periodicTTNormalEquationGramEntry_symm5
plain-language theorem explainer
On the finite N=5 periodic torus, Gram matrix entries built from the combined conformal and longitudinal TT generators are symmetric under index swap. Anyone assembling the discrete TT normal-equation Gram operator needs this identity. The proof is a one-line appeal to symmetry of the edge-space inner product.
Claim. For combined indices $i,j$ ranging over conformal vertex generators and longitudinal gauge generators on the $N=5$ periodic torus, if $G_i$ and $G_j$ are the corresponding edge perturbations, then $\langle G_i,G_j\rangle_{\mathrm{edge}}=\langle G_j,G_i\rangle_{\mathrm{edge}}$.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on a periodic Regge lattice. The conformal (vertex-scalar) ansatz alone cannot carry pure shear, so it misses transverse-traceless modes; independent edge perturbations and longitudinal gauge generators are therefore adjoined.
The finite edge-space inner product on the $N=5$ torus is the plain $\ell^2$ pairing $\sum_e \varepsilon(e)\eta(e)$ over periodic edges. The combined generator family maps each index in the sum type (conformal vertex index $\oplus$ longitudinal gauge index) to an edge perturbation: conformal generators come from vertex deltas, longitudinal ones from vertex-vector gauge modes.
The Gram entry is exactly that inner product evaluated on a pair of generators. Symmetry of the pairing is the elementary algebraic fact used here.
proof idea
One-line term proof: apply the already-proved symmetry lemma for the $N=5$ edge inner product to the two generator images. No case split on the sum-type index is required; the pairing is symmetric for every pair of edge perturbations.
why it matters
This identity is the entrywise step toward self-adjointness of the finite TT Gram operator on coefficient space. The immediate parent is the theorem that the Gram map is self-adjoint for the coefficient inner product, which needs $\langle G_i,G_j\rangle=\langle G_j,G_i\rangle$ at every matrix entry.
In the broader Recognition gravity track, a self-adjoint discrete TT Gram is the linear-algebraic prerequisite for a clean kernel/cokernel analysis of shear modes on the periodic torus, separating pure tensor degrees of freedom from conformal and longitudinal gauge junk. It sits inside the scaffold that aims to exhibit genuine TT content beyond the Track 1.B conformal slice.
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