Static Exteriors and the Black-Hole Problem on the Recognition Lattice

A static vacuum metric does not automatically give a vacuum of the recognition lattice: fifteen squared-edge classes carry ten metric components and five additional combinations. We give the exact uniform tidal correction in the timelike-hinge regime and an explicit obstruction at the next order in amplitude. At the flat identification κ = 1, a static cubic harmonic field with its internal correction solves every linearized equation on ℤ⁴, but its second-order vertex Ward defect is 756√3. No quadratic correction to that fixed tangent solves the thirty equations incident to the selected record. This result also holds for the Lorentzian action extended to spacelike hinges near that flat field. It does not exclude every static exterior. We distinguish this obstruction from the unresolved general radial class-Jacobian formula, describe the corresponding determinant polynomial, and state the further conditions required for a lattice black hole. Horizon geometry, entropy normalization and radiation on prescribed backgrounds remain distinct from a solution of those conditions. Companion to Quantum Gravity from the Recognition Ledger (https://doi.org/10.5281/zenodo.23839819). Preprint of the Recognition Physics Institute, 12 pages. Not peer reviewed.