Track1DTTGramLoadSolverReductionEndpoint
plain-language theorem explainer
Track 1.D packages the claim that a finite Gram-load solver for every edge-perturbation load yields Gram-system and normal-equation solutions, TT projectors for the longitudinal gauge, and closes the transverse-traceless orthogonal split at N=5. Gravity auditors cite it when wiring the shear-sector handoff into Track 7. The declaration is a pure Prop: an implication from load-solver data to four nonempty downstream witnesses.
Claim. If finite load-solver data exist for the periodic TT Gram operator at $N=5$, then the Gram system admits a solution, the normal equations admit a solution, the TT projectors exist relative to the concrete longitudinal gauge map, and the Freudenthal TT orthogonal decomposition target holds: every edge perturbation splits into conformal, gauge, and TT-orthogonal parts.
background
This module is the Track 7 fork-handoff integration lane. It records what parallel gravity endpoints prove without upgrading the discovery claim, and keeps remaining Track 1 displacement-class leaves open.
In the tensor-shear sector at period $N=5$, edge perturbations of the discrete metric are to be split into conformal-log, longitudinal-gauge, and transverse-traceless (TT) pieces. TT is interpreted honestly as finite orthogonality to the conformal and gauge subspaces; the remaining load is constructing three projectors. The concrete longitudinal gauge is indexed by vertex-vector pairs and generated by delta basis elements on periodic vertices.
Upstream, load-solver data isolate the finite linear-algebra step: for every load vector induced by an edge perturbation (the matrix-vector product of a $4\times 4$ edge block against a mode), solve the TT Gram normal equations. Gram-system solution data and normal-equation solution data are the explicit coefficient projectors that make those equations hold pointwise.
proof idea
Definitional packaging only: the Prop is the implication from periodic TT Gram load-solver data at $N=5$ to the conjunction of (i) nonempty Gram-system solution data, (ii) nonempty normal-equation solution data, (iii) nonempty TT projector data for the longitudinal gauge index type and map, and (iv) the Freudenthal TT orthogonal decomposition target at $N=5$ for that same gauge. No tactics; the body is the interface contract. The sibling theorem that discharges it builds Gram-system data from the load solver, then normal-equation data, then longitudinal coefficient and projector data, and finally the split.
why it matters
Closes the Track 1.D reduction leaf in the gravity master-theorem handoff: a finite solver for every edge-induced load supplies the Gram system and finishes the TT decomposition. Downstream, the holding theorem asserts this endpoint, and the fork handoff integration certificate consumes the Track 1 reduction/interface package alongside many-body, Schläfli-reduction, and residual forks. The module doc is explicit that this is not a closure of open Schläfli leaves and does not upgrade the structural master theorem's discovery claim. In RS gravity terms it is the finite linear-algebra bridge from edge loads to the orthogonal TT sector used by later shear and channel arguments.
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