periodicLongitudinalTTSubspace5_inner_generatorMap_eq_zero
plain-language theorem explainer
Longitudinal TT edge perturbations on the period-5 Freudenthal torus are orthogonal to every combined conformal-plus-longitudinal normal-equation generator. Anyone assembling the finite Fredholm alternative for the TT Gram operator cites this. The proof splits the generator into conformal and gauge summands and applies the two halves of the TT orthogonality hypothesis, then cancels.
Claim. Let $\varepsilon$ be a real edge perturbation on the period-5 periodic Freudenthal edges. Suppose $\varepsilon$ is TT-orthogonal: under the finite periodic-edge inner product it annihilates every conformal log-strain generator and every longitudinal gauge generator built from vertex-vector potentials. Then for every coefficient assignment on the combined conformal-and-longitudinal normal-equation index set, $\langle\varepsilon,\,G(\mathrm{coeff})\rangle=0$, where $G$ is the corresponding generator map.
background
Track 1.D opens the tensor/shear sector of the weak-field Regge calculus. The older Track 1.B conformal ansatz puts one scalar at each vertex and induces edge strains by endpoint averaging; that slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes. This module treats independent edge perturbations and separates them from vertex-conformal ones.
A period-5 edge perturbation is a real function on the typed periodic Freudenthal edges. TT orthogonality means the finite edge inner product of $\varepsilon$ vanishes on the conformal log-strain subspace and on a supplied gauge subspace. Here the gauge slice is the longitudinal one: potentials indexed by (periodic vertex)$\times{1,2,3}$, mapped to edges by the longitudinal gauge map.
The normal-equation generators combine two families: conformal vertex-delta generators and longitudinal vertex-vector generators, indexed by a sum type. Their generator map is the linear combination that appears as the right-hand side of the discrete TT normal equations on this fixed finite torus.
proof idea
Extract conformal and gauge coefficient projections of the combined coefficient vector. Rewrite the normal-equation generator map as the sum of the conformal generator map on the conformal coefficients and the longitudinal gauge map on the gauge coefficients (via the in-module split lemma). Expand the edge inner product by additivity on the right. The conformal summand vanishes by the first conjunct of TT orthogonality together with membership of the conformal generator image in the conformal subspace. The gauge summand vanishes by the second conjunct, witnessing gauge-subspace membership by the coefficient itself. Both terms are zero, so the sum is zero by ring.
why it matters
This is the elementary annihilation step that lets TT loads sit in the orthogonal complement of the normal-equation range. Downstream it is packaged into PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5, whose residual-kernel vanishing on TT perturbations is described as "the most compact finite calculation target for the TT Hessian-to-Lichnerowicz comparison." That comparison is the finite-surface stand-in for showing the discrete Regge Hessian agrees with the lattice Lichnerowicz operator on the physical TT sector.
In the broader Recognition gravity track, the conformal-only ansatz is known to be incomplete for shear; this lemma is part of closing the shear gap so that eight-tick, three-dimensional weak-field modes can be treated without residual gauge or conformal contamination. It does not itself force $D=3$ or the eight-tick octave (those sit in the T0–T8 chain), but it is infrastructure those continuum limits need once the discrete TT Gram operator is under control.
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