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plain-language theorem explainer
In the tensor/shear sector, a residual kernel is claimed to vanish on transverse-traceless (TT) edge perturbations once a separate orthogonality condition is imposed. Gravity and Regge-calculus workers would cite it when isolating pure shear from the vertex-conformal ansatz. The supplied extract has no proof body; the text only records that this orthogonality field is the gap left by Regge–Schläfli candidate diagnostics.
Claim. The residual kernel vanishes on transverse-traceless edge perturbations of the discrete geometry once a separate orthogonality field is imposed. That orthogonality condition is exactly the mathematical gap left open by the Regge–Schläfli candidate diagnostics in the tensor/shear sector.
background
Track 1.D builds the tensor/shear sector that Track 1.B's conformal ansatz cannot reach. The conformal ansatz assigns one scalar potential per vertex and induces edge-length changes by averaging endpoint potentials. That scalar slice cannot represent pure shear, so it cannot cover transverse-traceless gravitational-wave modes.
This module therefore separates independent edge perturbations from vertex-conformal ones and records elementary obstructions (for example, nontrivial rectangle shear is not vertex-conformal). The surrounding geometry imports Regge first-variation and periodic Freudenthal-torus constructions, so the natural language is edge-length strains on a periodic simplicial complex rather than continuum metric components.
Upstream names appearing in the dependency list (residual targets, BIT/ILG kernels, gap displays, hinge-aware zero modes) are ambient RS constants and kernel families; they fix notation for residuals and kernels but are not themselves the TT vanishing statement.
proof idea
No proof body is supplied for this declaration (zero body lines in the extract). The surrounding comment only asserts that some companion lemma, together with a separate orthogonality field, would force the residual kernel to vanish on TT perturbations, and that this orthogonality field is precisely the gap left by Regge–Schläfli candidate diagnostics. There is no tactic script, term proof, or wrapper application to walk.
why it matters
Pure shear and TT modes are the missing half of the weak-field metric sector once the vertex-conformal ansatz is fixed. Without a residual kernel that vanishes on those modes (under an explicit orthogonality condition), the discrete gravity track cannot claim that conformal diagnostics are cleanly separated from tensor/shear degrees of freedom.
No downstream uses are recorded for this declaration, so it currently sits as a local marker in the tensor/shear scaffold rather than a lemma consumed by a parent theorem. It touches the open gap between Regge–Schläfli hinge diagnostics and a full TT-orthogonal residual calculus on the periodic complex. Framework landmarks T0–T8, RCL, and the phi-ladder are not directly invoked here; the content is geometric and discrete-gravitational.
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