Track1DTTHessianLichnerowiczKernelEntryReductionEndpoint
plain-language theorem explainer
Entrywise equality of the Regge TT Hessian edge kernel and the lattice Lichnerowicz edge kernel is a stronger stencil-level sufficient condition for the full TT Hessian/Lichnerowicz match package on the N=5 periodic complex. Track 1.D and the Fork Handoff Integration certificate cite this endpoint. The declaration is a pure Prop: entry data implies nonempty row data, nonempty match data, and the bilinear reduction endpoint.
Claim. If the Regge TT Hessian and lattice Lichnerowicz operators agree entrywise on every pair of edges of the $N=5$ periodic edge complex, then there exist nonempty rowwise kernel data and operator-match data, and the bilinear TT energy identity holds on longitudinal TT modes.
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; remaining Track 1 displacement-class leaves stay open.
Track 1.D concerns matching the Regge second-variation TT Hessian to the spin-2 lattice Lichnerowicz operator on the $N=5$ periodic edge complex. Entrywise kernel data packages two finite edge kernels (Regge Hessian and lattice Lichnerowicz) together with the assertion that they agree on every edge pair. That is a stencil-level strengthening of the rowwise route, which only requires kernel rows to agree when restricted to longitudinal TT modes.
Upstream, the bilinear reduction endpoint states that once the two operators are identified pointwise on TT modes, the bilinear and quadratic TT energy matches follow. The present Prop chains entry data into that bilinear package via intermediate row and match witnesses.
proof idea
Pure definitional packaging, not a proved implication. The body is the implication arrow from entrywise kernel data to the conjunction of three goals: nonempty rowwise kernel data, nonempty match data, and the bilinear reduction endpoint. The companion theorem track1D_tt_hessian_lichnerowicz_kernel_entry_reduction_endpoint_holds discharges it by building row data from entry data via ofEntryData and assembling the remaining witnesses.
why it matters
This is the Track 1.D entrywise edge-kernel reduction endpoint consumed by Track 7. It feeds the companion holds theorem and is listed among the Track 1 reduction/interface packages inside ForkHandoffIntegrationCert, which integrates Forks A through F.
In the gravity master plan, matching the discrete Regge TT Hessian to the continuum Lichnerowicz operator on TT modes is the stencil bridge between combinatorial curvature and continuum spin-2 dynamics. Entrywise kernel equality is the strongest finite-kernel sufficient condition on that bridge: if every edge-pair coefficient agrees, row agreement on TT modes and the bilinear energy identities follow.
The module doc is explicit that this does not close the open Schläfli leaves; it only records a stronger handoff fact for the integration certificate.
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