PeriodicTTHessianLichnerowiczResidualEntryFormulaData5
plain-language theorem explainer
Data package for the raw scalar TT residual identity on the canonical 5×5×5 periodic Freudenthal torus: Regge Hessian kernel minus lattice Lichnerowicz kernel equals a normal-equation generator-map reconstruction. Gravity auditors comparing discrete TT stencils to continuum Lichnerowicz would cite it. It is a structure bundling two edge kernels, residual row coefficients, and the entrywise equality axiom.
Claim. A package of a Regge Hessian edge kernel $H$, a lattice Lichnerowicz edge kernel $L$, and residual row coefficients $c$ on the $5\times 5\times 5$ periodic Freudenthal torus, such that for all edges $e,f$, $H(e,f)-L(e,f)$ equals the TT normal-equation 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 potential per 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.
The comparison surface is a finite edge-operator kernel: a real matrix indexed by pairs of edges of the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus. The Regge TT Hessian stencil and the lattice Lichnerowicz stencil are both written in that form. Residual rows are indexed by a combined normal-equation index (conformal vertex-delta generators plus longitudinal vertex-vector generators). Spatial dimension is the forced $D=3$ of the forcing chain.
proof idea
No proof body: this is a structure definition. It packages two PeriodicEdgeOperatorKernel5 fields (Regge Hessian and lattice Lichnerowicz), a residual-row coefficient map from edges to normal-equation indices, and a single Prop field asserting the unwrapped finite identity that the kernel difference equals the generator-map reconstruction of that residual row. Instantiation later supplies concrete kernels and coefficients and discharges the identity.
why it matters
In the continuum, the Lichnerowicz operator governs TT metric perturbations; on the Regge side the second variation of the action supplies a discrete Hessian. Their difference, projected orthogonal to conformal and longitudinal gauge generators, is the TT residual that must vanish (or be controlled) for the discrete theory to reproduce linearized GR shear modes. This structure is the certificate surface for that raw scalar residual formula on the $N=5$ encoded torus, with edge indexing over the finite Fin edge set of the triangulation. No downstream consumers are wired yet; it is scaffolding for later kernel-extraction and residual-vanishing theorems in Track 1.D. It sits downstream of the eight-tick / $D=3$ geometry (T7–T8) only insofar as the Freudenthal torus and dimension constants are fixed.
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