periodicConformalGeneratorMap5_mem
plain-language theorem explainer
Any real coefficient assignment on the vertices of the canonical 5×5×5 periodic Freudenthal torus yields, via the encoded vertex-delta conformal generator map, an edge perturbation inside the vertex-conformal log-strain subspace. Gravity-track authors cite it when building the conformal projector and proving TT orthogonality to gauge generators. The proof is constructive membership: the input coefficients themselves witness the existential after unfolding and ring simplification.
Claim. For every coefficient vector $c$ on the vertices of the canonical $5\times 5\times 5$ periodic Freudenthal torus, the edge perturbation produced by the encoded vertex-delta conformal generator map applied to $c$ lies in the periodic conformal log-strain subspace: there exists a vertex potential $\xi$ such that the perturbation equals the encoded conformal edge log-strain of $\xi$.
background
Track 1.D isolates the tensor/shear sector of weak-field gravity on a discrete lattice. The older Track 1.B conformal ansatz puts one scalar potential at each vertex and induces edge-length changes by averaging the endpoint potentials. That scalar slice cannot represent pure shear, so it cannot cover transverse-traceless gravitational-wave modes by itself.
The ambient geometry is the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus. The periodic conformal log-strain subspace consists of those periodic edge perturbations that arise as the encoded conformal edge log-strain of some vertex potential $\xi$. The generator map under study packages a free coefficient vector on the encoded vertices into such an edge perturbation via vertex-delta potentials.
Membership in that subspace is the elementary closure fact needed before one can project edge data onto the conformal slice and prove orthogonality of longitudinal TT modes to the combined conformal-plus-gauge generator image.
proof idea
Term-mode constructive proof. After classical, refine supplies the input coefficient vector itself as the existential witness for membership in the conformal log-strain subspace. A funext on edges reduces the equality of edge perturbations to a pointwise identity. Unfolding the generator map, the gauge map, the vertex-delta potential, the encoding into periodic edge perturbations, and the conformal log-strain definition exposes a finite linear combination. A single simp with distributivity, association, and commutation of multiplication (plus div_eq_mul_inv) closes the algebra.
why it matters
This is the generator-image closure step for the conformal slice on the $5^3$ torus. Downstream it feeds the concrete projector data structure for the periodic longitudinal gauge basis (the finite-dimensional decomposition input still owed by Track 1.D), the theorem that every longitudinal TT perturbation is orthogonal to every combined conformal-plus-longitudinal normal-equation generator map, and the relative-frame orthogonality proposition that every TT mode is orthogonal to every row-frame translate of that generator map.
In the broader Recognition scaffold it keeps the conformal ansatz cleanly separated from pure shear, so the remaining TT residual can be identified with the tensor sector rather than being polluted by residual gauge. It does not itself force $D=3$ or the eight-tick octave; those sit upstream in the forcing chain. It closes a local scaffolding obligation: the image of the conformal generators is, by construction, inside the conformal subspace.
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