Track1DTTGramLoadImageReductionEndpoint
plain-language theorem explainer
Track 1.D endpoint: if every physical TT load at the N=5 periodic stencil lies in the image of the finite Gram operator, then a load solver, Gram-system solution, TT projectors, and the Freudenthal orthogonal TT split all exist. Gravity auditors cite it as the handoff receipt that packages the TT-load image reduction for Track 7. The body is a pure Prop abbreviation chaining those nonempty witnesses from the image hypothesis.
Claim. If every physical transverse-traceless load generated by a periodic edge perturbation at $N=5$ lies in the image of the finite TT Gram operator, then there exist a load solver on that Gram system, a full Gram-system solution, TT projector data relative to the concrete longitudinal gauge map, and an orthogonal decomposition of raw edge perturbations into conformal, gauge, and TT parts.
background
This module is the Track 7 fork-handoff integration lane. It records exactly what the parallel fork endpoints prove (Tracks 1.B stationarity, physical residual/Bianchi, many-body amplitude lift, Page capacity, dark-energy $w(z)$, falsifier sensitivity) without upgrading the structural master-theorem discovery claim. Remaining Track 1 displacement-class leaves stay as the next dependency.
The finite TT Gram operator maps coefficient vectors on the normal-equation index set to load vectors via the edge-TT load map (matrix-vector action of a $4\times4$ block on a mass vector). Image data asserts every load arising from an edge perturbation lies in that image; this is weaker and more geometric than choosing a solver. Load-solver data isolates the remaining finite linear algebra: produce coefficients that reproduce every such load under the Gram apply map.
The Freudenthal TT orthogonal decomposition target at $N=5$ asks for a splitting of raw edge perturbations into conformal, gauge, and TT parts, with TT read as finite orthogonality to the conformal and gauge subspaces. The concrete longitudinal gauge is indexed by periodic vertices times three vector components; the gauge map is generated by vertex-vector delta basis elements.
proof idea
Definitional Prop packaging, not a tactic proof. The body is the single implication from Gram-load image data at $N=5$ to the conjunction of four conclusions: nonempty load-solver data, nonempty Gram-system solution data, nonempty TT projector data for the concrete longitudinal gauge index and map, and the Freudenthal orthogonal TT decomposition target at those same gauge data. Downstream, the companion theorem discharges the Prop by building solver data from image data, then system solution from the solver, then normal-equation solution, and finally the projectors and split.
why it matters
Supplies the Track 1.D Gram-load image reduction endpoint consumed by Track 7. The companion theorem track1D_tt_gram_load_image_reduction_endpoint_holds proves the Prop, and ForkHandoffIntegrationCert records it among the fork A–F handoff facts. Per the integration certificate, the Track 1 result is a reduction/interface package, not a closure of the open Schläfli leaves; structural master-theorem witnesses remain where the master plan requires them. In the gravity lane this is the finite-algebra bridge that turns “every TT load is in the Gram image” into existence of solvers, projectors, and the conformal/gauge/TT split on the periodic $N=5$ stencil.
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