E

The module builds the smallest nontrivial multi-cube Freudenthal example: two unit cubes that share one square face, each cut into the standard six Freudenthal tetrahedra with matching face triangulation. After identifying the shared face, the global edge set is the disjoint union of the two cubes' edge slots with the five shared-face edges counted once.

Sibling abbreviations in the same file name the other incidence sorts (vertices $V$, tetrahedra $T$) and the maps that send local cube edges and tet edges into this global edge index. The construction is the first concrete multi-cube incidence instance beyond the single-cube sanity check; later lemmas prove the local-edge-slot partition and the local-to-global square-edge equalities against this $E$.

(The import path touches a cosmology lattice-edge Finset also named $E$, but that object is a filtered adjacency set on a diamond ball, not this finite index type.)

No proof. The declaration is a one-line abbrev equating the edge sort with Fin 33, fixing the cardinality of the deduplicated global edge set by definition.

Every incidence statement in the two-cube strip is indexed over this $E$: edge endpoints, membership of an edge in a tet, the local-to-global square-edge map, and the partition of edge slots after face identification. Downstream geometry in the same module (edge-in-tet characterizations, strip assembly) is typed against it.

In the broader Recognition geometry stack this is the first multi-cube test of Freudenthal compatibility: shared-face edge counts must match so that larger lattice assemblies (and any continuum or action limits built on them) inherit a coherent 1-skeleton. Framework landmarks that care about discrete 3-space and eight-tick structure ultimately need such clean multi-cell incidence; this abbreviation is the edge carrier for that smallest example. Name collisions with energy functionals and other E symbols elsewhere in the monolith are unrelated.