Auditing Entanglement Equilibrium: Quantitative Anatomy of the Matter First-Law Mismatch

Jacobson's entanglement-equilibrium argument derives the Einstein equation from three inputs on the entropy of small geodesic balls: an area law with finite Newton constant, a first law for the matter modular Hamiltonian, and stationarity of the vacuum under volume-preserving variations. This paper is an audit of that argument on an exactly solvable model: every assumption beyond established mathematics is isolated as one of seven explicit hypotheses (H1 to H7) on a family of quantum network states, and each hypothesis is proved, checked numerically, or refuted on a composite of a two-qubit-pair sector (whose entanglement encodes a Riemannian metric) and the Gaussian ground state of a conformally coupled scalar field on the same weighted graph. The audit yields: (i) H1 holds with an analytic contamination bound under an explicit hierarchy involving the regulator mass. (ii) Spectral convergence of the graph Laplacian, previously an assumption, is verified quantitatively: on regular grids the exact convolution spectrum satisfies lambda(k) / c-squared |k|-squared = 1 minus (0.36 plus or minus 0.04) times (k times xi)-squared plus higher orders. (iii) The geometric area law holds with an explicit Newton constant, and an extension to quasi-uniform point sets identifies density fluctuations as the leading uncontrolled systematic, converted into a pre-registered control protocol. (iv) The matter first law fails on the lattice, and the mismatch is explained quantitatively by a two-component boundary-layer model for the lattice modular Hamiltonian whose leading behaviour R1 = 1 + a1/(omega times l/c) + a2/(omega times l/c)-squared, fitted on a single ball radius, predicts the response at two further radii to 3.2 percent and 1.0 percent. (v) All error models are documented explicitly, resolving a discrepancy between naive and conservative parameter errors. The model realizes the equilibrium hypothesis only in a restricted sense (static observers, linear order in G) and by construction; the audit locates the quantitative obstruction precisely in the matter sector, in the tension between exponentially decaying mutual information and the massless limit required by the first law.