track1D_tt_normal_equation_reduction_endpoint_holds
plain-language theorem explainer
A solution of the combined normal equations on the fixed conformal-plus-longitudinal generator family at stencil size N=5 supplies the finite transverse-traceless (TT) decomposition package. Track 7 fork-handoff integration cites this as the Track 1.D endpoint. The proof is a short term-mode constructor tower: normal-equation data is rewrapped through longitudinal coefficient, projector, generator-map, and finite-generator packages, then the Freudenthal orthogonal-decomposition target is discharged.
Claim. If $D$ solves the periodic TT normal equations at stencil size $N=5$, then there exist nonempty longitudinal coefficient solution data, longitudinal coefficient projector data, and TT projector data for the periodic longitudinal gauge map, and the Freudenthal TT orthogonal decomposition target at $N=5$ holds for real-valued maps on the longitudinal gauge index set.
background
Track 1.D in the Recognition Science gravity stack treats the finite transverse-traceless (TT) sector on a periodic stencil. Normal-equation solution data encode an explicit finite Gram system for the fixed conformal-plus-longitudinal generator family. From a solution one builds successive packages: longitudinal coefficients, coefficient projectors, longitudinal projectors, generator maps, gauge generators, and finite-generator projectors, ending at the Freudenthal-style orthogonal split of metric perturbations into gauge, conformal, and TT pieces.
The ambient module is the Track 7 fork-handoff integration lane. It records what each parallel fork endpoint proves and does not upgrade discovery claims. Spatial dimension is fixed at $D=3$ by the forcing chain (T8). The TT analysis lives in the tensor shear sector on the $N=5$ periodic geometry.
The endpoint proposition is exactly the implication from normal-equation solution data to the four-part package: nonempty longitudinal coefficient and coefficient-projector data, nonempty TT projector data for the longitudinal gauge map, and the Freudenthal TT orthogonal decomposition target.
proof idea
Pure packaging, no new analysis. Given normal-equation data $D$:
- build longitudinal coefficient solution data from $D$;
- build coefficient projector data from that solution;
- build longitudinal projector data from the coefficient projectors;
- lift to generator-map projector data;
- lift to gauge-generator projector data;
- lift to finite-generator projector data.
The return value is the four conjuncts: nonempty witnesses for the longitudinal coefficient solution and coefficient projector, a nonempty TT projector constructed from the finite-generator package, and the Freudenthal orthogonal-decomposition target obtained from $D$ by the dedicated of_normalEquationData lemma. The whole argument is a term-mode constructor tower.
why it matters
This is the Track 1.D receipt consumed by the fork handoff integration certificate, which bundles Track 1 Schläfli and displacement reductions, Track 2 many-body amplitude linearity, and the other fork endpoints into one integration-lane instance. Closing the finite TT decomposition via the combined normal equations is the algebraic step that lets shear-sector projectors feed higher gravity tracks without an open Gram-system leaf.
In the framework the TT split is the discrete analogue of the continuum transverse-traceless gauge of linearized gravity, forced here on the $N=5$ periodic stencil with $D=3$. The module doc is explicit that remaining Track 1 displacement-class leaves stay as the next dependency, so this endpoint narrows but does not finish the full Track 1 stationarity program. Downstream packaging treats it as a proved handoff, not a discovery upgrade.
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