firstEndpointVal_edgeAB

Module Crux-2 (Seven Gaps, path-sum) develops class pushforward bookkeeping for the labeled path-sum partition function $Z$. The standing protocol is quotient bookkeeping: sums over labeled complexes decompose over fibers of the triangulation-class quotient, with class mass equal to fiber cardinality times the symmetry factor $1/|\mathrm{Aut}|$, not bare $1/|\mathrm{Aut}|$.

At $B=2$, two labeled one-edge complexes serve as witnesses: one with ordered edge $(0,1)$ and one with $(1,0)$. The first-endpoint observable reads the numeric first vertex of edge $0$ (or $0$ if there is no edge). It is the concrete labeled invariant used to separate those two complexes before any quotient identification.

One-line definitional proof. Unfolding the first-endpoint observable on the $B=2$ complex with a single edge $(0,1)$ reduces the conditional (there is an edge) to the first component of that edge, which is the numeral $0$; rfl closes.

Feeds the distinctness theorem for the two $B=2$ edge witnesses: that result assumes equality of the two labeled complexes, applies congruence of the first-endpoint observable, rewrites with this evaluation and its counterpart on the swapped edge, and obtains a numeric absurdity. That distinctness is the first half of the fork verdict detonating the C1 landmine: the two complexes are related by an explicit vertex-swap relabeling yet remain unequal as labeled objects, so the corresponding quotient fiber has cardinality greater than one and class mass strictly exceeds bare $\mu$. Scope is the labeled path-sum $Z$ only, not the exact-shell quotient-sum convention.