EncodedTTHessianLichnerowiczResidualOriginRowTableFormulaData5
plain-language theorem explainer
Packages the seven origin-row residual certificate for the encoded TT Hessian versus lattice Lichnerowicz comparison on the 5×5×5 periodic Freudenthal torus. A generator emits only those seven residual rows plus displacement coefficients; translation identities recover every matrix row. Downstream Track 1.D handoff endpoints cite this as the row-table surface in the residual reduction chain.
Claim. A data package on the encoded $5\times5\times5$ periodic Freudenthal torus consisting of three edge-operator kernels (Regge Hessian, lattice Lichnerowicz, and their residual), a seven-row origin residual table $R:\mathrm{Fin}\,7\times E\to\mathbb{R}$, and residual displacement coefficients $c_d$ on the combined TT normal-equation index, such that: the residual kernel equals the pointwise difference of the two operator kernels; each origin row is the residual kernel evaluated at the origin edge of that displacement; every residual-kernel row equals the origin-row table entry for the row edge's displacement class; and each origin-row entry equals the periodic TT normal-equation generator map applied to $c_d$ at the column edge.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on a finite triangulation. The older Track 1.B conformal ansatz puts one scalar at each vertex and averages endpoints to edge lengths; that slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes. This module therefore treats independent edge perturbations separately from vertex-conformal ones.
The ambient lattice is the canonical encoded $5\times5\times5$ periodic Freudenthal torus. An encoded edge-operator kernel is simply a real matrix indexed by the finite edge set $E$ of that torus. The residual kernel is the difference between the discrete Regge Hessian kernel and the lattice Lichnerowicz kernel on those edges.
The seven origin rows correspond to the seven edge-displacement classes relative to a fixed origin. Combined normal-equation indices mix conformal vertex-delta generators with longitudinal gauge generators. The structure records both the residual table and the coefficient data that regenerate each table entry via the periodic TT normal-equation generator map.
proof idea
No proof body: this is a structure (definitional certificate bundle), not a theorem. Inhabitants are built by supplying the three kernels, the seven-row table, the displacement coefficients, and four propositional fields that lock the algebraic relations (residual as kernel difference; origin-row extraction; row translation by displacement class; entry formula via the generator map). Downstream constructors such as ofRawOriginColumnData and ofOriginColumnTableData assemble instances from coarser residual certificates.
why it matters
This is the generator-facing seven-row residual surface in the Track 1.D TT Hessian/Lichnerowicz reduction. Master-theorem handoff endpoints require nonempty instances of this structure as part of the full residual chain: the coefficient-translated full-chain endpoint, the raw origin-column reduction endpoint, and the residual origin-column table reduction endpoint all list it among their audit targets, and the corresponding ..._holds theorems construct it from upstream data.
In framework terms it sits inside the gravity tensor/shear scaffold that aims to cover TT modes the conformal ansatz cannot reach, on the $D=3$ spatial lattice forced by the forcing chain. It does not itself compute masses, $\alpha$, or continuum GR limits; it only certifies the finite residual table that later tracks consume when closing the discrete Lichnerowicz comparison.
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