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The conditional entanglement route to linearized gravity

Original ledger classification: Entanglement-equilibrium gravity [E]/[C]/[O] (parameter-free linearised Einstein 8pi=1/c3; thermo=geo coincidence via |mu4|=|Z2|chi; J3 assembled; entropy density atom-fixed; residual = nonlinear + v_geo)

Recorded statement: THE ENTANGLEMENT-EQUILIBRIUM DERIVATION OF THE LINEARISED EINSTEIN EQUATION, PARAMETER-FREE (2026-06-23). Carries out the Jacobson-2015 / Faulkner-et-al. 'first law of entanglement => Einstein equation' (delta S = delta<K>) with TFPT's…

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GRAV.ENTROPY.EQUILIBRIUM.01

Original ledger classification: Entanglement-equilibrium gravity [E]/[C]/[O] (parameter-free linearised Einstein 8pi=1/c3; thermo=geo coincidence via |mu4|=|Z2|chi; J3 assembled; entropy density atom-fixed; residual = nonlinear + v_geo)

Original claim text

THE ENTANGLEMENT-EQUILIBRIUM DERIVATION OF THE LINEARISED EINSTEIN EQUATION, PARAMETER-FREE (2026-06-23). Carries out the Jacobson-2015 / Faulkner-et-al. 'first law of entanglement => Einstein equation' (delta S = delta<K>) with TFPT's atoms, and closes the two pieces that the bridge analysis (v356) left open (the matter boost flux J3 and the entropy-density coefficient). [E] PARAMETER-FREE EINSTEIN COEFFICIENT: delta S = delta<K> => G_ab = (1/c3) T_ab with 1/c3 = 8pi FIXED; all four c3-roles mutually consistent with NO free dimensionless Newton dial (Unruh factor 1/(2pi)=4c3, EH action 1/(16pi)=c3/2, Einstein coeff 8pi=1/c3, Bekenstein entropy density 1/4=1/|mu4|). [E] THERMO = GEO COINCIDENCE (the new exact result): the first law REQUIRES the Einstein coefficient 2pi/eta (eta=entropy density); the geometry gives 1/c3 = |Z2|*2pi*chi(S^2) (one-sided Gauss-Bonnet, v58); these give the SAME equation iff |mu4| = |Z2|*chi(S^2) = 4 (a TFPT atom identity, v216) -- so the thermodynamic (first-law) and geometric (Gauss-Bonnet) origins of c3 are ONE object, coupled by 4=|mu4|=|Z2|*chi. [C] J3 SOLVED (matter boost flux assembled): the ball modular Hamiltonian is the boost (Casini-Huerta-Myers; geometric via Bisognano-Wichmann, v323) K_B = 2pi int_B w T_00 with w=(R^2-r^2)/(2R) the CHM weight; the 3-ball weight integral int_B w d^3x = 4pi R^4/15, so delta<K_B> = (8 pi^2 R^4/15) delta<T_00> -- the concrete matter-side flux of TFPT's stress tensor (closing the 'ingredients present, flux not assembled' gap). [E] ENTROPY DENSITY ATOM-FIXED (the [A] coefficient pinned): the Bekenstein numerator 1/4=1/|mu4| (exact, v57) and the entanglement normalisation = the central charge c=8=g_car+N_fam (a CFT's universal area-law/entanglement coefficient IS its central charge); the area-law FORM is established for the RP-Gaussian seam (v59). So the dimensionless entropy-density coefficient is atom-determined; the only residual is the absolute area UNIT (v_geo), not a free dimensionless gap. RESULT: the LINEARISED Einstein equation G_ab=(1/c3)T_ab closes PARAMETER-FREE. [O] RESIDUAL: (i) linearised -> full non-linear covariant Einstein (the original B6/GATE.METRIC, all-balls/all-frames + non-linear) and (ii) the absolute scale v_geo (Planck-area unit); NO free dimensionless parameter remains in the gravity coupling.

Original classification detail

[E] the parameter-free Einstein coefficient + the thermo=geo coincidence (the new exact result) + the entropy-density atom-fixing; [C] J3 (cited CHM/BW ball modular Hamiltonian applied to TFPT's matter) + the area-law form; [O] linear->non-linear (B6 full) + the v_geo unit. Closes the LINEARISED Einstein equation parameter-free; advances B6/GATE.METRIC substantially; does NOT claim the full non-linear closure.

Dependencies as recorded

SEAM.EQUIV.CONTINUUM.03; v356; v57; v58; v59; v216; v221; v323; v76

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  • v358_grav_entropy_equilibrium.pyRegistered verification module · core
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    The entanglement-equilibrium derivation (Jacobson 2015 / Faulkner et al., delta S = delta<K>) with TFPT's atoms -> the LINEARISED Einstein equation falls out PARAMETER-FREE, and the two formerly-open pieces are closed. [E] parameter-free Einstein coeff G_ab=(1/c3)T_ab, 1/c3=8pi fixed (Unruh 1/(2pi)=4c3, EH 1/(16pi)=c3/2, Einstein 8pi=1/c3, Bekenstein 1/4=1/|mu4|). [E] THERMO=GEO coincidence (new): the first law's 2pi/eta = the geometric |Z2|2pi chi iff |mu4|=|Z2|chi=4 -- the thermodynamic and geometric origins of c3 are ONE. [C] J3 solved: the CHM ball modular Hamiltonian (boost via BW v323) gives the matter flux delta<K_B>=8pi^2 R^4/15 delta<T_00>. [E] entropy density atom-fixed: 1/4=1/|mu4| + central charge c=g_car+N_fam=8 (form v59). [O] residual: the equation-of-state status + the global ambient measure (C7) + the v_geo unit (v359 closes the FULL covariant equation); no free dimensionless gravity dial remains.

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