track1D_tt_hessian_lichnerowicz_kernel_row_reduction_endpoint_holds
plain-language theorem explainer
Rowwise equality of the Regge TT Hessian edge kernel and the lattice Lichnerowicz edge kernel on transverse-traceless perturbations yields operator-match data and the bilinear TT energy match. Track 7 gravity integration cites this as the Track 1.D kernel-row reduction endpoint. The proof is a short term package: build match data from kernel-row data, then conjoin the already-proved bilinear reduction endpoint.
Claim. In the $N=5$ periodic setting, if the Regge TT Hessian and lattice Lichnerowicz operators agree entrywise on their finite edge kernels for transverse-traceless perturbations, then the TT Hessian/Lichnerowicz operator-match data is inhabited and the bilinear/quadratic TT energy reduction endpoint holds.
background
This module is the Track 7 fork-handoff integration lane: it records what the parallel gravity endpoints prove without upgrading the discovery claim. Track 1.D sits in the tensor-shear sector, comparing the discrete Regge TT Hessian to the lattice Lichnerowicz operator on transverse-traceless metric perturbations.
The endpoint proposition says that rowwise (entrywise) equality of the two finite edge kernels is a stencil-level sufficient condition for the operator match. From that match one obtains the bilinear and quadratic TT energy identities used downstream. Spatial dimension is the forced $D=3$ of the Recognition forcing chain (T8).
Upstream, the bilinear reduction endpoint is already available in-module; the kernel-row data type packages the entrywise kernel equality hypothesis that this theorem consumes.
proof idea
Term-mode packaging, not a computation. Introduce the kernel-row data hypothesis. Apply the sector constructor that turns kernel-row data into inhabited periodic TT Hessian/Lichnerowicz match data at $N=5$. Pair that witness with the already-proved bilinear reduction endpoint. The resulting pair is exactly the endpoint proposition (match data nonempty and bilinear endpoint).
why it matters
Closes the Track 1.D rowwise kernel-reduction leaf consumed by Track 7. The sole downstream consumer is the fork handoff integration certificate, which aggregates Track 1 stationarity/Schlaefli leaves, Track 2 many-body amplitude linearity, Page-capacity transfer, dark-energy $w(z)$ bands, and falsifier-sensitivity packaging.
In the Recognition gravity program this is the stencil-level bridge from discrete Regge TT curvature energy to continuum Lichnerowicz form on the shear sector. It strengthens the operator-match story beyond pure bilinear comparison: entrywise kernel equality is the finer sufficient condition. Remaining open work in the module is the displacement-class leaves of Track 1, not this kernel row.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.