periodicConformalGeneratorMap5
plain-language theorem explainer
Maps real coefficients on the encoded vertices of the 5×5×5 periodic Freudenthal torus to edge-length perturbations via the conformal (endpoint-average) generator. Track 1.D workers cite it whenever they build conformal projectors, longitudinal residuals, or TT orthogonality statements on the periodic lattice. The body is a one-line specialization of the gauge-generator map to the conformal generator.
Claim. Define the linear map sending a coefficient vector $c$ on the vertices of the canonical $5\times 5\times 5$ periodic Freudenthal torus to an edge perturbation on the typed periodic edges by applying the conformal generator through the gauge-generator construction. On each edge the induced value is the average of the coefficients at the two endpoints.
background
Track 1.D isolates the tensor/shear sector of weak-field edge perturbations on a discrete lattice. The older Track 1.B conformal ansatz assigns one scalar potential per vertex and induces edge-length changes by averaging the two endpoint potentials. That scalar slice cannot represent pure shear, so it cannot cover transverse-traceless gravitational-wave modes by itself.
The ambient geometry here is the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus. Edge perturbations are real functions on the typed periodic edges. The conformal generator is the discrete map that turns a vertex coefficient vector into such an edge function by endpoint averaging; the gauge-generator map is the generic wrapper that turns any fixed generator into a map from coefficient space to edge perturbations.
This definition simply names the conformal instance of that wrapper, so later lemmas can talk about the conformal image, its two-point support, and its orthogonal complement without repeating the construction.
proof idea
One-line definitional wrapper: apply the generic gauge-generator map to the already-defined conformal generator. No tactics, no algebraic reduction. Downstream endpoint and membership lemmas unfold this abbreviation and compute on a concrete edge.
why it matters
This is the named conformal image map for the entire Track 1.D projector stack. Downstream, the endpoint formula records two-point support (only base and head coefficients contribute), and the membership lemma places every image vector in the conformal log-subspace. Residual constructions subtract this image (together with a longitudinal-gauge image) from a general edge perturbation. Orthogonality theorems for longitudinal TT modes and the relative-frame generator-closure targets all quantify over this map, so the conformal half of the normal-equation generators is fixed here.
In the broader Recognition gravity program this is scaffolding for separating pure shear from the vertex-conformal slice, a prerequisite for discrete TT modes on the eight-tick / $D=3$ lattice geometry. It does not yet close the TT completeness or shifted-generator orthogonality props; those remain open Prop-valued targets that consume this map.
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