Track1DTTHessianLichnerowiczEncodedResidualEntryFormulaReductionEndpoint
plain-language theorem explainer
Track 1.D handoff proposition: given encoded Fin-indexed scalar residual formulas for the TT Hessian versus lattice Lichnerowicz comparison on the N=5 periodic torus, the typed periodic residual-entry, row-coefficient, row-span, kernel, and match packages are inhabited, and the bilinear TT energy match endpoint follows. Gravity auditors cite it as the encoded-to-periodic residual bridge into Track 7. It is a pure Prop packaging of implication and Nonempty conjunctions, not a proved theorem.
Claim. If encoded residual-entry formula data for the transverse-traceless (TT) Hessian–Lichnerowicz comparison at $N=5$ are given, then there exist witnesses for: the typed periodic residual-entry formula, the entrywise and row residual-coefficient tables, the residual-row span, the kernel residual vanishing on TT modes, the rowwise kernel match data, and the full operator match data; and the bilinear TT energy-match endpoint holds (Regge TT Hessian bilinear equals lattice Lichnerowicz bilinear on longitudinal TT edge perturbations).
background
Module context is Gravity Track 7 fork-handoff integration: a receipt lane that records what parallel forks prove without upgrading the discovery claim. Fork material here is Track 1.D, the discrete TT shear sector on the $N=5$ periodic torus, comparing the Regge second-variation edge Hessian to the spin-2 lattice Lichnerowicz stencil.
Upstream, the bilinear reduction endpoint states that once Regge and lattice operators are identified pointwise on TT modes, bilinear and quadratic TT energies match for longitudinal TT edge perturbations. Kernel-row data package two edge-operator kernels with a TT-restricted row match. Residual-row coefficient (and entrywise) data give the finite table of conformal/longitudinal generator coefficients whose map reconstructs each residual kernel row or entry.
The encoded side supplies raw scalar formulas indexed by edge cardinality; the periodic side is the typed route through the torus edge equivalence. This definition chains those packages as a single Prop implication.
proof idea
No proof body: this is a def equating a name to a proposition. The proposition is an implication from encoded residual-entry formula data to a conjunction of seven Nonempty structure packages (periodic residual-entry formula, residual-row coeff entry, residual-row coeff, residual-row span, kernel residual TT-zero, kernel row, match) plus the already-named bilinear reduction endpoint.
The companion theorem that discharges it builds the periodic packages via ofEncodedData constructors from the encoded witness, then feeds the match data into the bilinear endpoint. This declaration only names that interface shape.
why it matters
Sits in the Track 1.D residual-formula route that feeds Track 7. Downstream, the holding theorem asserts this endpoint, and ForkHandoffIntegrationCert consumes Track 1 reduction/interface packages alongside many-body, Schläfli, displacement, Page-capacity, dark-energy $w(z)$, and falsifier-sensitivity handoffs.
Doc-comment frames it as encoded Fin-scalar formulas entering the typed periodic-edge residual path. It does not close open Schläfli or displacement-class leaves; the module explicitly keeps those as remaining dependencies. In RS gravity terms it is bookkeeping for the discrete spin-2 operator match on the eight-tick-compatible lattice sector, not a new forcing-chain step (T5–T8) or a mass/alpha claim.
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