Dorau and Much (2026) [1] have shown, using Tomita—Takesaki modular theory, that the Araki—Uhlmann relative entropy between the vacuum and a coherent excitation of a scalar field on a local Rindler horizon equals the boost-energy flux across the horizon, and that the semi-classical Einstein equations follow once this relative entropy is identified with one quarter of the horizon area variation. Two limitations remain: the Rindler bifurcation surface has infinite area, so only area variations are meaningful, and the derivation leaves open what the Bekenstein—Hawking normalization S rel = δA/4 is counting. We address both by transplanting the construction from the local Rindler wedge to the Nariai spacetime dS2 × S 2 , the degenerate limit of Schwarzschild—de Sitter in which the black-hole and cosmological horizons coincide.Nariai possesses a global bifurcate Killing horizon with a compact bifurcation surface of area A = 4π/Λ, so every quantity in the derivation — the horizon two-point function, the modular flow, the relative entropy, and the total area — is finite. We compute the relative entropy for coherent excitations on the Nariai horizon — exhibiting the modular Hamiltonian explicitly, factorizing the transverse sphere out of the Araki—Uhlmann formula mode by mode, and evaluating the entropy in closed form for an explicit family of s-wave excitations — and verify that the identification S rel = δA/4 reproduces the Einstein coupling α = 8π with no infinite-area subtraction. The linear-response model through which the area variation enters — the one postulate of the flat-space derivation — is here derived rather than assumed: the s-wave back-reaction of the throat is exactly linearized de Sitter Jackiw—Teitelboim gravity, whose dilaton equation admits the response model as its unique zero-mode-free solution, with the coupling 8π fixed by the dimensional reduction.Because the Nariai background is itself unstable to vacuum polarization, we quantify the domain of validity of the equilibrium construction: the Ginsparg—Perry/Bousso—Hawking instability evolves at a rate suppressed relative to the modular frequency by the inverse horizon entropy, Γ/κ ∼ 1/SN , so the Kubo—Martin—Schwinger structure holds as a quasi-equilibrium over ∼ SN thermal times — self-consistent at precisely the order at which the semiclassical Einstein equations are derived.