EncodedTTHessianLichnerowiczCoeffRelativeTranslatedTTZeroData5
plain-language theorem explainer
Packages relative-frame translated residual formula data with the shifted-generator orthogonality property on transverse-traceless modes. Anyone closing the Track 1.D Hessian-to-Lichnerowicz residual on TT cites this bundle. It is a pure data structure: two fields, no proof obligation inside the declaration itself.
Claim. A data package consisting of (i) relative-frame coefficient-only translated residual formula data (encoded edge-kernel residual equals a row-rebased normal-equation generator map on the 5-torus) and (ii) the property that every transverse-traceless edge perturbation is orthogonal, under the periodic edge inner product, to every row-frame translate of the combined conformal/longitudinal generator map.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on the periodic 5-torus. The older Track 1.B conformal ansatz assigns one scalar potential per vertex and averages endpoints to get edge strains; that slice cannot represent pure shear, so it misses transverse-traceless gravitational-wave modes. This module therefore separates independent edge perturbations from vertex-conformal ones.
The first field records relative-frame coefficient-only translated residual formula data: after each residual row is re-based at its own edge, the encoded Regge-Hessian minus lattice-Lichnerowicz residual equals a periodic TT normal-equation generator map applied to a dispersion-indexed coefficient vector. The second field is the missing shifted-generator lemma: every TT perturbation is orthogonal to every row-frame translate of the combined conformal/longitudinal generator. Spatial dimension is the forced $D=3$ of the forcing chain (T8/T9).
proof idea
No proof body: the declaration is a structure (definition). It simply pairs the already-defined relative-frame translated formula data with the proposition asserting relative TT generator orthogonality on TT. Downstream consumers pattern-match on the two fields; nothing is discharged here.
why it matters
This is the preferred mathematical target for the relative-frame route: prove generator closure once, then orthogonality follows from the existing TT definition, and residual-kernel vanishing on TT follows. The master-theorem handoff consumes it as the hypothesis of the conditional endpoint Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedTTZeroEndpoint, which yields a nonempty residual-zero certificate (PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5). That residual-zero statement is the compact finite calculation target for the TT Hessian-to-Lichnerowicz comparison. The separate orthogonality field is exactly the mathematical gap left by the Regge–Schläfli candidate diagnostics; closing it finishes the relative-frame bridge into Track 7.
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