Einstein's Metric Field Equation at Every Order from the Recognition Lattice

The leading two-derivative metric field equation of the Lorentzian Kuhn recognition lattice is Einstein's equation, with its full nonlinear dependence on the metric. The identity holds at every symmetric background in the regular star-shaped chamber of a flat reference metric. It follows from the linear identity at each constant background and the vanishing of both residuals on jets of flat coordinate changes. A further proof extends the identity to every regular background; independent verification of that extension is pending. We distinguish this field-equation identity from the existence and convergence of lattice solutions. For a specified static worldline matter model, the continuum equation and the unique coupled band solutions agree through second order in source amplitude, with the uniform virial stress supplied by the support itself. This result does not determine whether the physical comparison cost prices particle counts or stresses. At third order, the metric-projected gravitational source matches the cubic Einstein coefficient with an O(s³) error for bounded finite-mode long-wave data; the complete cubic solution limit remains unfinished, and independent verification of the source calculation is pending. We give precise targets for that limit and for the nonlinear solution problem, retaining the finite-spacing consistency conditions that an exact construction must satisfy. Companion to Quantum Gravity from the Recognition Ledger (https://doi.org/10.5281/zenodo.23839819). Preprint of the Recognition Physics Institute, 23 pages. Not peer reviewed.