PeriodicTTHessianLichnerowiczKernelRowData5
plain-language theorem explainer
Packages two finite edge-operator kernels (Regge second-variation Hessian and lattice spin-2 Lichnerowicz) on the 5³ periodic Freudenthal torus with the demand that they act identically on every longitudinal-gauge TT edge perturbation. Track 1.D gravity handoff cites it as the rowwise sufficient condition for the TT Hessian–Lichnerowicz operator match. Pure structure: no proof body, only field obligations.
Claim. A data package of two real edge-to-edge kernels $K_{\mathrm{Regge}}, K_{\mathrm{Lich}} : E_5 \to E_5 \to \mathbb{R}$ on the $5\times 5\times 5$ periodic Freudenthal torus, together with the requirement that for every longitudinal-gauge transverse-traceless edge perturbation $\varepsilon$ and every edge $e$, $(K_{\mathrm{Regge}}\varepsilon)(e)=(K_{\mathrm{Lich}}\varepsilon)(e)$.
background
Track 1.D opens the tensor/shear sector that Track 1.B's vertex-conformal ansatz cannot reach. Conformal edge strains are averages of endpoint scalar potentials; they miss pure shear and therefore miss transverse-traceless gravitational-wave modes. The module isolates independent edge perturbations on the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus and equips them with a concrete longitudinal-gauge TT predicate.
An edge-operator kernel is a finite matrix $E_5\to E_5\to\mathbb{R}$ acting on periodic edge perturbations. That matrix surface is where the Regge second-variation edge Hessian stencil is meant to be compared with the lattice Lichnerowicz (spin-2) stencil. The TT subspace here is the orthogonal complement, inside edge space, to the image of the longitudinal gauge map generated by vertex-vector deltas.
Spatial dimension is fixed at $D=3$ by the forcing chain (T8/T9). The present structure does not yet name residual or BIT-kernel constants; those appear only in adjacent residual-formula routes.
proof idea
Definitional structure, not a proved theorem. Three fields: (i) a Regge Hessian edge kernel, (ii) a lattice Lichnerowicz edge kernel, both of type edge-to-edge real matrix on the $5^3$ torus, and (iii) a Prop asserting that the two kernels, applied as operators to any longitudinal-gauge TT perturbation, return identical values at every edge. Instantiation is the work: supply the two stencils and discharge the row-match obligation. Downstream constructors such as ofEntryData lift stronger entrywise kernel data into this rowwise package.
why it matters
This is the rowwise kernel interface that MasterTheoremHandoffIntegration consumes for Track 1.D. The kernel-row reduction endpoint takes an inhabitant of this structure and produces nonempty TT Hessian–Lichnerowicz match data plus the bilinear/quadratic TT energy matches. The stronger entrywise kernel endpoint reduces through this structure (ofEntryData) before handing off the same match data. Residual-entry and residual-row-coeff endpoints sit on parallel routes that ultimately feed the same operator-match and Track 7 handoff.
Physically it is the stencil-level bridge from discrete Regge calculus second variation to continuum spin-2 Lichnerowicz on TT modes, the missing shear sector after the conformal Track 1.B slice. It does not itself close the match; it names the data a future proof must fill. Framework landmarks: $D=3$ torus geometry and the eight-tick/octave discrete setting that forces the Freudenthal triangulation used here.
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