Track1DTTHessianLichnerowiczKernelRowReductionEndpoint
plain-language theorem explainer
Defines the Track 1.D handoff proposition: rowwise equality of the Regge TT Hessian edge kernel with the lattice Lichnerowicz edge kernel on TT modes yields nonempty operator-match data and the bilinear/quadratic TT energy match. Gravity Track 7 cites it as the kernel-row reduction leaf in the fork integration certificate. The body is a pure Prop abbreviation (hypothesis implies match data and the bilinear endpoint).
Claim. The Track 1.D endpoint proposition asserts: if one is given rowwise kernel data equating the Regge transverse-traceless Hessian edge kernel with the lattice Lichnerowicz edge kernel on longitudinal TT edge perturbations (at period $N=5$), then the corresponding operator-match data is nonempty and the bilinear/quadratic TT energy-match endpoint holds.
background
Module context is Gravity Track 7 fork-handoff integration: a receipt lane that records what parallel forks prove without upgrading the discovery claim. Fork A covers Track 1.B Schläfli stationarity at $N=5$; Track 1.D sits among the remaining displacement-class and operator-match leaves that feed the structural master theorem only as reduction/interface facts.
The hypothesis package is rowwise kernel data: a Regge second-variation edge Hessian kernel and a spin-2 lattice Lichnerowicz stencil kernel, together with a proof that their rows agree on the periodic longitudinal TT subspace. Upstream, the bilinear reduction endpoint states that once the Regge TT Hessian and lattice Lichnerowicz operators are identified pointwise on TT modes, bilinear and quadratic TT energy matches follow for pairs of TT edge perturbations.
Operator-match data is the intermediate witness that converts kernel-row agreement into the pointwise operator identification those energy identities need.
proof idea
Definitional Prop, not a proved theorem. It is the implication from PeriodicTTHessianLichnerowiczKernelRowData5 to the conjunction of nonempty PeriodicTTHessianLichnerowiczMatchData5 and the already-named bilinear reduction endpoint. Discharge is elsewhere: the companion ..._holds theorem intros the kernel-row data, builds match data via ofKernelRowData, and reuses the bilinear endpoint theorem.
why it matters
Closes the stated Track 1.D kernel-row leaf in the handoff stack. Downstream, track1D_tt_hessian_lichnerowicz_kernel_row_reduction_endpoint_holds asserts the proposition, and ForkHandoffIntegrationCert consumes the Track 1 reduction/interface package alongside Track 2 many-body and Track 6 sensitivity facts. Doc-comment on the cert: the structural master theorem still uses structural witnesses where required; Track 1 here is reduction/interface, not closure of open Schläfli leaves. In RS gravity terms this is the discrete TT shear-sector step that aligns Regge second variation with the lattice Lichnerowicz operator before energy identities are handed upward. It does not finish the master theorem; it packages the kernel-row route into Track 7.
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