PeriodicTTHessianLichnerowiczResidualRowSpanData5
plain-language theorem explainer
Packages a Regge TT Hessian edge kernel and a lattice Lichnerowicz edge kernel on the 5×5×5 periodic Freudenthal torus, plus the requirement that every residual row lies in the combined conformal-plus-longitudinal generator span. Track 1.D handoff endpoints cite it as the sharp finite stencil target for the TT Hessian/Lichnerowicz comparison. As a structure it only states the data obligation; constructors downstream discharge it from entry formulas.
Claim. A data package of two real edge-to-edge kernels $K_R$ (Regge TT Hessian) and $K_L$ (lattice Lichnerowicz) on the $5\times 5\times 5$ periodic Freudenthal torus such that, for every edge $e$, the residual row $(K_R-K_L)(e,\cdot)$ equals the image of some coefficient map on the combined conformal vertex-delta and longitudinal gauge index under the normal-equation generator map.
background
Track 1.D opens the tensor/shear sector of weak-field gravity on the encoded periodic Freudenthal torus. Track 1.B's conformal ansatz assigns one scalar potential per vertex and induces edge strains by endpoint averaging; that slice cannot represent pure shear, so it misses transverse-traceless gravitational-wave modes. This module separates independent edge perturbations from vertex-conformal ones and records the elementary rectangle obstruction for the conformal ansatz.
An edge-operator kernel is a real matrix indexed by pairs of periodic edges: the concrete finite surface on which the Regge TT Hessian stencil and the lattice Lichnerowicz stencil are compared. The residual kernel is their difference. The combined normal-equation index is the disjoint sum of encoded vertex indices (conformal generators) and longitudinal gauge indices (vertex-vector generators). The generator map sends a coefficient vector on that index to an edge-row vector.
Spatial dimension is fixed at $D=3$ by the forcing chain (T8/T9). The $5\times 5\times 5$ torus is the canonical finite triangulation used for Track 1.D stencil work.
proof idea
No proof body: this is a structure definition. It declares three fields: the Regge Hessian kernel, the lattice Lichnerowicz kernel (both of type edge-to-edge real matrix on the $5\times 5\times 5$ torus), and a Prop requiring that every residual row equals some generator-map image. Downstream ofFormulaData / ofEntryCoeffData constructors build inhabitants from entrywise residual formulas; the handoff layer only needs Nonempty of this type.
why it matters
This is the sharp finite stencil target for the TT Hessian versus lattice Lichnerowicz comparison in Track 1.D. Master-theorem handoff endpoints treat it as a required consequence of residual entry formulas: once every residual row is in the conformal-plus-longitudinal span, the residual annihilates the TT subspace and the kernel-row comparison closes.
Cited by Track1DTTHessianLichnerowiczResidualEntryFormulaReductionEndpoint, the encoded residual-entry endpoint, and the residual-row-coefficient endpoints in MasterTheoremHandoffIntegration. Those props feed Track 7's gravity handoff chain. The structure sits between raw stencil arithmetic and the claim that the residual vanishes on transverse-traceless modes, completing the shear sector that pure conformal Track 1.B cannot reach. Framework landmark: $D=3$ spatial dimensions on the eight-tick-compatible lattice geometry.
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