EncodedTTHessianLichnerowiczResidualDispRowFormulaData5
plain-language theorem explainer
Displacement-row certificate for the encoded residual kernel of the Regge TT Hessian minus the lattice Lichnerowicz operator on the 5×5×5 periodic Freudenthal torus. Translation collapses every matrix row onto one of seven origin-edge families indexed by displacement. Track 1.D gravity cites it as the handoff between a seven-row origin table and the full residual-kernel formula. The structure only packages kernels, the Fin-7 coefficient table, and three witnessing equalities.
Claim. A bundle of three encoded edge-operator kernels $K_{\mathrm{Regge}}$, $K_{\mathrm{Lich}}$, $K_{\mathrm{res}}$ on the $5\times 5\times 5$ periodic Freudenthal torus, with coefficients $c:\mathrm{Fin}\,7\to I\to\mathbb{R}$ ($I$ the combined conformal and longitudinal gauge generator index), such that $K_{\mathrm{res}}=K_{\mathrm{Regge}}-K_{\mathrm{Lich}}$ pointwise, every residual row equals the origin-edge row of the same displacement class, and each origin residual row equals the TT normal-equation generator map applied to $c(\mathrm{disp})$.
background
Track 1.D opens the tensor/shear sector of discrete gravity. Track 1.B assigns one scalar potential per vertex and induces edge strains by averaging endpoints; that conformal 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.
The ambient complex is the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus. An encoded edge-operator kernel is simply a real matrix indexed by Fin nE on that triangulation. The TT normal-equation index is the sum type of conformal vertex-delta generators and fixed longitudinal vertex-vector generators.
The residual under study is the difference of the Regge TT Hessian kernel and the lattice Lichnerowicz kernel. Edges fall into seven displacement classes relative to a chosen origin edge; translation equivariance of the residual lets every row be rewritten as the corresponding origin-edge row.
proof idea
No proof body: this is a structure (certificate bundle), not a theorem. The four data fields hold the Regge Hessian kernel, the lattice Lichnerowicz kernel, their residual kernel, and a seven-row coefficient table Fin 7 → I → ℝ. Three propositional fields enforce: (i) residual equals the pointwise encoded difference of Regge and Lichnerowicz; (ii) every residual row equals the origin-edge residual row of the same displacement; (iii) each origin residual row equals the periodic TT normal-equation generator map applied to the matching coefficient row. Downstream constructors such as ofOriginRowTableData inhabit the structure from a seven-row origin-table certificate.
why it matters
This is the intermediate certificate surface in the Track 1.D residual-kernel reduction ladder. The master-theorem handoff endpoint Track1DTTHessianLichnerowiczEncodedResidualDispRowFormulaReductionEndpoint consumes an inhabitant and produces nonempty residual-kernel, residual-entry, periodic residual-entry, and Lichnerowicz-match certificates. The dual origin-table endpoint reduces a seven-row table into this displacement-row form (and then onward). In-module, the residual-entry, residual-kernel, and periodic match structures all reference it.
Framework role: it is the finite, translation-normalized witness that the discrete TT Hessian on the D=3 Freudenthal torus (T8) differs from lattice Lichnerowicz by a generator-reconstructible residual, the algebraic step needed before matching continuum TT wave operators. It does not yet close the continuum or small-residual claims; those sit further down the handoff chain.
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