EncodedTTHessianLichnerowiczCoeffRelativeOriginColumnFormulaData5
plain-language theorem explainer
Packages two encoded edge kernels (Regge Hessian and lattice Lichnerowicz), residual displacement coefficients, and the origin-column identity that their difference equals the TT normal-equation generator map on origin-based rows. Track 1.D gravity diagnostics and the MasterTheorem handoff cite it as the weaker relative-frame certificate surface. As a structure, the content is the field bundle itself, not a derived proof.
Claim. A data package on the $5\times5\times5$ periodic Freudenthal torus consisting of: (i) an encoded Regge-Hessian edge kernel $K_R$, (ii) an encoded lattice Lichnerowicz edge kernel $K_L$, and (iii) residual displacement coefficients $c:\mathrm{Fin}\,7\to I_{\mathrm{TT}}\to\mathbb{R}$, such that for every origin-row displacement $r$ and every periodic edge column $(b,d)$, $K_R(e_0(r),e(b,d))-K_L(e_0(r),e(b,d))$ equals the TT normal-equation generator map of $c(r)$ at that edge. No absolute translated formula for non-origin rows is required.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on a discrete triangulation. The older Track 1.B conformal ansatz puts one scalar potential at each vertex and averages endpoints to get edge strains; that slice cannot realize pure shear, so it misses transverse-traceless gravitational-wave modes. This module therefore treats independent edge perturbations separately from vertex-conformal ones and works on the canonical encoded $5\times5\times5$ periodic Freudenthal torus.
An encoded edge-operator kernel is a real matrix indexed by the finite edge set of that torus. The combined TT normal-equation index mixes fixed conformal vertex-delta generators with fixed longitudinal vertex-vector generators. Origin-column data means: only rows whose edge base has been rebased to a fixed origin displacement are constrained, matching the existing origin-row generator map.
Upstream dimension facts fix spatial $D=3$ (T8/T9), which sets the three-torus geometry underlying the $5^3$ lattice.
proof idea
Definitional structure, not a proved theorem. The four fields are the certificate payload: two EncodedEdgeOperatorKernel5 matrices, a residual coefficient table on Fin 7 times the TT normal-equation index, and a single universal equality originColumn_entry_formula equating kernel difference on origin-based rows to periodicTTNormalEquationGeneratorMap5. No tactics or lemmas discharge anything here; inhabitants are built elsewhere (e.g. via ofCoeffRelativeTranslatedData).
why it matters
This is the origin-column slice of the relative-frame translated residual certificate for the TT Hessian versus lattice Lichnerowicz comparison. Downstream, Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedDiagnosticEndpoint is the implication "full relative translated formula data $\Rightarrow$ nonempty origin-column formula data," and the handoff theorem proves that endpoint by applying ofCoeffRelativeTranslatedData. The doc-comment stresses the intentional weakness: physical stencil covariance after row-rebasing is recorded without claiming the stronger absolute translated formula for every row. That gap is exactly what still blocks converting relative covariance into a full absolute row formula or a shifted-generator theorem before Track 7 consumption. In the broader RS gravity program it sits on the shear/TT path opened once the conformal rectangle obstruction is known, on the $D=3$ forced lattice.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.