The foundational paper of the Absolute Frame Theory (AFT) retains the Einstein–Hilbert term of its written action rather than deriving it, and defers "a thermodynamic derivation of the field equations from the M–A channel" to separate work. This paper supplies that derivation. The instrument is the Clausius relation applied to all local causal horizons, due to Jacobson; the content of every hypothesis of the argument is native to the framework. The horizon temperature is the channel temperature, derived once at the level of the M–A channel from the Euclidean period, with the factor 2π arising from that period rather than postulated. The entropy is the entanglement of Axiom IV with the inaccessible fiber, carried by the Nyquist cells that straddle the horizon, with capacity ln (nₘax + 1) per cell; that the straddling cells operate at capacity is derived rather than assumed — at cell resolution the boost weight of a straddling cell vanishes and its reduced state is the maximally mixed state — which also discharges, at cell resolution, the maximal-vacuum-entanglement hypothesis of the entanglement-equilibrium formulation of the argument, whose nonconformal obstruction is in turn reduced, for the framework's matter content, to a bounded trace-sector dressing. The universality of the entropy density η — an assumption in the original argument — is here derived from the uniqueness and translation invariance of the interface. The conservation input of the Bianchi step is clause (i) of Axiom III, the first law of the framework performing first-law work. The integration constant is the cosmological term, and the framework identifies it independently as Λ = 8πG ρA / c⁴, with ρA the substratum density; the derivation adopts that identification. The Einstein field equation follows as the equation of state of the channel, with G = 1/ (4ħη) ; the value η = 1/ (4 ℓP²) — the Bekenstein–Hawking coefficient — enters as the framework's declared Gödelian input and is not derived. Two structural by-products are recorded. An interface relation, (2π/k₀) ² = 4 ln (nₘax + 1) ℓP², connects the Nyquist momentum of Axiom I, the channel capacity, and the Planck length, and is calibrable through the coherence bound Ncrit. A coefficient dictionary reconciles the two gravitational coefficients of the framework, and under it the interaction tension emerges at the order of the Planck density from the measured G alone. The result is a reduction in the programme's declared pattern — structure derived, scale Gödelian — and the Einstein–Hilbert term of the written action is retained as the variational encoding of the derived equation of state. The equilibrium scope is declared, and the relation of the nonequilibrium extension to the emergent R² sector of the inflation companion is settled at the structural level: the coefficient dictionary identifies the scalar–tensor field of that companion with the channel entropy density, whose curvature dependence is the Wald density of f (R) gravity, so the two R² sectors are one object; the entropy production of the generalized balance is nonnegative exactly on the stability domain of the scalaron, the scale M remains fixed on the inflation side, and the microscopic derivation of the density correction is left open.
Patricio E. Valenzuela (Sun,) studied this question.