PeriodicTTHessianLichnerowiczResidualRowCoeffEntryData5
plain-language theorem explainer
Packages a certificate that every residual kernel entry (Regge TT Hessian minus lattice Lichnerowicz) on the 5×5×5 periodic edge set equals the corresponding generator-map reconstruction from a residual-row coefficient table. Gravity Track 1.D and MasterTheorem handoff endpoints cite it as the entrywise scalar form of the residual-row target. It is a pure data structure: kernels, coefficients, and the entrywise identity field.
Claim. A bundle of data on the $5\times 5\times 5$ periodic Freudenthal torus consisting of: a Regge TT Hessian edge-operator kernel $K_{\mathrm{Regge}}$, a lattice Lichnerowicz edge-operator kernel $K_{\mathrm{Lich}}$, and a residual-row coefficient map $c$ from edges to normal-equation generator indices, such that for all edges $e,f$, the residual $K_{\mathrm{Regge}}(e,f)-K_{\mathrm{Lich}}(e,f)$ equals the generator-map reconstruction of the row $c(e)$ at $f$.
background
Track 1.D opens the tensor/shear sector of weak-field gravity on the Recognition lattice. Track 1.B's conformal ansatz assigns one scalar per vertex and averages endpoint potentials onto edges; 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 works on the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus.
An edge-operator kernel is a real matrix indexed by pairs of periodic edges: the concrete surface on which the Regge TT Hessian stencil and the lattice Lichnerowicz stencil are compared. The residual kernel is their entrywise difference. Normal-equation indices combine fixed conformal vertex-delta generators with fixed longitudinal vertex-vector generators; the generator map rebuilds an edge-row from coefficients on that index set.
Spatial dimension $D=3$ (forced in the RS chain) underlies the torus geometry. The residual language here is geometric (kernel mismatch), not the fine-structure residual against CODATA.
proof idea
No proof body: this is a structure definition. It packages two PeriodicEdgeOperatorKernel5 fields (Regge Hessian and lattice Lichnerowicz), a residual-row coefficient table of type PeriodicEdge5 → PeriodicTTNormalEquationIdx5 → ℝ, and a Prop field asserting the entrywise identity that the residual kernel equals the generator-map reconstruction of each row. Downstream constructors (e.g. ofFormulaData, ofEncodedData) inhabit the structure by supplying kernels, coefficients, and discharging that identity.
why it matters
This is the certificate-friendly scalar form of the residual-row coefficient target on Track 1.D: every residual kernel entry equals the corresponding generator-map entry. MasterTheorem handoff endpoints require nonempty instances of this structure along the reduction chain from encoded residual kernels and raw scalar formulas down to residual-row span and TT-kernel residual vanishing (PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5).
Downstream docs state the route explicitly: raw formulas Regge(e,f) - Lichnerowicz(e,f) = generatorCoeff(e)(f) suffice to close the TT Hessian/Lichnerowicz consequence path consumed by Track 7. The structure sits between entry-formula data and row-coefficient/span data, so inhabiting it is a mandatory handoff step for matching discrete Regge shear to continuum Lichnerowicz on the periodic torus. It does not itself force $D=3$ or the eight-tick octave; those enter via the ambient geometry and forcing chain.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.