Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosedTTZeroEndpoint
plain-language theorem explainer
Defines the Track 1.D closed relative endpoint: a relative-frame translated coefficient certificate for the TT Hessian versus Lichnerowicz residual implies residual-zero and kernel-match data on the periodic TT sector at N=5. Gravity auditors and Track 7 integrators cite it as the Prop-shaped handoff gate. It is a pure implication Prop, not a proved theorem.
Claim. The Track 1.D closed relative TT-zero endpoint is the proposition: if relative-frame coefficient-only translated residual formula data exist for the encoded TT Hessian versus lattice Lichnerowicz kernels at $N=5$ (translation-covariant after each row is re-based at its edge base), then both the periodic TT residual-zero kernel data and the periodic TT Hessian–Lichnerowicz match data are nonempty.
background
This module is the Track 7 fork-handoff integration lane. It records what parallel forks prove without upgrading the discovery claim: Fork A (Track 1.B Schläfli stationarity at $N=5$), Fork B (physical residual/Bianchi), Fork C (many-body amplitude-linear lift), and related Track 3–6 packages. Remaining Track 1 displacement-class leaves stay open.
Upstream, the relative-frame coefficient structure packages encoded Regge Hessian and lattice Lichnerowicz edge-operator kernels together with a residual dispersion coefficient table and an entrywise relative residual formula. Its doc states that generated physical stencils are translation-covariant after each row is re-based at its own edge base, recorded separately from any stronger absolute translated certificate.
TT here is the transverse-traceless shear sector; residual-zero means the Hessian–Lichnerowicz mismatch vanishes on that sector; match data certify kernel agreement. The endpoint is the Prop that turns the relative coefficient certificate into those two nonempty data packages.
proof idea
No proof body: this is a def equating a name to an implication Prop. The antecedent is the relative-frame translated coefficient formula data structure; the consequent is the conjunction of nonempty residual-zero data and nonempty match data on the periodic TT Hessian–Lichnerowicz kernels.
The companion theorem discharges the Prop by introducing the coefficient data and packaging two constructors: residual-zero data of the relative translated certificate, and match data of the same certificate. The definition itself only names that implication gate.
why it matters
Track 7 consumes this endpoint as the closed relative-frame translated TT-zero handoff after shifted-generator closure. The holding theorem feeds the fork integration certificate and the one-statement A–F handoff theorem, which packages many-body lift, Schläfli-to-stationarity reduction, physical residual/Bianchi interface, Page tick-capacity, $w(z)$ falsifier bands, and falsifier-sensitivity without asserting the unconditional discovery theorem.
In the gravity master plan this is a Track 1.D interface leaf: coefficient-level relative translation covariance on the TT shear sector supplies residual-zero, tightening the discrete Lichnerowicz comparison used in stationarity and residual tracks. It does not close open Schläfli or displacement-class leaves; those remain the next dependency noted in the module doc.
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