PeriodicTTHessianLichnerowiczResidualRowCoeffData5
plain-language theorem explainer
Packages the residual between a Regge TT Hessian edge-kernel and a lattice Lichnerowicz edge-kernel on the 5³ periodic Freudenthal torus as an explicit coefficient table: each residual row equals the generator-map of a conformal/longitudinal coefficient vector. Track 1.D handoff endpoints and the residual-row-span layer cite this as the finite certificate surface. It is a structure definition (fields plus one equality axiom), not a proved identity.
Claim. A residual-row coefficient datum on the $5\times5\times5$ periodic Freudenthal torus consists of two edge-operator kernels $K_R$ (Regge Hessian) and $K_L$ (lattice Lichnerowicz), together with a map $c$ sending each edge $e$ to a real coefficient vector indexed by the combined conformal vertex-delta and longitudinal gauge generators, such that for every edge $e$ the residual row $(K_R-K_L)(e,\cdot)$ equals the generator-map reconstruction of $c(e)$.
background
Track 1.D opens the tensor/shear sector of the weak-field metric. Track 1.B's conformal ansatz assigns one scalar potential per vertex and induces edge-length changes by averaging endpoints; 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 $5\times5\times5$ periodic Freudenthal torus.
An edge-operator kernel is a finite matrix on periodic edges: the concrete surface where the Regge TT Hessian stencil and the lattice Lichnerowicz stencil are compared. The residual kernel is their difference. The combined normal-equation index runs over conformal vertex-delta generators plus longitudinal vertex-vector generators; the generator map turns a coefficient vector on that index into an edge-row.
The structure records the finite table the remaining TT stencil calculation is expected to produce: for each edge-row, coefficients whose generator map recovers the residual row exactly.
proof idea
Definitional structure, not a tactic proof. Four fields: the Regge Hessian kernel, the lattice Lichnerowicz kernel, the residual-row coefficient map (edge to combined conformal/longitudinal coefficients), and the axiom that every residual kernel row equals the generator-map image of that row's coefficient vector. Downstream constructors (ofEncodedData, ofFormulaData, ofEntryData) inhabit the structure from coarser residual formulas or entrywise identities; those inhabitations live in the handoff layer, not here.
why it matters
This is the explicit coefficient form of the residual-row span target on Track 1.D. Master-theorem handoff endpoints treat it as an intermediate certificate: raw or encoded residual entry formulas reduce through entrywise row-coeff data into this structure, then into residual-row span and kernel residual TT-zero data. Downstream docs state that entrywise residual-row coefficient identities (and the coarser formula layers above them) are enough to close the TT Hessian/Lichnerowicz consequence route consumed by Track 7.
In the Recognition gravity program this is the finite stencil surface where discrete Regge curvature second variation is matched against the lattice Lichnerowicz operator on pure shear, the sector missing from the conformal ansatz. Spatial dimension $D=3$ (forcing chain T8/T9) is already baked into the torus encoding. The structure does not itself prove residual vanishing on TT modes; it packages the coefficient table those vanishing theorems require.
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