EncodedTTHessianLichnerowiczCoeffOriginColumnFormulaData5
plain-language theorem explainer
Certificate packaging the Regge Hessian and lattice Lichnerowicz edge kernels on the 5×5×5 periodic Freudenthal torus with seven residual-generator coefficient rows. It requires that both the origin-column scalar residual and the fully translated residual equal the TT normal-equation generator map on those coefficients. Gravity auditors cite it as the smallest generator-facing surface for the TT Hessian/Lichnerowicz residual. The declaration is a structure definition bundling kernels, coefficients, and two formula obligations.
Claim. A coefficient-only origin-column formula package consists of two encoded edge-operator kernels $H$ (Regge Hessian) and $L$ (lattice Lichnerowicz) on the $5\times 5\times 5$ periodic Freudenthal torus, residual-generator coefficient rows $c:\mathrm{Fin}\,7\to I\to\mathbb{R}$ (with $I$ the combined conformal vertex-delta and longitudinal gauge index), such that for every displacement row and typed edge column the origin-row residual $H-L$ equals the normal-equation generator map of $c$, and the full encoded residual matrix equals that same generator map after edge translation.
background
Track 1.D opens the tensor/shear sector of the weak-field metric. Track 1.B's conformal ansatz assigns one scalar potential per vertex and induces edge strains by averaging endpoints; that slice cannot represent pure shear, so it misses transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and works on the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus.
An encoded edge-operator kernel is a real matrix on the finite edge index set of that triangulation. The combined normal-equation index packages fixed conformal vertex-delta generators with fixed longitudinal vertex-vector generators. The residual of interest is the difference between the discrete Regge Hessian and the lattice Lichnerowicz operator on edges; the certificate reduces that residual to seven coefficient rows against the generator map, which is the smallest generator-facing surface described in the doc-comment.
proof idea
Structure definition, not a proved theorem. An inhabitant supplies the two kernels, the coefficient table of seven residual-generator rows, and two propositional fields: the origin-column entry formula (residual of the kernels on origin-edge rows equals the generator map of those coefficients on the typed column edge) and the encoded residual entry formula (full residual matrix equals the generator map after reading the row displacement from the edge equivalence). No tactics run at definition time. Obligations discharge when a concrete package is built, typically by specializing a translated coefficient certificate via the ofCoeffTranslatedData constructor used in the handoff endpoints.
why it matters
Smallest generator-facing surface for the TT Hessian/Lichnerowicz residual in Track 1.D. Downstream Master Theorem handoff endpoints consume it: the origin-column reduction endpoint turns this package into the raw typed-column residual route without storing a residual origin table, and the translated full-chain endpoint exposes the whole finite reduction chain once a translated coefficient certificate is given (deriving this structure, then raw origin-column and residual table data). Sibling match lemmas and the raw origin-column structure sit one layer out. Framework setting is discrete gravity on the forced $D=3$ spatial lattice (T8) inside the Recognition scaffold, not a continuum Einstein identity. It closes the shear-sector gap left by the conformal ansatz's rectangle obstruction.
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