track1D_tt_hessian_lichnerowicz_kernel_entry_reduction_endpoint_holds
plain-language theorem explainer
Entrywise equality of the finite Regge-TT and Lichnerowicz-TT edge kernels is a stencil-level sufficient condition for the TT Hessian/Lichnerowicz match on the N=5 periodic sector. Gravity auditors cite this as the Track 1.D kernel-entry reduction endpoint consumed by the Track 7 fork handoff. The proof builds row and match witnesses from the entry data and reuses the bilinear reduction endpoint.
Claim. Given periodic TT Hessian/Lichnerowicz kernel entry data at $N=5$, there exist nonempty row data and match data for those kernels, and the bilinear TT Hessian/Lichnerowicz reduction endpoint holds. Equivalently, entrywise equality of the two finite edge kernels is a stronger stencil-level sufficient condition for the TT Hessian/Lichnerowicz match (it is enough that the residual edge kernel $\mathrm{ReggeTT}-\mathrm{LichnerowiczTT}$ annihilates every TT perturbation).
background
Track 7 is the integration-lane receipt for parallel gravity fork handoffs. It records what the new endpoints prove without upgrading the discovery claim, and leaves remaining Track 1 displacement-class leaves as the next dependency. Track 1.D sits in the tensor-shear sector: one compares the discrete Regge TT Hessian on a periodic stencil against the continuum Lichnerowicz operator on transverse-traceless metric perturbations.
The endpoint proposition says that kernel entry data (edgewise equality of the two finite kernels at $N=5$) already yields nonempty row data, nonempty match data, and the bilinear reduction endpoint. That is a stronger stencil-level sufficient condition than working only with bilinear forms. Spatial dimension $D=3$ is the forced value from the unified forcing chain (T8/T9), which fixes the local geometry underlying the periodic torus cells and hinge-aware edge slots.
Upstream, the bilinear reduction endpoint is the immediate sibling this packages; hinge-aware zero-mode structure supplies the actual periodic edge assignment used by the stencil.
proof idea
Term-mode packaging, not a fresh analytic argument. Introduce the kernel entry data $D$. Build row data by PeriodicTTHessianLichnerowiczKernelRowData5.ofEntryData D, and match data by PeriodicTTHessianLichnerowiczMatchData5.ofKernelEntryData D. Return the triple of nonempty witnesses together with the already-proved bilinear reduction endpoint track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds. The constructors discharge the two Nonempty conjuncts; the third conjunct is the sibling endpoint.
why it matters
This is the Track 1.D entrywise edge-kernel reduction endpoint consumed by Track 7. Downstream it is wired into forkHandoffIntegrationCert, the integration-lane certificate that aggregates fork handoffs (Tracks 1.B stationarity, physical residual/Bianchi, many-body amplitude lift, Page capacity, dark-energy $w(z)$, and falsifier sensitivity).
In the Recognition gravity program the claim tightens the discrete-to-continuum bridge: matching Regge TT and Lichnerowicz TT at the edge-kernel stencil is enough to hand off the TT Hessian comparison, rather than re-proving the full bilinear identity at every use site. It sits under the forced $D=3$ geometry from the forcing chain and does not itself close remaining Track 1 displacement-class leaves; those stay open as the next dependency noted in the module doc.
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