Track1DTTHessianLichnerowiczEncodedResidualOriginRowTableReductionEndpoint
plain-language theorem explainer
Track 1.D reduction endpoint: a seven-row origin residual table for the encoded Regge TT Hessian minus lattice Lichnerowicz residual implies the full displacement-row residual route and the bilinear TT energy match. Gravity auditors cite it when packaging the Hessian–Lichnerowicz identification into the Track 7 fork handoff. Pure Prop definition chaining certificate structures; the companion holds theorem discharges it by constructing intermediate certificates from the origin table.
Claim. If a seven-row origin-table certificate is given for the encoded residual (Regge TT Hessian kernel, lattice Lichnerowicz kernel, residual kernel, and origin residual rows indexed by $\mathrm{Fin}\,7$), then there exist nonempty certificates for the displacement-row residual form, the residual-kernel form, the residual-entry form, the periodic residual-entry form, and the periodic TT Hessian–Lichnerowicz match data, and the bilinear TT energy reduction endpoint holds.
background
Module setting is Gravity Track 7 fork-handoff integration: a receipt lane that records what parallel forks prove without upgrading the discovery claim. Track 1.D sits in the tensor-shear sector on the $N=5$ periodic torus, comparing the Regge TT Hessian operator to the lattice Lichnerowicz operator on transverse-traceless modes.
The seven-row origin table is the most compressed residual certificate: the generator emits only origin residual rows, then translation invariance reduces every matrix row to one of seven families. Upstream, the displacement-row form normalizes every encoded row to the origin edge with the same displacement; the residual-kernel form emits the already-subtracted residual matrix and proves it equals Regge minus Lichnerowicz. The bilinear reduction endpoint then asserts that once the two operators agree pointwise on TT modes, bilinear and quadratic TT energy matches follow.
proof idea
Definitional Prop, not a tactic proof. The body is an implication: assume the seven-row origin-table residual certificate, conclude five nonempty intermediate certificate types (displacement-row, residual-kernel, residual-entry, periodic residual-entry, periodic match data) conjoined with the bilinear TT energy reduction endpoint.
The companion holds theorem discharges this by intro on the origin-table data, then constructing each intermediate certificate via the ofOriginRowTableData constructors in the tensor-shear sector, and finally invoking the bilinear endpoint. No algebraic computation lives here; the work is certificate chaining.
why it matters
Packages the most compressed residual certificate (seven origin rows) into the full residual route that Track 7 consumes. Downstream, the holds theorem proves the endpoint, and ForkHandoffIntegrationCert records Track 1 material as a reduction/interface package alongside Forks A–F (Schläfli stationarity, physical residual/Bianchi, many-body amplitude lift, Page-capacity transfer, dark-energy $w(z)$ band, falsifier sensitivity).
Per the module doc, this does not upgrade the discovery claim and leaves remaining Track 1 displacement-class leaves open. It is the handoff hinge between finite residual-table generation and the bilinear TT energy match once Regge and Lichnerowicz are identified on TT modes. Framework role is bookkeeping inside the gravity master-theorem lane, not a T0–T8 forcing step.
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