Second-Order Entanglement Response in Lattice Quantum Media: A Measured Linearized Static Field Equation in 1+1, 2+1, and 3+1 Dimensions

The literature descending from Jacobson's equation-of-state argument connects gravity to quantum information through derivations; to our knowledge it contains no measured relation, in a microscopic system, linking entanglement data to signal propagation. This paper reports such measurements. On lattice fermion models in 1+1, 2+1, and 3+1 dimensions, three static response kernels are measured the same way, by linear response to weak static sources from equilibrium data alone: the entanglement-entropy response, the density response, and the response of the local modular clock, which defines an effective metric field. On one 3+1 lattice the three close into a linearized static field equation: the metric field, reconstructed from the medium's modular data alone, inverts to the independently imposed source at 1.9 percent and reproduces the measured density response, with every constant fixed before comparison. Consistent with its status as an equation of state, the relation reduces within its regime of validity to an identity among equilibrium response functions; its empirical content is that the couplings are computed rather than fitted (a held-out model agrees at 2.5 percent) and that the regime's boundary is measured: the relation fails exactly where its premises fail, at the Kane-Fisher line, under non-adiabatic driving, and for sub-wavelength sources. A matched-pair experiment, both outcomes frozen in advance, separates the entanglement functional from low-order acoustics: two sources with identical low-order hydrodynamic observables split the entanglement measure as predicted while their transit delays stay degenerate. Determinant quantum Monte Carlo excludes two alternative entanglement Hamiltonians and shows the coupling surviving interactions. All results are measurements on named lattice model classes against pre-registered acceptance bands and kill conditions; the failures are reported alongside the successes, each with its mechanism identified. Everything here is static, linearized, and lattice-scale: no nonlinear self-sourcing, no propagating tensor dynamics, no continuum extrapolation. It is not the Einstein field equation, and section 10 tabulates the logical status of every claim. Reproduction scripts accompany the paper.