Track1DTTHessianLichnerowiczResidualRowSpanReductionEndpoint
plain-language theorem explainer
Track 1.D residual-row endpoint: if every residual edge-kernel row of the Regge TT Hessian versus lattice Lichnerowicz comparison lies in the conformal-plus-longitudinal generator span, then the residual vanishes on TT modes and the bilinear/quadratic TT energy match follows. Gravity auditors cite it as the sharp finite-stencil handoff into Track 7. It is a pure Prop packaging of that implication chain, not a proved closure.
Claim. The Track 1.D residual-row reduction endpoint holds when the following implication is true: given residual-row span data at $N=5$ (every residual edge-kernel row lies in the combined conformal plus longitudinal generator span), one obtains nonempty residual-TT-zero kernel data, nonempty rowwise kernel-match data, nonempty pointwise TT operator-match data, and the bilinear TT energy-match endpoint (Regge TT Hessian bilinear form equals lattice Lichnerowicz bilinear form on longitudinal-TT pairs).
background
This module is the Track 7 fork-handoff integration lane. It records what parallel gravity forks prove without upgrading the discovery claim; Track 1 displacement-class leaves remain open. Track 1.D concerns matching the Regge second-variation TT Hessian to the spin-2 lattice Lichnerowicz operator on a periodic five-edge stencil.
Residual-row span data asserts that for every edge, the residual kernel row (Regge Hessian kernel minus lattice Lichnerowicz kernel) belongs to the finite-dimensional span of conformal and longitudinal generators. Upstream rowwise kernel data packages the two kernels and the claim that they agree on longitudinal-TT modes. The bilinear reduction endpoint then says: once the operators match pointwise on TT modes, bilinear and quadratic TT energies match for longitudinal-TT pairs.
The physical idea is standard TT gauge: residual components in the conformal/longitudinal span are annihilated by TT orthogonality, so a generator-span certificate for residual rows is enough to kill the residual on the physical shear sector.
proof idea
This declaration is a definition of a proposition, not a proof. It packages a single implication: residual-row span data at $N=5$ implies the conjunction of (i) nonempty residual-TT-zero data, (ii) nonempty rowwise kernel data, (iii) nonempty pointwise match data, and (iv) the bilinear reduction endpoint.
The companion theorem that discharges the Prop builds the chain by constructors: residual-row span data yields residual-TT-zero data, which yields rowwise kernel data, which yields match data, which feeds the bilinear endpoint. No analytic estimate is performed here; the content is the reduction interface itself.
why it matters
In the Recognition gravity stack this is the sharp finite-stencil target for the TT Hessian/Lichnerowicz comparison on Track 1.D. Downstream, the companion holds-theorem and the Fork Handoff Integration certificate consume it as part of the Track 1 reduction/interface package. The fork cert doc is explicit: Track 1 here is a reduction package, not a closure of the open Schläfli leaves.
That matters for the master theorem handoff: once residual rows sit in the conformal-plus-longitudinal span, TT orthogonality finishes the residual, and the bilinear energy match is free. The remaining work is instantiating the residual-row span witness from the physical Regge Hessian and lattice stencil, not inventing a new continuum identity. It does not touch T0–T8 forcing, RCL, or the alpha band; it is pure discrete gravity operator matching on the periodic $N=5$ shear sector.
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