Track1DTTHessianLichnerowiczResidualRowCoeffEntryReductionEndpoint
plain-language theorem explainer
Entrywise residual-row coefficient identities on the N=5 periodic TT sector suffice to discharge the full TT Hessian/Lichnerowicz reduction chain. Gravity auditors cite this as the Track 1.D handoff predicate consumed by Track 7 fork integration. The body is a pure Prop package: one entry-data hypothesis implies nonempty coarser residual, span, kernel-zero, kernel-row, and match witnesses, plus the bilinear energy-match endpoint.
Claim. If the entrywise residual-row coefficient certificate for the $N=5$ periodic TT Hessian versus lattice Lichnerowicz comparison is inhabited, then the residual-row coefficient data, residual-row span data, residual TT-kernel zero data, kernel-row data, and operator-match data are all nonempty, and the bilinear TT energy-match endpoint holds.
background
This module is the Track 7 integration-lane receipt for parallel fork handoffs in the gravity master theorem. It records what the new endpoints prove without upgrading the discovery claim, and keeps remaining Track 1 displacement-class leaves as open dependencies.
Track 1.D concerns the transverse-traceless (TT) sector on the $N=5$ periodic edge complex. The Regge second-variation Hessian on edge lengths is compared to the spin-2 lattice Lichnerowicz stencil. Residual rows are the difference of those two kernels; the sharp finite target is that every residual row lies in the combined conformal-plus-longitudinal generator span.
The entrywise certificate supplies a scalar table residualRowCoeff such that each residual kernel entry equals the corresponding generator-map reconstruction. Coarser structures package the same data as row coefficients, row-span membership, kernel-row match on TT modes, and full operator match. The bilinear endpoint then asserts that once the operators agree pointwise on TT modes, the bilinear and quadratic TT energies match.
proof idea
Definitional Prop, not a proved theorem. The right-hand side is an implication whose hypothesis is inhabited entrywise residual-row coefficient data. The conclusion is the conjunction of five Nonempty witnesses (coefficient data, row-span data, residual TT-kernel zero data, kernel-row data, match data) with the already-defined bilinear reduction endpoint. No tactics or lemmas fire here; the sibling ..._holds theorem later discharges the implication by constructing each coarser structure from the entry data via the ofEntryData / ofEntryCoeffData constructors.
why it matters
This is the Track 1.D entrywise residual-row coefficient endpoint that Track 7 consumes. Downstream, track1D_tt_hessian_lichnerowicz_residual_row_coeff_entry_reduction_endpoint_holds proves the predicate, and ForkHandoffIntegrationCert packages it among the Fork A–F handoff facts. The cert doc is explicit: the Track 1 result is a reduction/interface package, not a closure of the open Schläfli leaves.
In the Recognition gravity stack this sits on the discrete TT shear route that links Regge calculus second variation to continuum Lichnerowicz form on the eight-tick lattice geometry. It does not touch T5–T8 forcing, RCL, or the alpha band directly; it is infrastructure for the master-theorem gravity fork. The remaining open work is inhabiting the entry data itself and closing the displacement-class Schläfli leaves still flagged by the module doc.
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