encodedVertexDeltaPotential5
plain-language theorem explainer
Kronecker delta on vertices of the encoded 5×5×5 periodic Freudenthal torus: unit potential at a chosen vertex index and zero at every other vertex. Anyone assembling the finite conformal log-strain subspace for Track 1.D cites this as the elementary generator. The body is the standard indicator map on the finite vertex set, typed as a vertex potential.
Claim. For each vertex index $v$ on the canonical $5\times 5\times 5$ periodic Freudenthal torus, the encoded vertex delta potential is the real-valued map on vertices given by $w \mapsto 1$ if $w = v$ and $w \mapsto 0$ otherwise.
background
Track 1.D separates pure edge (shear) perturbations from the older Track 1.B conformal ansatz. That ansatz assigns one real scalar to each vertex and induces edge-length variations by averaging the two endpoint potentials. The scalar slice cannot represent pure shear, so it cannot by itself cover transverse-traceless weak-field modes; this module begins the tensor/shear track by isolating the conformal generators on a concrete finite complex.
A vertex potential on a finite 3D triangulation $K$ is simply a map from the finite vertex index set to $\mathbb{R}$. The ambient complex here is the canonical encoded $5\times 5\times 5$ periodic Freudenthal torus (external vertex order $(x\cdot 5+y)\cdot 5+z$). The present definition supplies the unit basis vectors in that finite-dimensional potential space.
proof idea
Pure definition: the body is the indicator function fun w => if w = v then 1 else 0 on Fin PeriodicTorus5.K.nV, returned at type VertexPotential PeriodicTorus5.K. No lemmas are applied.
why it matters
This delta is the seed for the concrete $N=5$ conformal generators. Downstream, each encoded vertex delta is pushed through the conformal edge-log-strain map and retyped as a periodic edge perturbation, yielding the family of conformal generators. Those generators span the conformal log subspace on the $5\times 5\times 5$ torus, which is exactly the conformal half of the finite-generator TT projector data for Track 1.D. Without an explicit finite basis of vertex deltas, the spanning statement and the membership of linear combinations in the conformal subspace would have no concrete generators to sum over.
In the broader Recognition gravity scaffold this is bookkeeping, not a forcing-chain step: it equips the shear-versus-conformal split on a fixed periodic complex so later work can isolate pure shear (and eventually TT) modes orthogonal to the conformal slice.
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