Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosureEndpoint
plain-language theorem explainer
Defines the Track 1.D conditional endpoint: if relative-frame translated TT-Hessian/Lichnerowicz coefficient data carry generator closure, then a nonempty residual TT-zero package exists. Gravity Track 7 cites it as the relative-frame residual-zero handoff. The body is a pure Prop abbreviation (implication to Nonempty), not a proved theorem.
Claim. The Track 1.D relative-frame translated closure endpoint is the proposition: whenever one has encoded TT-Hessian/Lichnerowicz coefficient data on relative row-frame translates together with periodic relative TT generator closure (at the $N=5$ instance), the set of periodic TT-Hessian/Lichnerowicz kernel residual TT-zero data is nonempty.
background
Module context is Gravity Track 7 fork-handoff integration: a receipt lane that records what parallel forks prove without upgrading the unconditional discovery claim. Track 1 leaves (displacement-class reductions) remain open dependencies; this endpoint is the sharper Track 1.D relative-frame residual-zero route.
Upstream, EncodedTTHessianLichnerowiczCoeffRelativeTranslatedClosureData5 packages relative-frame translated formula data with a periodic relative TT generator-closure witness. Its doc states the preferred target: "prove closure once, then orthogonality follows from the existing TT definition." Generator closure is what supplies the orthogonality package for the conditional relative-frame TT-zero route.
In linearized gravity language, the Lichnerowicz operator and TT (transverse-traceless) Hessian control shear-sector residuals on a discrete stencil; residual TT-zero means the kernel residual vanishes in the TT sector after the relative-frame translation.
proof idea
No proof: this is a def equating a name to a Prop. The proposition is the implication from the upstream closure-data structure to Nonempty of the periodic residual TT-zero data type. Downstream, the companion ..._endpoint_holds theorem discharges it in one intro step by applying PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5.ofCoeffRelativeTranslatedClosureData to the given closure datum.
why it matters
Track 7 consumes this endpoint as the relative-frame translated closure handoff inside the fork A–F integration certificate and the one-statement integration theorem. Those parents deliberately "do not assert the fully unconditional discovery theorem"; they record stronger handoffs while structural master witnesses stay structural where the plan requires.
Within Recognition gravity, this sharpens Track 1.D: generator-closure on every relative row-frame translate closes the residual-zero route without finishing open Schläfli/displacement leaves. It sits beside Fork A stationarity reduction, Fork B physical residual/Bianchi, and the shear-sector TT packaging, feeding the master handoff rather than claiming full TT-kernel vanishing unconditionally.
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