EncodedTTHessianLichnerowiczResidualOriginColumnTableFormulaData5
plain-language theorem explainer
Packages a typed origin-column residual certificate for the Regge Hessian minus lattice Lichnerowicz operator on the encoded 5×5×5 Freudenthal torus. Each residual entry is indexed by origin-row displacement, column base vertex, and column displacement, with a translation law recovering the full edge kernel. Gravity auditors cite it as the generator-facing surface that drops the explicit residual matrix. The body is a pure data structure of kernels, coefficient tables, and equational fields.
Claim. A residual origin-column table certificate on the canonical $5\times5\times5$ periodic Freudenthal torus consists of encoded edge kernels $H$ (Regge Hessian), $L$ (lattice Lichnerowicz), and residual $R=H-L$, together with a typed origin-column table $C:\mathrm{Fin}\,7\times V\times\mathrm{Fin}\,7\to\mathbb{R}$ and displacement coefficients $a$, such that $C$ equals $R$ on origin-row edges, every residual matrix entry is the translate of an origin-column entry, and each origin-column entry equals the TT normal-equation generator map applied to $a$.
background
Track 1.D opens the tensor/shear sector of the weak-field metric. Track 1.B's conformal ansatz puts one scalar potential at each vertex and averages endpoints to vary edge lengths; 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.
The ambient lattice is the canonical encoded $5\times5\times5$ periodic Freudenthal torus. Edges and vertices carry typed periodic indices (PeriodicEdge5, PeriodicVertex5). An encoded edge-operator kernel is a real matrix on the finite edge set Fin K.nE. The residual of interest is the difference between the Regge Hessian kernel and the lattice Lichnerowicz kernel on that edge set.
The seven origin-row displacements label a fundamental set of edge directions. The combined TT normal-equation index mixes fixed conformal vertex-delta generators with longitudinal vertex-vector generators. The generator map turns a coefficient row and a typed edge into a scalar residual entry.
proof idea
No proof obligation: this is a structure bundling data and equational fields. Inhabitants must supply three encoded edge kernels, the typed origin-column table, and residual displacement coefficients, then discharge four identities: residual equals the encoded Regge-minus-Lichnerowicz difference; the origin column matches the residual kernel on origin-row edges; every residual matrix entry is the translate of an origin-column entry via the periodic edge equivalence; and each origin-column entry equals the TT normal-equation generator map applied to the residual displacement coefficients.
why it matters
This is the generator-facing residual surface for Track 1.D: finite calculations need only the origin-column residual table and a translation law, not an explicit full residual matrix. Downstream, MasterTheorem handoff endpoints treat nonempty instances as reduction steps. The raw origin-column reduction endpoint consumes a raw typed-column certificate and produces this table form (then residual kernels and entry formulas). The residual origin-column table reduction endpoint starts from this structure and feeds the seven-row origin-table route, residual kernel, and entry certificates. The full-chain translated coefficient endpoint lists it among the audit targets exposing the whole TT Hessian/Lichnerowicz reduction. In the broader scaffold it closes the shear-sector residual bookkeeping needed before Track 7 consumes the handoff props. It does not itself force $D=3$ or the eight-tick octave; those enter only as ambient RS geometry.
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