periodicConformalGenerator5
plain-language theorem explainer
For each encoded vertex on the canonical 5×5×5 periodic Freudenthal torus, this builds the conformal edge generator given by unit scalar potential at that vertex and zero elsewhere. Track 1.D workers cite it as the spanning family for the conformal log-strain slice and as input to the finite TT projector data. The body is a thin composition: vertex-delta potential, then conformal edge log-strain, then the typed periodic-edge encoding.
Claim. For each encoded vertex index $v$ on the canonical $5\times 5\times 5$ periodic Freudenthal torus, the associated conformal generator is the periodic edge perturbation obtained by applying the conformal edge log-strain map (endpoint-average of vertex potentials) to the unit delta potential supported at $v$, then transporting into typed periodic-edge coordinates.
background
Track 1.D separates pure edge (tensor/shear) perturbations from the older Track 1.B conformal ansatz. That ansatz assigns one scalar potential per vertex and induces edge-length variations by averaging the two endpoint potentials; it cannot represent pure shear, so it cannot alone cover transverse-traceless weak-field modes.
The ambient geometry here is the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus. Periodic edge perturbations are real-valued functions on the typed periodic edges of that torus. The conformal edge log-strain of a vertex potential is the endpoint average of that potential along each edge; a unit delta potential at one encoded vertex is the Kronecker spike at that index.
This definition packages those pieces into one generator per encoded vertex, so the conformal slice becomes an explicit finite linear family rather than an abstract subspace.
proof idea
Pure definition, not a proof. Compose three maps already in scope: form the encoded unit delta potential at vertex index $v$; push it through the conformal edge log-strain on the torus complex (endpoint averaging); transport the resulting encoded edge field into typed PeriodicEdgePerturbation5 coordinates via the encoding map. No tactics, no lemmas discharged beyond the abbreviations those maps rest on.
why it matters
This is the concrete spanning set for the $N=5$ conformal log-strain subspace. Downstream, the subspace is proved equal to the linear span of these generators, and the coefficient map built from them lands in that subspace. That supplies the conformal half of the finite-generator TT projector data and the Master Theorem handoff endpoint stating that every conformal edge field is a finite linear combination of these vertex generators.
In the broader Recognition gravity track, the conformal slice is the gauge/scalar part that must be quotiented or projected away before pure shear (tensor) modes can be isolated. The module's rectangle obstruction already shows nontrivial shear is not vertex-conformal; these generators make the complementary conformal half explicit and finite, which is what a discrete TT projector needs on a periodic complex.
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