classInSimplexNat

This module enumerates the Freudenthal/Kuhn triangulation of the 4-cube: twenty-four monotone 4-simplices (axis permutations), each with five nested vertices and ten local edges drawn from the fifteen nonzero 0/1 edge classes of the imported 4D Regge edge stencil. The campaign is combinatorial support for a flat-Hessian assembly skeleton; per-hinge deficit and area kernels remain open parameters.

Membership of a class in a simplex is already decided by the Boolean predicate that asks whether some local edge index $e\in{0,\ldots,9}$ has local edge class equal to $d$. The present definition only reifies that Boolean as a $\mathbb{N}$-valued indicator so it can sit under finite sums over the fifteen classes.

Definition, not a proof. The body is the standard indicator conversion: return $1$ when the Boolean membership predicate holds for $(s,d)$, else $0$. No lemmas are applied beyond that predicate's existence check over the ten local edges.

Supplies the summand for the theorem that every Freudenthal 4-simplex carries exactly ten edge classes: the sum over all fifteen classes of this indicator equals $10$. That count is part of deliverable B in the Regge 4D hinge-incidence campaign (combinatorial support of the flat-Hessian class form), after the edge stencil and before any numeric second-variation kernels. It does not touch Einstein–Hilbert recovery, gap-action recovery, or RS convergence of the 4D Regge action; those remain explicitly open in the module scope.