regularBoundaryCWEdges

The module bridges raw cubical positive-excursion boundaries to a desingularized readout: the boundary of a regular neighborhood of the exact positive region. Phase 25 saw nonmanifold edges; Phase 26 and later phases work with this canonical surface and prove only the algebraic half of the genus bridge used by the foam-interface scripts.

A BettiTriple packages integer Betti numbers $(b_0,b_1,b_2)$ so Euler algebra is literal: the region Euler characteristic is $\chi=b_0-b_1+b_2$. The module then defines a canonical CW skeleton on the regular-neighborhood boundary (vertices, edges, faces) whose Euler number is meant to match the doubled-region law $2\chi$.

Sibling counts fix components as $b_0+b_2$ and boundary Euler as $2(b_0-b_1+b_2)$, forcing total desingularized genus $b_1$. The edge count here is the middle term of that CW Euler formula.

Definitional, not a proof. The one-cell count is set equal to $2\cdot b_1$ by fiat as part of the canonical regular-boundary CW model. No lemmas or tactics fire; downstream Euler identities unfold this abbreviation and cancel by ring arithmetic.

Supplies the edge term in the canonical CW Euler characteristic (vertices minus edges plus faces). That CW Euler is then proved equal to the regular-boundary Euler $2\chi(\mathrm{region})$, the Phase-28 doubled-Euler law.

Together with the component and genus bridges, this locks the algebraic statement that total desingularized boundary genus equals $b_1$. It does not close the still-open geometric realization: embedded digital-cubical collapse and homeomorphism of corrected cellulations to the true regular-neighborhood boundary remain outside this file.

In the broader RS cosmology stack this is bookkeeping for the foam-interface desingularization pipeline, not a forcing-chain (T0–T8) step.