Track1DTTHessianLichnerowiczEncodedRawOriginColumnReductionEndpoint
plain-language theorem explainer
A raw typed-column residual certificate for the Regge TT Hessian versus lattice Lichnerowicz operators on the N=5 periodic lattice yields residual origin-column/row tables, residual kernel, residual-entry and pointwise match packages, plus the bilinear TT energy-match endpoint. Track 1.D and Track 7 handoff code cite it as the matrix-free residual route. The declaration is a Prop implication packaging Nonempty certificate witnesses with the bilinear endpoint.
Claim. If a raw typed-column residual certificate is given for the encoded Regge TT Hessian and lattice Lichnerowicz operators on the $N=5$ periodic lattice (origin-column residual table and translation law only; no full residual matrix), then residual origin-column table, origin-row table, residual-kernel, residual-entry, and pointwise operator-match certificates exist, and the bilinear/quadratic TT energy-match endpoint holds.
background
Module setting is Gravity Track 7 fork-handoff integration: it records what parallel fork endpoints prove without upgrading the discovery claim, and leaves open Track 1 displacement-class leaves as the next dependency.
Track 1.D compares the Regge transverse-traceless (TT) Hessian to the lattice Lichnerowicz operator on TT modes of the $N=5$ periodic edge lattice. The raw typed-column residual certificate supplies only the origin-column residual table and a translation law for the encoded Regge-minus-Lichnerowicz residual, removing any explicit residual matrix from the generator-facing input.
Upstream, the residual origin-column table, origin-row table, and residual-kernel structures are the generator surfaces that reconstruct the full residual from those origin data. The bilinear reduction endpoint then asserts that once the two operators match pointwise on TT modes, the bilinear and quadratic TT energy forms agree on longitudinal TT subspace perturbations.
proof idea
Definitional Prop, not a proved theorem. The body is an implication whose hypothesis is the raw typed-column residual certificate structure, and whose conclusion is the conjunction of five Nonempty residual/match certificate types together with the bilinear reduction endpoint Prop. No tactics or lemmas fire here; the companion holds theorem later discharges the implication by building each certificate from the raw data via the ofRawOriginColumnData constructors and then invoking the bilinear endpoint.
why it matters
This is the Track 1.D residual handoff surface consumed by Track 7. It feeds the fork handoff integration certificate as part of the Track 1 reduction/interface package (explicitly not a closure of the open Schläfli leaves). The coefficient-only origin-column endpoint reduces into this one, and the companion holds theorem asserts the Prop is inhabited.
In the Recognition gravity stack it sits on the discrete TT shear sector that underwrites continuum Lichnerowicz matching on the eight-tick / D=3 lattice geometry. It does not touch mass ladders or alpha bands directly; its job is to keep the residual route matrix-free so finite generator output can still reach the bilinear energy identity used downstream in master-theorem structural witnesses.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.