PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5
plain-language theorem explainer
Packages two finite edge-operator kernels (Regge Hessian and lattice Lichnerowicz) with the demand that their residual annihilates every longitudinal-gauge transverse-traceless edge perturbation on the 5-periodic setting. Discrete-gravity workers comparing TT stencils cite this as the compact data carrier for that match. It is a structure definition: no proof body, only the residual-vanishing field to be filled by an instance.
Claim. A data package of two periodic edge-operator kernels $K_{\mathrm{Regge}}$ and $K_{\mathrm{Lich}}$ together with the assertion that for every edge perturbation $\varepsilon$ in the longitudinal-gauge transverse-traceless subspace, the residual operator built from $K_{\mathrm{Regge}}$ and $K_{\mathrm{Lich}}$ sends $\varepsilon$ to the zero edge field.
background
Track 1.D opens the tensor/shear sector of the discrete gravity program. Track 1.B's conformal ansatz puts one scalar at each vertex and induces edge-length changes by averaging endpoints; that slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and records elementary rectangle obstructions for the conformal ansatz.
A periodic edge-operator kernel is a real matrix indexed by pairs of periodic edges: the concrete finite stencil surface on which the Regge TT Hessian and the lattice Lichnerowicz operator are compared. The longitudinal-gauge TT subspace is the concrete TT predicate for edge fields orthogonal to the image of the longitudinal gauge map (vertex-vector delta generators).
The ambient spatial dimension is the forced value $D=3$ from the Recognition forcing chain (linking requires three spatial dimensions). The residual kernel is the difference of the two stencils; vanishing of that residual on TT modes is the sharp finite comparison target.
proof idea
Definitional structure, not a proved theorem. The three fields are: the Regge Hessian kernel, the lattice Lichnerowicz kernel (both of edge-to-edge real-matrix type), and a Prop field requiring that the residual kernel operator annihilates every edge field in the longitudinal-gauge TT subspace. Instantiators must supply both kernels and discharge residual vanishing; there is no tactic script or wrapper lemma inside the declaration itself.
why it matters
Residual-kernel vanishing on TT perturbations is, per the doc-comment, the most compact finite calculation target for the TT Hessian-to-Lichnerowicz comparison in the Recognition gravity track. Without a shear/tensor sector, the conformal ansatz cannot represent TT gravitational-wave modes; this structure is the data shape that makes the discrete stencil match checkable on the 5-periodic Freudenthal torus.
It sits inside Track 1.D after the rectangle obstruction lemmas that show nontrivial shear is not vertex-conformal. Framework landmarks in play are $D=3$ (forcing chain) and the move from scalar conformal edge strains to independent edge perturbations. No downstream consumers are wired yet (used_by empty), so the structure is presently a scaffold interface waiting for concrete kernel instances and the residual-zero proof.
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