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Original ledger classification: Numerical [N] (high-precision lattice exhibits) + Formal [E] (single-scale corollary) + [C] (meeting-point reading) + [O] (both legs open)

Recorded statement: FIRST COMPUTABLE STEPS ON THE TWO REMAINING LEGS OF THE ENTROPIC-ACTION BRIDGE (2026-07-13; after v473-v477 the open items were the AP2 continuum statement and the derivation of the KMS scale measure) -- both legs stay [O]; closes NOTHING.…

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GRAV.ENTROPIC.LEGS.01

Original ledger classification: Numerical [N] (high-precision lattice exhibits) + Formal [E] (single-scale corollary) + [C] (meeting-point reading) + [O] (both legs open)

Original claim text

FIRST COMPUTABLE STEPS ON THE TWO REMAINING LEGS OF THE ENTROPIC-ACTION BRIDGE (2026-07-13; after v473-v477 the open items were the AP2 continuum statement and the derivation of the KMS scale measure) -- both legs stay [O]; closes NOTHING. LEG A (AP2 CONTINUUM, exhibited on the exactly solvable critical chain, infinite-volume sinc kernel, 50-digit precision): [N] Calabrese-Cardy -- the compressed interval state has S(L) = (c/3) ln L + k with c_est = 1.00089, 1.00038, 1.00021 (pairs (8,16), (12,24), (16,32); monotone convergence to the Dirac c = 1 = 2 Majoranas x 1/2, consistent with the v450 reading), while a gapped control gives c_est ~ 6e-8 (area law) -- the compressed state carries the conformal/BW structure quantitatively. [N] CHM PARABOLA -- the nearest-neighbour couplings of the Peschel modular Hamiltonian K = ln((1-C_A)/C_A) organise on the CHM weight beta(x) = x(L-x)/L with correlation 0.866 -> 0.967 -> 0.986 (L = 8, 16, 24) and shrinking mid-plateau drift; constants deliberately NOT asserted (the lattice EH is only asymptotically local). [N] QUASI-LOCALITY -- at half filling the even-range bands of K vanish identically (particle-hole structure, max |K_{i,i+2}| ~ 1e-36) and the odd tail is small (mean ratio r=3/r=1 = 1.8%) with parabolic envelope. READING [C]: the state-side Delta_Sigma (the reading FORCED by v476) flows to the CHM/Bisognano-Wichmann geometric form in the continuum limit -- exactly the object TFPT's Einstein derivation consumes (the CHM ball modular Hamiltonian, v323/v358): the bridge's modular side and TFPT's gravity side MEET. NOT a continuum proof. LEG B (THE MEASURE): [E] single-scale corollary -- for dmu = delta(t - t0) dt the moments give mu2/mu1^2 = e^{t0} EXACTLY, so the v477 one-moment condition forces t0 = ln(72/17) + 9 ln(8 pi) = 30.460996... -- ONE dimensionless KMS time with an exact closed form; DECLINED explicitly: t0 is NOT h(E8) = 30 (difference 0.461; the near-miss is recorded and rejected per the v354/v355 no-free-pattern rule); no TFPT atom equals t0 -- deriving the measure from the entropic structure stays OPEN. [O] both legs remain the named open research items; no status marker moves anywhere

Original classification detail

Two-leg exhibit [N]/[E]/[C]/[O]: leg A -- the compressed critical state's modular data flows to the CHM/BW geometric form (c_est -> 1 at 2e-4, CHM parabola corr -> 0.99, even bands exactly zero), meeting TFPT's Einstein-derivation input (v323/v358); leg B -- the measure condition reduced to one exact KMS time t0 = ln(72/17) + 9 ln(8 pi) = 30.461 with the h(E8) = 30 near-miss explicitly declined; both legs stay [O]

Dependencies as recorded

GRAV.ENTROPIC.COMPRESS.01; GRAV.ENTROPIC.SCALEFLOW.01; v323; v358; v450; v473; v476; v477

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  • v478_entropic_continuum_legs.pyRegistered verification module · frontier
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    GRAV.ENTROPIC.LEGS.01: first computable steps on the two remaining legs of the entropic-action bridge. LEG A (AP2 continuum): [N] the compressed critical interval has S(L) = (c/3) ln L + k with c_est = 1.00089/1.00038/1.00021 (monotone -> Dirac c = 1; gapped control ~6e-8, area law); the Peschel modular Hamiltonian's NN couplings organise on the CHM weight beta(x) = x(L-x)/L (corr 0.87 -> 0.99, plateau drift shrinking); even-range bands vanish identically and the odd tail is 1.8% with parabolic envelope -- the state-side Delta_Sigma (v476's forced reading) flows to the CHM/BW geometric form that TFPT's Einstein derivation consumes (v323/v358): the bridge's modular side and TFPT's gravity side meet in the continuum limit. LEG B (measure): [E] single-scale corollary -- mu2/mu1^2 = e^{t0} exactly, so the v477 condition forces t0 = ln(72/17) + 9 ln(8 pi) = 30.461 (exact closed form); the near-miss to h(E8) = 30 is explicitly DECLINED (v354/v355 discipline); deriving the measure stays open. [C] meeting point; [O] both legs, no marker moves. High-precision numerical (mpmath 50 digits), Python-only by nature.

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