Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedTTZeroEndpoint
plain-language theorem explainer
The Track 1.D conditional endpoint asserts that relative-frame translated TT Hessian/Lichnerowicz coefficient data, plus shifted-generator orthogonality on TT modes, yields a nonempty residual-zero certificate for the periodic TT Hessian kernel. Track 7 handoff consumers cite it as the named bridge from physical translation covariance into the TT residual route. As a Prop abbreviation it packages that implication; the companion holds theorem discharges it via the sector constructor.
Claim. If relative-frame translated encoded TT Hessian/Lichnerowicz coefficient data at $N=5$ is available together with the shifted-generator orthogonality lemma on TT perturbations, then there exists a residual-zero certificate for the periodic TT Hessian/Lichnerowicz kernel.
background
This module is the Track 7 integration-lane receipt for parallel fork handoffs (A through F). It records what the new endpoints prove without upgrading the discovery claim, and keeps remaining Track 1 displacement-class leaves as the next dependency.
In the shear sector, the discrete gravity Hessian on transverse-traceless (TT) metric perturbations is compared to the Lichnerowicz operator (the linearized Einstein operator on TT tensors). Residual-zero means that comparison kernel vanishes on the TT subspace once coefficient data and generator constraints are in place.
The upstream package supplies relative-frame translated formula data plus a shifted-generator orthogonality field on TT modes. Its doc states: "Relative-frame translated data plus the shifted-generator orthogonality lemma is enough to prove the residual kernel vanishes on TT perturbations. The separate orthogonality field is the exact mathematical gap left by the Regge Schläfli candidate diagnostics."
proof idea
This is a Prop definition, not a proved theorem. The body is the bare implication from the relative-frame translated TT-zero data package to nonempty residual-zero kernel data on the periodic TT Hessian/Lichnerowicz side.
The companion holds theorem discharges the Prop in one step: introduce the data hypothesis and apply the sector constructor ofCoeffRelativeTranslatedTTZeroData, which builds the residual-zero witness from that package. No independent algebraic work lives here.
why it matters
Names the exact remaining bridge from physical translation covariance to the TT Hessian/Lichnerowicz residual route (Track 1.D), as the doc-comment states. Downstream, the holds theorem, ForkHandoffIntegrationCert, and the one-statement fork A/B/C/D/E/F integration consume this endpoint as a named handoff fact.
The integration one-statement deliberately does not assert the fully unconditional discovery theorem; it only records what the forks now supply. This definition keeps the Track 1.D leaf explicit so Track 7 can cite a stable Prop while the shifted-generator orthogonality gap (left by Regge–Schläfli diagnostics) is closed upstream in the tensor shear sector.
It sits among the Track 1 displacement-class endpoints that the module flags as the next dependency after the structural master certificate.
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