A registered result in its source context

GRAV.NONLINEAR.01

Original ledger classification: Full covariant Einstein equation [E]/[C]/[O] (parameter-free: 8pi=1/c3 + Lambda from alpha; Einstein tensor via fixed-volume equilibrium + Lovelock conservation; B6 closed at local level; residual = equation-of-state status + C7 global measure)

Original claim text

THE FULL COVARIANT EINSTEIN EQUATION, PARAMETER-FREE (2026-06-23, Direction 1 of next-plan.md, extends GRAV.ENTROPY.EQUILIBRIUM.01/v358). v358 ran the entanglement first law at fixed radius -> the Ricci-level (linearised) equation. Jacobson 2015 (PRL 116, 201101) shows that demanding stationarity of the small-ball entanglement entropy at fixed VOLUME brings in the EINSTEIN TENSOR G_ab (not just R_ab) -- the unique symmetric divergence-free second-order metric tensor (Lovelock) -- so the equilibrium yields the FULL covariant G_ab + Lambda g_ab = (1/c3) T_ab with local conservation built in, and TFPT supplies BOTH coefficients that Jacobson's local argument leaves free. [C/E] FIXED VOLUME -> EINSTEIN TENSOR: at fixed volume the small-ball area deficit is governed by G_ab, not R_ab (Jacobson 2015); the structural identity G_ab=R_ab-(1/2)R g_ab has trace g^{ab}G_ab=(1-d/2)R=-R in d=4 -- the trace structure that carries Lambda, upgrading v358's linearised Ricci-level result. [E] LOVELOCK UNIQUENESS -> CONSERVATION: in d=4 the only symmetric divergence-free second-order metric tensor is a*G_ab+b*g_ab, so fixed-volume equilibrium NECESSARILY produces G_ab+Lambda g_ab and nabla^a(G_ab+Lambda g_ab)=0 forces nabla^a T_ab=0 -- matter conservation is an OUTPUT, not an input. [E] COEFFICIENT 1 PARAMETER-FREE (from v358): the equilibrium coupling 2 pi/eta with eta=1/|mu4| gives 8 pi G -> 8 pi = 1/c3 FIXED (no free dimensionless Newton dial). [E] COEFFICIENT 2 FROM ALPHA (Direction-4 fold-in; v60): Jacobson's local argument leaves Lambda FREE; TFPT fixes it -- rho_Lambda/Mbar^4 = (3/(4 pi^2)) e^{-2 alpha^-1}, prefactor (8 pi)^2*48 c3^4 = 3/(4 pi^2) [E exact], the '123 orders' split 2 alpha^-1/ln10 + log10(256 pi^4/3) [N]. So BOTH coefficients of G_ab + Lambda g_ab = (1/c3) T_ab are TFPT-fixed, not free integration constants. [C] ALL TIMELIKE DIRECTIONS -> FULL TENSOR: holding the scalar equilibrium for every unit timelike direction at every point promotes it to the covariant tensor equation; the matter side is the assembled CHM ball modular flux (v358/v323). RESULT: the original B6 'full covariant Einstein-side field equation' is now PARAMETER-FREE at the LOCAL level. [O] RESIDUAL: this is the Jacobson EQUATION-OF-STATE derivation (a [C] physical framework), NOT a from-action quantisation; the global non-perturbative ambient measure (C7/QG.AMB.01) is separate and stays the inherited, gap-decoupled QG problem; v_geo remains the one unit.

Original classification detail

[E] the Einstein-tensor structure (trace=-R d=4) + Lovelock conservation + both parameter-free coefficients (8pi=1/c3, Lambda prefactor 3/(4pi^2)); [C] the Jacobson-2015 fixed-volume machinery + the all-directions quantifier; [O] the equation-of-state status + the global measure (C7). Delivers the full LOCAL covariant equation parameter-free; advances B6/GATE.METRIC from 'linearised (v358)' to 'full covariant local'; does NOT claim a from-action QG quantisation.

Dependencies as recorded

GRAV.ENTROPY.EQUILIBRIUM.01; v358; v60; v323; v57; v58; v216