8,\ 2^{\gcar}{=}\dim\Lambda^\bullet\} — minimal uniform solution q{=}\cthree^2{=}1/(64\pi^2) per Fock mode, bonus identity \operatorname{Tr}Q^2/\delta_{\mathrm{top}}{=}32/48{=}2/3{=}\dim\Lambda^\bullet/ \Oadm; the two-form block 10 (the problem-note “pair sector” conjecture) and the fiber 16 need irrational multipliers \sqrt{16/5}, \sqrt2 — killed in uniform form. [C]the physical choice among the three admissible supports; [O]unchanged (AP2 compression + the R^2 mechanism). Keybox extended in tfpt_research_contracts (Research Contract 2, the v473 keybox); Wolfram mirrors the enumeration and the weight-conjugation identity (373/373); ledger row GRAV.ENTROPIC.HODGE.01; no gate closes, no marker moves.

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• v473_entropic_action_bridge.py (GRAV.ENTROPIC.ACTION.01). TFPT's classical field equation G_{ab}+\Lambda g_{ab}=\cthree^{-1}T_{ab} is derived thermodynamically (v358/v359) and honestly typed “equation of state, not a from-action quantisation” [O]. Bianconi's entropic action S_{\mathrm B}=-\operatorname{Tr}_F\ln(\widetilde G\widetilde g^{-1}) (the quantum relative entropy between the spacetime metric and the matter-induced metric, Araki-related via \Delta^{1/2}=\widetilde G\widetilde g^{-1}) is a local covariant action whose variation gives Einstein + an emergent nonnegative \Lambda at low coupling — an explicit external candidate for exactly that missing variational layer. Machine-checked: [E]the carrier Hodge count (her \Omega^{0,1,2} fiber on the five-slot carrier \mathbb C^5 counts 1{+}5{+}10{=}16{=}2^{\gcar-1} {=}\dim S^+_{D_5}, while on 4d spacetime it is only 11 — the 16 requires the carrier); [E]the one quantitative pin — matching her weak-coupling 3\beta R to 1/(16\pi G)=\cthree/(2G) fixes her free constant exactly, \beta'_{\mathrm B}=\cthree/6=1/(48\pi); [E]her \Lambda_{\mathcal G}=\tfrac1{2\beta}\operatorname{Tr}(\mathcal G-I-\ln\mathcal G) is nonnegative and quadratic at the fixed point, and with \varepsilon_\star=e^{-\alpha^{-1}}Q it reproduces the closed v60 branch (3/4\pi^2)e^{-2\alpha^{-1}} with the exact target \operatorname{Tr}Q^2= 32\cthree^4=1/(128\pi^4)=\tfrac23\delta_{\mathrm{top}} ([C]until Q is constructed); [E]the R^2 kill test — the raw log expansion sits exactly 3\cthree^{-9}=3(8\pi)^9\approx 1.2\times10^{13} below the TFPT Starobinsky coefficient 1/(12\cthree^7), so a direct identification is false without a mechanism (pre-registered); [C]the compression conjecture P_\Sigma(\widetilde G\widetilde g^{-1})P_\Sigma=\Delta_\Sigma^{1/2} (the bridge proper) stays open, with Lorentzian positivity flagged (TFPT's OS route is the sturdier construction). New keybox in tfpt_research_contracts (Research Contract 2, after the v359 keybox); residual pointers updated in introduction/tfpt_4_frontier/status card. Wolfram mirrors the three exact blocks (371/371); ledger row GRAV.ENTROPIC.ACTION.01; no gate closes, no marker moves, zero TFPT knobs added.