EncodedTTHessianLichnerowiczCoeffRelativeTranslatedFormulaData5
plain-language theorem explainer
Packages a relative-frame residual certificate comparing the Regge Hessian to the lattice Lichnerowicz operator on the 5×5×5 periodic Freudenthal torus. Every residual matrix entry equals a fixed displacement-coefficient table evaluated on the column edge rewritten in the row edge's base frame. Track 1.D gravity workers cite it to separate translation-covariant stencil diagnostics from stronger absolute translated certificates. Pure structure: data fields plus one universal equality, no derivation.
Claim. A relative-frame translated residual formula datum consists of two encoded edge-operator kernels $K_{\mathrm{Regge}}$ and $K_{\mathrm{Lich}}$ on the $5\times 5\times 5$ periodic Freudenthal torus, together with residual displacement coefficients $c:\{0,\ldots,6\}\to I\to\mathbb{R}$ ($I$ the combined conformal vertex-delta and longitudinal gauge index set), such that for all edges $e,f$, $(K_{\mathrm{Regge}}-K_{\mathrm{Lich}})_{e,f}=G\bigl(c(\mathrm{disp}(e)),\,\mathrm{rel}_e(f)\bigr)$, where $\mathrm{rel}_e(f)$ is the column edge expressed in the coordinate frame of $e$, and $G$ is the periodic TT normal-equation generator map.
background
Track 1.D opens the tensor/shear sector of weak-field gravity on the encoded periodic Freudenthal triangulation. The earlier conformal ansatz only varies edge lengths by averaging endpoint vertex potentials, so it cannot represent pure shear and therefore cannot cover transverse-traceless gravitational-wave modes. This module isolates independent edge perturbations from vertex-conformal ones and builds discrete Hessian comparisons against a lattice Lichnerowicz operator.
An encoded edge-operator kernel is a real matrix on the finite edge index set of the canonical $5\times 5\times 5$ torus. The residual kernel is their entrywise difference. The relative-column construction rewrites a column edge in the base frame of a chosen row edge, converting translation covariance of physical stencils into a single coefficient table on relative indices. Combined normal-equation indices package fixed conformal vertex-delta generators with fixed longitudinal vertex-vector generators.
proof idea
No proof body: the declaration is a structure (definition). An inhabitant supplies two encoded edge kernels, a coefficient table on the seven edge displacements times the normal-equation index set, and one universal equality: every residual entry equals the periodic TT normal-equation generator map applied to those coefficients at the relative column of the row. Downstream lemmas unpack the fields; nothing is derived at this site.
why it matters
This is the coefficient-only relative diagnostic surface for Track 1.D. It feeds origin-column formula data, the relative translated TT-zero package (paired with a shifted-generator orthogonality lemma), and the preferred closure package (paired with generator closure). Master-theorem handoff endpoints treat it as the hypothesis of both a weak diagnostic endpoint (origin-column consequence shared with the absolute route) and a closed TT-zero endpoint once shifted-generator closure is available. The doc-comment stresses that generated physical stencils are translation-covariant after each row is re-based at its own edge base, recorded separately from stronger absolute translated certificates. It sits on the path toward residual vanishing on transverse-traceless modes in the discrete Regge-versus-Lichnerowicz comparison.
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