Track1DTTHessianLichnerowiczEncodedResidualOriginColumnTableReductionEndpoint
plain-language theorem explainer
Track 1.D reduction endpoint: a typed-column origin-table residual certificate for the Regge TT Hessian versus lattice Lichnerowicz operators implies nonempty seven-row, kernel, entry, and periodic match certificates, plus the bilinear TT energy match. Gravity auditors cite it when wiring residual-table handoffs into Track 7. It is a pure implication Prop packaging certificate existence, not a proved closure.
Claim. If a typed-column origin-table residual certificate exists for the $N=5$ Regge TT Hessian versus lattice Lichnerowicz residual (entries indexed by origin-row displacement, column base vertex, and column displacement), then there exist nonempty certificates for the seven-row origin residual table, the residual kernel, the residual entry form, the periodic residual entry form, and the periodic TT Hessian–Lichnerowicz match data, and the bilinear/quadratic TT energy-match endpoint holds.
background
Module context is Gravity Track 7 fork-handoff integration: receipts for parallel forks (Schläfli stationarity, physical residual/Bianchi, many-body amplitude lift, Page capacity, $w(z)$ bands, falsifier sensitivity). It records what endpoints prove without upgrading the discovery claim; open Track 1 displacement-class leaves remain.
In the tensor-shear sector at $N=5$, the residual is Regge Hessian minus lattice Lichnerowicz on TT modes. The typed-column origin-table certificate stores kernels plus a table residualOriginColumn : Fin 7 → PeriodicVertex5 → Fin 7 → ℝ (generator-facing surface). The seven-row form stores only origin residual rows so every matrix row can be reduced by translation. The residual-kernel form emits the already-subtracted residual matrix for direct comparison to generator-map reconstruction.
Upstream, the bilinear reduction endpoint states that once the Regge TT Hessian and lattice Lichnerowicz operators agree pointwise on TT modes, bilinear and quadratic TT energy matches follow on the periodic longitudinal TT subspace.
proof idea
Definitional packaging only: the Prop is the implication from a typed-column origin-table residual certificate to the conjunction of five Nonempty certificate types (seven-row origin table, residual kernel, residual entry, periodic residual entry, periodic match) and the bilinear reduction endpoint. No tactics or lemmas live in the body; the companion theorem discharges it by constructing each certificate from the column table via ofOriginColumnTableData converters and then invoking the bilinear endpoint.
why it matters
Places the Track 1.D residual-table route into the Track 7 handoff ledger. Downstream, the companion theorem proves the endpoint holds, and ForkHandoffIntegrationCert consumes Track 1 reduction/interface packages alongside many-body and sensitivity facts. The fork cert doc stresses that the structural master theorem still uses structural witnesses where required and that Track 1 here is reduction/interface, not closure of open Schläfli leaves.
In RS gravity work this is bookkeeping for discrete TT shear: matching Regge Hessian to lattice Lichnerowicz residual tables so energy identities can feed continuum-limit or master-theorem arguments. It does not itself force $D=3$, eight-tick structure, or the J-cost; those sit earlier in the forcing chain. It keeps the residual origin-column surface available as a typed handoff for later displacement-class work.
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