periodicTTGramKernel_generatorMap_zero5
plain-language theorem explainer
A coefficient vector in the kernel of the finite TT Gram operator on the 5-cell periodic torus induces the zero edge perturbation. Anyone assembling the Fredholm alternative for the TT normal equations cites this. The proof shows the self-inner-product of the generated map vanishes by linearity and the kernel hypothesis, then invokes positive-definiteness of the edge inner product.
Claim. Let $c$ be a real coefficient vector on the combined conformal-plus-longitudinal index set of the 5-cell periodic torus. If $c$ lies in the kernel of the finite TT Gram operator, then the edge perturbation generated by $c$ is identically zero on every edge.
background
Track 1.D opens the tensor/shear sector of weak-field gravity on discrete complexes. The older conformal ansatz puts one scalar at each vertex and averages to edges; that slice cannot carry pure shear, so it misses transverse-traceless gravitational-wave modes. This module isolates independent edge perturbations and builds a finite TT normal equation on a periodic 5-cell torus.
The combined index set packages fixed conformal vertex-delta generators together with fixed longitudinal vertex-vector generators. The generator map sends a coefficient vector on that index set to a linear combination of those generators, viewed as an edge perturbation. The Gram operator is the matrix of the periodic edge inner product in that generator basis; a kernel vector is one whose Gram image vanishes pointwise on the index set.
The edge inner product is the natural $L^2$-type pairing on edge loads of the periodic complex. Vanishing of the self-pairing forces the perturbation itself to zero (positive-semidefinite nondegeneracy already proved for this pairing).
proof idea
Apply the lemma that an edge perturbation with vanishing self-inner-product is the zero map. It remains to show the self-pairing of the generator map is zero.
Unfold the generator map as the linear combination $\sum_i c_i g_i$. Pull the sum out of the right slot by right-linearity of the edge inner product. Symmetrize each summand so the generator sits on the left. Rewrite each pairing as the Gram apply of $c$ at index $i$. The kernel hypothesis sets every Gram apply to zero, so each term is $c_i\cdot 0$. The finite sum collapses to zero by simp.
why it matters
This is the kernel-to-nullspace half of the finite Fredholm alternative for the TT Gram operator. Downstream it is packaged into PeriodicTTGramKernelCriterionData5, whose doc-comment states the criterion surface: loads lie in the Gram image once they annihilate every kernel vector and the finite-range criterion holds.
In the Recognition gravity track this closes a discrete obstruction step toward representing pure shear (hence TT wave modes) outside the conformal vertex ansatz. It does not yet force continuum GR or the eight-tick octave; it only guarantees that Gram-kernel coefficients produce no residual edge strain on the 5-cell periodic model. Parent structures use it to reduce solvability of the TT normal equations to an orthogonality check against the kernel.
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