periodicRelativeTTGeneratorClosure5_holds
plain-language theorem explainer
On the period-5 torus, every row-frame translate of a combined TT normal-equation generator splits exactly into a fixed conformal image plus a longitudinal-gauge image. Gravity and Regge analysts cite this when reducing relative-frame Hessian kernels to the absolute conformal/gauge span. The proof shifts coefficients by the row-base index equivalence, then applies the unshifted generator split.
Claim. For every periodic edge $e$ (the row frame) and every coefficient function $c$ on the TT normal-equation index set, there exist conformal coefficients $\alpha$ on vertices and longitudinal-gauge coefficients $\beta$ such that for every column index the row-translated combined generator equals the sum of the fixed conformal generator of $\alpha$ and the longitudinal-gauge map of $\beta$.
background
Track 1.D isolates the tensor/shear sector of weak-field Regge gravity. The older Track 1.B conformal ansatz puts one scalar at each vertex and averages endpoints to edge strains; that slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes. This module therefore treats independent edge perturbations separately from vertex-conformal ones.
The period-5 periodic torus supplies a finite discrete geometry on which TT normal equations, conformal generators, and longitudinal gauge maps are fully enumerated. The relative (row-frame) generator map is the combined normal-equation generator after translating indices by the base vertex of a chosen edge. Closure asserts that every such translate still lands in the sum of the fixed conformal image and the fixed longitudinal-gauge image.
Upstream, the unshifted generator already splits into conformal plus gauge pieces, and an index equivalence implements translation of normal-equation slots by a base vertex. Those two facts are the only algebraic inputs.
proof idea
Tactic proof, classical. Fix a row edge and a coefficient function $c$. Define shifted coefficients by pulling $c$ back along the inverse of the normal-equation index equivalence for the row base vertex. Feed those shifted coefficients into the existing conformal-coefficient and gauge-coefficient extractors, and take those as the existential witnesses.
For an arbitrary column, rewrite the relative generator map as the ordinary generator map of the shifted coefficients (via the shifted-map identity). Then apply the unshifted split lemma, which decomposes that map into conformal plus longitudinal-gauge images. No new analysis is required beyond transport of coefficients under the translation equivalence.
why it matters
Closure is the bridge to shifted-generator orthogonality: once every row-frame translate splits into conformal plus gauge images, any edge perturbation already orthogonal to both fixed images is automatically orthogonal to all relative generators. The immediate parent is the structure packing relative-frame translated Hessian data with that orthogonality lemma; its doc-comment states that relative data plus orthogonality is enough to prove the residual kernel vanishes on TT perturbations, and that the separate orthogonality field is the exact gap left by Regge–Schläfli candidate diagnostics.
In the Recognition gravity track this is discrete Lichnerowicz/TT kernel control on a finite torus, not continuum GR. It does not invoke the T0–T8 forcing chain or the mass ladder directly; it is infrastructure for showing that pure shear (TT) modes sit outside the conformal ansatz and that the encoded Hessian residual on those modes is zero once generator closure holds.
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