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

The entanglement-thermodynamics literature descending from Jacobson's argument connects gravity to quantum information through derivations; to our knowledge it contains no measured relation, in a microscopic system, from information-theoretic data to propagation (the search boundary is stated in section 1.1). This paper reports the measurements that fill that gap, with their logical structure stated as plainly as their numbers. On lattice quantum media in 1+1, 2+1, and 3+1 dimensions, three static kernels are measured by one method, the response to weak cosine sources, from equilibrium data with nothing dynamical in the lineage: the information kernel F, through which the medium's entanglement books (its ledger, in this paper's usage) weigh a source; the matter kernel Π of the density response; and the clock kernel K_v of the metric field, read from the medium's own modular data. On one 3+1 substrate the three close into a linearized static field equation: the reconstructed metric field inverts to the independently imposed source at 1.9 percent and, passed through the matter kernel, reproduces the measured density response, with every constant frozen in advance. The equation has exactly the logical structure Jacobson's thesis requires: inside its window it reduces to an identity among the medium's equilibrium response functionals, a field equation that is an equation of state, and its empirical content lies in the computed exchange rates (nothing fitted, one medium fully held out at 2.5 percent), in the kernels' closure, in the ledger-side structure of the pair nonlinearity, and in the measured boundary of the window, where the relation fails exactly where its premises fail: at the Kane-Fisher line, under non-adiabatic driving, for sub-wavelength sources. The constitutive core Δt = 𝒢 · D, with D the BKM quadratic response of the equilibrium cells, is verified at 1.6 to 3.3 percent in 1+1, where it is a genuine one-constant relation; in higher dimensions it is presented as what its formula shows it to be, a decomposition whose two factors, dispersion optics and the kernel bookkeeping, are separately verified at 2 and 0.1 percent. ("Second order" refers to the quadratic order of the relative-entropy functional in the state perturbation, not to second-order metric dynamics.) The strongest discriminator is a matched pair with both edges frozen in advance: two sources identical in the registered low-order hydrodynamic observables split the information functional exactly as the kernel predicts while their measured transits stay degenerate, the degeneracy itself a frozen prediction, because the delay reads only the geometry the source builds. A determinant quantum Monte Carlo extension excludes the two registered alternative forms of the entanglement Hamiltonian and shows the coefficient surviving interactions with a symmetry-forced residual. All results are measurements on named lattice model classes against pre-registered acceptance bands and kill conditions, with fired kills and their mechanism chases reported alongside the successes. 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.