Track1DTTHessianLichnerowiczEncodedCoeffTranslatedFullChainEndpoint
plain-language theorem explainer
Track 1.D packages the full finite TT Hessian versus lattice Lichnerowicz residual ladder into one audit proposition. From a coefficient-only translated residual certificate it demands nonempty witnesses for every intermediate form (origin-column, raw column, residual tables, residual kernel, residual entry, periodic match) and the bilinear energy-match endpoint. Gravity auditors and Track 7 fork-handoff consumers cite it. The body is a pure implication of Nonempty certificates, not a computational argument.
Claim. If a coefficient-only translated residual formula certificate exists for the $N=5$ TT Hessian/Lichnerowicz residual (Regge and lattice Lichnerowicz edge-operator kernels, residual-displacement coefficients, and an encoded residual entry formula), then there exist certificates for the origin-column coefficient form, the raw origin-column form, residual origin column and row tables, residual-kernel form, residual-entry form, periodic residual-entry form, and periodic TT match data, and the bilinear/quadratic TT energy-match 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. Fork A covers Track 1.B Schläfli stationarity at $N=5$; Track 1.D sits in the TT shear sector of that gravity stack.
The objects are finite certificates comparing the Regge TT Hessian operator to the lattice Lichnerowicz operator on transverse-traceless edge modes at period $N=5$. Upstream, the translated coefficient-only surface stores two kernels and residual-displacement coefficients and proves the encoded residual entry formula; it is strictly smaller than the origin-column surface, which recovers origin-row scalars by specializing the translated formula to the origin edge. Raw origin-column data further removes the full residual matrix from the generator-facing input.
The terminal bilinear endpoint states that once Regge TT Hessian and lattice Lichnerowicz are identified pointwise on TT modes, bilinear and quadratic TT energy matches follow for longitudinal TT subspace perturbations.
proof idea
Definitional packaging only: the proposition is the implication from translated coefficient-only formula data to the conjunction of eight Nonempty intermediate certificate types plus the bilinear reduction endpoint. No tactics or algebraic steps live here. The companion theorem discharges the chain by constructing origin-column data from translated data via ofCoeffTranslatedData, then threading residual tables, residual-kernel, residual-entry, periodic residual-entry, and periodic match witnesses down to the bilinear endpoint.
why it matters
Gives Track 7 a single audit target for the whole finite TT Hessian/Lichnerowicz residual reduction, rather than a scatter of intermediate structures. Downstream, the holds theorem asserts the proposition, and ForkHandoffIntegrationCert consumes Track 1 reduction/interface packages alongside Track 2 many-body and Track 6 sensitivity handoffs. The module doc is explicit: this is a reduction/interface package, not a closure of the open Schläfli displacement-class leaves. In the RS gravity program it records exact operator-match depth on the discrete TT shear sector before continuum or continuum-limit claims are attempted. It does not touch T5–T8 forcing, RCL, or the alpha band; those sit upstream of the gravity stack.
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