Abstract
<title>Abstract</title> <p>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 semiclassical Einstein equations follow once this relative entropy is identified. 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 move the construction from the local Rindler wedge to the Nariai spacetime, with a global bifurcate Killing horizon and compact bifurcation surface area A = 4π/Λ. We compute the relative entropy for coherent excitations on the Nariai horizon — factorizing the transverse sphere out of the Araki–Uhlmann formula mode by mode, evaluating the entropy for a family of s-wave excitations — and verify that S rel = δA/4 reproduces the Einstein coupling α = 8π without infinite-area subtraction. The s-wave backreaction 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.</p>