EncodedTTHessianLichnerowiczCoeffRelativeTranslatedClosureData5
plain-language theorem explainer
Packages relative-frame translated residual formula data with the shifted-generator closure property on the period-5 torus. Anyone closing the Track 1.D TT Hessian–Lichnerowicz residual-zero route cites this as the preferred certificate: prove generator closure once, then TT orthogonality is free. It is a pure data structure, not a proved theorem.
Claim. A certificate consisting of (i) relative-frame coefficient data for the residual between the encoded Regge TT Hessian kernel and the lattice Lichnerowicz kernel on the period-5 torus (each residual entry equal to a row-rebased normal-equation generator map), together with (ii) the statement that every row-frame translate of a combined normal-equation generator splits into the fixed conformal image plus the longitudinal-gauge image.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on a discrete torus. The conformal (vertex-potential) ansatz cannot carry pure shear, so independent edge perturbations and transverse-traceless (TT) modes must be treated separately from vertex-conformal ones.
The relative-frame formula data records that residual stencil entries, after rebasing each row at its own edge, equal a fixed linear combination of the normal-equation generators. Generator closure asserts that every such row-frame translate still lies in the sum of the conformal generator image and the longitudinal-gauge image.
On the period-5 periodic edge complex, the comparison target is residual-kernel vanishing on the longitudinal-TT subspace: that is the compact finite check that the Regge second-variation TT block matches the lattice Lichnerowicz operator on physical TT modes.
proof idea
No proof: this is a structure bundling two fields. Downstream, ofClosureData builds the TT-zero package by keeping the relative formula data and obtaining shifted-generator orthogonality on TT modes from the closure field via the existing lemma that closure implies orthogonality. A further one-line path uses the already-proved global closure theorem so that relative translated formula data alone yields the same TT-zero package.
why it matters
This is the preferred mathematical target on the relative-frame residual-zero route: close generators once, then orthogonality follows from the TT definition rather than a separate ad-hoc check. The master handoff endpoint consumes exactly this package: given such closure data, one obtains a nonempty residual-kernel TT-zero certificate, which feeds the Hessian–Lichnerowicz match cascade (entrywise residual coefficients to row span to residual vanishing to operator match).
In the broader Recognition gravity track, matching the discrete Regge TT Hessian to the lattice Lichnerowicz operator on shear modes is the missing piece beyond the conformal Track 1.B slice. The structure sits inside TensorShearSector and is the sharper conditional endpoint quoted by Track 1.D handoff integration toward Track 7.
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