Track1DTTHessianLichnerowiczEncodedCoeffTranslatedReductionEndpoint
plain-language theorem explainer
Track 1.D reduction endpoint: a coefficient-only translated residual certificate for the TT Hessian/Lichnerowicz operator yields both nonempty origin-column coefficient data and the origin-column reduction endpoint. Gravity integrators cite it when routing the smaller translated surface into the origin-column chain toward Track 7. It is a pure implication Prop, not a proved existence claim.
Claim. If coefficient-only translated residual formula data exist for the transverse-traceless Hessian/Lichnerowicz residual (two edge-operator kernels, seven residual-generator coefficient rows, and the encoded residual entry formula), then the corresponding origin-column coefficient formula data are nonempty and the origin-column reduction endpoint holds.
background
This module is the Track 7 fork-handoff integration lane. It records what parallel gravity forks prove without upgrading the discovery claim, and leaves open Track 1 displacement-class leaves as the next dependency.
The transverse-traceless (TT) Hessian/Lichnerowicz residual is packaged in the tensor-shear sector by coefficient-only certificates. The translated residual formula data store the Regge Hessian kernel, the lattice Lichnerowicz kernel, seven residual-generator coefficient rows, and an encoded residual entry formula. That surface is strictly smaller than the origin-column package: once the translated formula is known, origin-column scalars follow by specializing the row to the origin edge.
Upstream, the origin-column reduction endpoint says a coefficient-only origin-column certificate feeds the raw typed-column residual route without storing a residual origin table. The present definition sits one step earlier: it takes the translated certificate as hypothesis and demands both nonempty origin-column data and that origin-column endpoint.
proof idea
Definitional packaging only: the declaration is a Prop abbreviation, not a tactic or term proof. It is the implication from translated coefficient-only formula data to the conjunction of (i) nonempty origin-column coefficient formula data and (ii) the already-defined origin-column reduction endpoint. The companion theorem discharges it by constructing origin-column data via the sector's of-translated constructor and invoking the origin-column endpoint theorem.
why it matters
This is the Track 1.D translated coefficient-only residual endpoint consumed by Track 7. Downstream, the holds theorem asserts the Prop, and the fork handoff integration certificate structure consumes the Track 1 reduction/interface package alongside many-body, Schläfli, and displacement endpoints. The module doc is explicit: the Track 1 result is a reduction/interface package, not a closure of the open Schläfli leaves. In the Recognition gravity stack it tightens the certificate surface for the TT shear residual so later forks can cite a smaller generator-facing handoff without re-proving origin-column algebra.
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