periodicConformalGenerator5_apply_endpoint
plain-language theorem explainer
On the canonical 5×5×5 periodic Freudenthal torus, the conformal vertex-delta generator at vertex v evaluates on a typed edge e as the average of the two endpoint indicators. Regge/TT analysts cite this when building the conformal half of the finite-generator projector. The proof unfolds the generator stack and rewrites endpoints via the torus edge-vertex identity.
Claim. For every vertex index $v$ on the canonical $5\times 5\times 5$ periodic Freudenthal torus and every typed periodic edge $e$, the conformal vertex-delta generator associated to $v$ evaluates on $e$ as $\tfrac12\bigl(\mathbf{1}_{v=e_1}+\mathbf{1}_{v=e_2}\bigr)$, where $e_1,e_2$ are the decoded endpoints of $e$.
background
Track 1.D separates pure edge (shear/tensor) perturbations from the older Track 1.B conformal ansatz. That ansatz assigns one scalar potential per vertex and induces edge-length log-strains by averaging the two endpoint values; it cannot represent pure shear and therefore cannot alone cover transverse-traceless gravitational-wave modes.
The ambient geometry is the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus (PeriodicTorus5), with typed edges PeriodicEdge5. The conformal generator at a vertex is the edge perturbation obtained by feeding a unit delta potential at that vertex through the conformal log-strain map (endpoint average). Endpoint decoding uses the torus vertex equivalence, so the generator is most useful once written in decoded endpoint coordinates.
Upstream infrastructure supplies the Kronecker-style indicator arithmetic and the edge-to-endpoint identification on the periodic torus; those are the only geometric facts needed for the pointwise formula.
proof idea
Short unfold-and-rewrite proof. Unfold the conformal generator, the encoding of vertex-delta potentials into edge perturbations, the conformal edge log-strain, and the encoded unit delta potential. After unfolding, the value on an edge is the average of two indicators on the encoded edge vertices. Rewrite those encoded vertices to the typed edge's decoded endpoints via periodicTorus5_edgeVerts_symm_eq_endpoints, which yields the stated two-point average.
why it matters
This is the concrete pointwise form of the $N=5$ conformal slice generators. Downstream, periodicConformalGeneratorMap5_apply_endpoint lifts it to arbitrary coefficient vectors (two-point support on each edge), and periodicConformalGenerator5_relativeColumn_eq_shift shows row-frame translates of generator columns are globally shifted generators. Together they supply the conformal half of the finite-generator TT projector data on the periodic torus, as flagged in the module's conformal-slice comment.
In the broader Recognition gravity track this sits inside the tensor/shear scaffold that must eventually complement the vertex-conformal ansatz so weak-field modes are not confined to pure dilatation. It does not yet close the shear sector or the full TT projector; it only pins the conformal columns in typed edge coordinates.
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