Track1DTTHessianLichnerowiczBilinearReductionEndpoint
plain-language theorem explainer
Whenever Regge TT Hessian and lattice Lichnerowicz operators are identified on longitudinal TT modes, their induced bilinear and quadratic TT energies agree. Gravity auditors cite this Track 1.D endpoint as the energy-level handoff from operator match data to TT energy equality. The declaration is a Prop packaging that equality; the companion holds theorem discharges it from the bilinear-of-match lemma.
Claim. For every periodic TT Hessian/Lichnerowicz match datum $D$ and every pair of periodic edge perturbations $\varepsilon,\eta$ in the longitudinal TT subspace, the bilinear form induced by the Regge TT Hessian equals that induced by the lattice Lichnerowicz operator on $(\varepsilon,\eta)$, and the quadratic energies on $\varepsilon$ likewise agree: $B_{\mathrm{Regge}}(\varepsilon,\eta)=B_{\mathrm{Lich}}(\varepsilon,\eta)$ and $B_{\mathrm{Regge}}(\varepsilon,\varepsilon)=B_{\mathrm{Lich}}(\varepsilon,\varepsilon)$.
background
Track 7 is the fork-handoff integration lane for gravity: it records what parallel forks prove without upgrading the discovery claim. Track 1.D sits in the tensor-shear sector on the fixed $N=5$ periodic Freudenthal triangulation (spatial dimension $D=3$ from the forcing chain T8).
Edge perturbations are real functions on typed periodic edges. The longitudinal TT subspace is the concrete TT-orthogonal complement to the longitudinal gauge image. An edge-space operator induces a bilinear form via the finite periodic-edge inner product: $B_{\mathrm{op}}(\varepsilon,\eta)=\langle\varepsilon,\mathrm{op},\eta\rangle$.
Match data packages a Regge TT Hessian operator and a lattice Lichnerowicz TT operator already identified on TT modes. The endpoint only asks that those operators yield equal bilinears (and equal quadratic energies) once both arguments lie in the longitudinal TT subspace.
proof idea
Pure Prop definition: no proof body beyond the quantified equality. It packages the statement that for any match datum $D$ and any longitudinal-TT pair $\varepsilon,\eta$, the periodic TT operator bilinears of $D$'s Regge Hessian and lattice Lichnerowicz operators agree on $(\varepsilon,\eta)$ and on $(\varepsilon,\varepsilon)$.
The companion holds theorem discharges it in one intro-and-exact step by applying the upstream bilinear-of-match lemma (twice: mixed pair and diagonal), which already knows that pointwise operator agreement on TT modes lifts to bilinear equality under the finite edge inner product.
why it matters
This is the energy-level receipt for Track 1.D inside the Track 7 handoff. Downstream, the holds theorem and the fork integration certificate consume it; a chain of encoded residual/column/row reduction endpoints (coeff-translated full chain, raw origin column, residual disp-row, residual entry, residual kernel) builds on the same match-data spine so auditors can see the finite TT Hessian/Lichnerowicz reduction in one place.
In the Recognition gravity program the Regge Hessian is the discrete second variation of the action on edge lengths; the lattice Lichnerowicz operator is the continuum TT wave operator discretized on the same complex. Matching their bilinears on longitudinal TT modes is the discrete stand-in for equal TT kinetic energy, a prerequisite for graviton-sector consistency on the eight-tick, $D=3$ lattice. It does not close open Schläfli or displacement-class leaves; those remain the next Track 1 dependency flagged by the module doc.
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