Track1DTTHessianLichnerowiczEncodedResidualDispRowFormulaReductionEndpoint
plain-language theorem explainer
Track 1.D reduction endpoint: a displacement-row residual-kernel certificate (seven translation-normalized row families at N=5) implies the encoded residual-kernel route, residual-entry formulas, periodic match data, and the bilinear TT energy-match endpoint. Track 7 fork-handoff integration and its companion holds theorem cite it. The declaration is a Prop packaging of implication arrows; constructors from the displacement-row data discharge the chain.
Claim. If a displacement-row form of the encoded residual-kernel certificate exists at lattice size $N=5$ (Regge Hessian, lattice Lichnerowicz, and residual kernels with coefficients on seven row families and TT normal-equation indices), then the encoded residual-kernel certificate, residual-entry formulas (encoded and periodic), periodic TT Hessian--Lichnerowicz match data, and the bilinear reduction endpoint are all inhabited: pointwise identification of the Regge TT Hessian with the lattice Lichnerowicz operator on longitudinal TT modes yields matching bilinear and quadratic TT energies.
background
This module is the Gravity Track 7 integration-lane receipt for parallel fork handoffs (A--F). It records what the new endpoints prove without upgrading the discovery claim, and keeps remaining Track 1 displacement-class leaves as open dependencies.
The premise structure is the displacement-row residual-kernel certificate: Regge Hessian, lattice Lichnerowicz, and residual kernels on encoded edges, with residual coefficients indexed by seven row families after translation normalizes every encoded row to the origin edge at fixed displacement. Upstream, the encoded residual-kernel form lets a finite calculation emit the already-subtracted residual matrix, prove it equals Regge minus Lichnerowicz, and compare it to generator-map reconstruction.
The terminal conjunct is the bilinear reduction endpoint: once the Regge TT Hessian and lattice Lichnerowicz operators agree pointwise on longitudinal TT modes, bilinear and quadratic TT energy matches follow. Spacetime displacement here is the usual 4-vector $(\Delta t,\Delta x_1,\Delta x_2,\Delta x_3)$.
proof idea
Pure Prop definition, not a proved theorem. The body is a single implication: from displacement-row residual formula data, assert Nonempty of four certificate structures (encoded residual-kernel, encoded residual-entry, periodic residual-entry, periodic match) conjoined with the bilinear reduction endpoint. No tactics; the companion holds theorem later builds those witnesses via ofDispRowData constructors and feeds the bilinear endpoint.
why it matters
Packages the Track 1.D displacement-row residual formula leaf that Track 7 consumes. Downstream, the holds theorem proves the Prop by constructing residual-kernel, residual-entry, periodic residual-entry, and match data from the displacement-row certificate, then the fork handoff integration certificate records the Track 1 result as a reduction/interface package (not a closure of open Schläfli leaves). In the RS gravity program this sits on the TT shear sector path that identifies the discrete Regge Hessian with the lattice Lichnerowicz operator, a prerequisite for residual control before continuum matching. It does not finish the master theorem; it only tightens the handoff surface between residual-row formulas and bilinear energy identity.
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