First Law of Proto-Area Entropy from Modular Spectral Geometry

Abstract

We derive a first law of proto-area entropy in the CCKLP--Witten framework for approximate holographic entanglement wedge reconstruction. The central spectral function admits a modular Hamiltonian representation with kernel $L(x)=x\coth(x/2)$. For near-maximally-mixed bulk states, and within the unstructured Gaussian-unitary-ensemble (GUE) model of the encoding perturbation, the ensemble-averaged proto-area entropy varies linearly with bulk entropy to leading order, with a response coefficient containing a universal factor~$1/3$, traced to $L''(0)/2=1/6$ and independent of the bulk spectrum's detailed shape within this model. Imposing the gravitational-scaling condition required for nonzero backreaction, the first-law coefficient is $O(1)$, parametrically matching the semi-classical relation $\delta({\rm Area}/4G_N)=\delta S_{\rm bulk}$.