periodicTTNormalEquationGramApply5_eq_inner_generatorMap
plain-language theorem explainer
On the finite N=5 periodic torus, the Gram operator of the combined TT normal equation equals the edge-space inner product of a fixed generator against the linear combination of all generators weighted by the coefficient vector. Anyone assembling the discrete TT residual identity or Gram-kernel vanishing cites this. The proof unfolds the Gram and generator-map definitions, then applies right-linearity of the edge inner product.
Claim. Let $c$ be a real coefficient vector on the combined conformal-plus-longitudinal index set of the $N=5$ periodic torus, and let $i$ be any index in that set. Then the Gram operator applied to $c$ at $i$ equals the finite edge inner product of the $i$-th normal-equation generator with the edge perturbation obtained by linearly combining all generators with weights $c$.
background
Track 1.D builds the tensor/shear sector missing from the Track 1.B conformal ansatz. That ansatz assigns one scalar potential per vertex and induces edge-length variations by averaging endpoints; it cannot represent pure shear, so it cannot cover transverse-traceless weak-field modes. This module separates independent edge perturbations from vertex-conformal ones on a finite periodic torus of size five.
The working inner product on edge perturbations is the Euclidean sum $\sum_e \varepsilon(e),\eta(e)$. The combined normal-equation index runs over fixed conformal vertex-delta generators together with fixed longitudinal vertex-vector generators. The generator map sends a coefficient vector to an edge perturbation by the finite linear combination of those generators; the Gram operator is the associated Gram matrix acting on coefficient space.
Upstream, right-linearity of the edge inner product on finite linear combinations is already proved, and the gauge generator map is the same linear-combination construction specialized to longitudinal generators.
proof idea
Short unfold-and-rewrite proof. Unfold the Gram-apply definition, the combined generator map, and the underlying gauge generator map so both sides become an inner product of a fixed generator against an explicit finite linear combination of generators. Then rewrite by the already-proved right-linearity lemma for the finite periodic-edge inner product, which pulls the coefficient sum out of the second slot and matches the Gram matrix entrywise.
why it matters
This identity is the algebraic hinge between the abstract Gram operator and concrete edge-space pairings. Downstream it feeds two results: (i) pairing the combined residual with a generator equals load minus Gram, and (ii) any coefficient vector in the Gram kernel produces the zero edge perturbation. Those steps are required to certify that the discrete normal equation really projects out conformal and longitudinal gauge directions, leaving a pure shear/TT sector on the finite torus.
In the broader Recognition scaffold this is infrastructure for weak-field gravity beyond the scalar conformal slice, toward modes that can carry tensor shear in $D=3$. It does not yet close continuum TT uniqueness or the full forcing chain, but it removes a definitional gap inside the finite normal-equation setup.
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