track1D_tt_gram_system_reduction_endpoint_holds
plain-language theorem explainer
Given a finite Gram-system solution for the periodic transverse-traceless sector at N=5, the normal equations, longitudinal coefficients, TT projector, and Freudenthal orthogonal decomposition target all become available. Gravity Track 7 cites this as the Track 1.D handoff receipt into the fork-integration certificate. The proof is a pure constructor chain through the shear-sector data types, ending at the Freudenthal target lemma.
Claim. If $D$ is solution data for the explicit finite Gram system of the periodic transverse-traceless (TT) sector at $N=5$, then the normal-equation solution data and longitudinal-coefficient solution data are nonempty, a TT projector exists for the periodic longitudinal gauge map, and the Freudenthal TT orthogonal decomposition target holds for maps from the periodic longitudinal gauge index set into $\mathbb{R}$.
background
Track 7 is the integration-lane receipt for parallel gravity fork handoffs. It does not upgrade discovery claims; it records what each new endpoint proves and leaves remaining Track 1 displacement-class leaves as open dependencies. Track 1.D is the Gram-system reduction leg: a finite solver for every load induced by an edge perturbation is supposed to supply the Gram system and close the TT decomposition.
The ambient objects live in the tensor shear sector. Periodic TT Gram-system solution data at $N=5$ is the input. From it one builds normal-equation data, longitudinal-coefficient data, coefficient and longitudinal projectors, generator-map and gauge-generator projectors, and finally finite-generator projector data. The target is the Freudenthal TT orthogonal decomposition at $N=5$ for real-valued maps on the periodic longitudinal gauge index set.
Spatial dimension $D=3$ is the forced value from the T8/T9 chain, and the eight-tick register sits in the background Clifford/Bott picture, but this endpoint is purely about closing the finite linear-algebra pipeline for the TT sector.
proof idea
Term-mode proof by successive constructors. Introduce Gram-system data $D$. Build normal-equation data via ofGramSystemData, then longitudinal-coefficient solution data via ofNormalEquationData. Lift through coefficient projector, longitudinal projector, generator-map projector, gauge-generator projector, and finite-generator projector data. Package nonempty witnesses for normal equations, longitudinal coefficients, and the TT projector (ofFiniteGeneratorData), and discharge the Freudenthal orthogonal decomposition target by the dedicated lemma that takes Gram-system data directly. No tactics beyond intro/let/exact; pure data-flow reduction.
why it matters
This is the explicit Gram-system reduction endpoint consumed by Track 7. It feeds forkHandoffIntegrationCert, the integration-lane certificate that bundles Track 2 many-body, Track 1 Schläfli and displacement reductions, and sibling handoffs. Closing the finite TT decomposition from Gram data is the algebraic gate between edge-load perturbations and the projector/orthogonal-split infrastructure used downstream in shear-sector gravity arguments.
In the Recognition framework this sits on the gravity analysis side of the master theorem, not on the T0–T8 forcing chain itself. It records that the N=5 periodic TT Gram solver is enough to inhabit the normal equations, longitudinal coefficients, projector, and Freudenthal target, so the fork integration can treat Track 1.D as discharged rather than scaffolding.
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