EncodedEdgePerturbation5

Track 1.D opens the tensor/shear sector of the weak-field Regge surface. The older Track 1.B conformal ansatz puts one scalar at each vertex and induces edge strains by averaging endpoint potentials; that slice cannot carry pure shear or transverse-traceless modes.

An edge perturbation is a map from the finite edge index set of a 3D triangulation $K$ into $\mathbb{R}$: one independent length variation per global edge. That is the natural finite surface for anisotropic shear. The $N=5$ periodic Freudenthal torus supplies a concrete triangulation $K$ with a known edge count; the present abbreviation simply names the edge-perturbation space of that $K$.

Upstream, the same edge-perturbation structure appears in deficit linearization as the first-order variation of flat edge lengths that feeds hinge-deficit derivatives.

Definitional abbreviation only: specialize the generic edge-perturbation type (maps from $\mathrm{Fin},n_E$ to $\mathbb{R}$) to the triangulation of the $N=5$ periodic torus. No proof obligations.

This type is the ambient space for the Lane 3 edge-tensor gap on the $N=5$ torus. Downstream theorems record that its real dimension is 875, that the image of the conformal strain map has rank at most 125, and that the gap is witnessed by an explicit localized rectangle face shear with no vertex-conformal realization.

Existence and constructive non-conformality statements are phrased as elements of this space outside the conformal range (or failing the conformal predicate). The pullback into typed periodic edges also lands here. In the broader Recognition gravity track it marks the finite Regge surface on which pure shear, hence TT-like modes, can live once the conformal ansatz is left behind.