regularBoundaryCWVertices

This module builds the algebraic bridge between a compact cubical 3D region's Betti data and the topology of the boundary of a regular neighborhood of the exact positive excursion set. After Phase 25 found nonmanifold edges on the raw cubical boundary, Phase 26 switched to that desingularized readout.

A BettiTriple packages integer Betti numbers $(b_0,b_1,b_2)$ so Euler algebra is literal. The predicted regular-neighborhood boundary component count is $b_0 + b_2$ (Alexander-duality style). The companion Euler law doubles the region Euler characteristic: $\chi(\partial N) = 2(b_0 - b_1 + b_2)$.

The CW model is purely combinatorial cell counts for Phase-28 certificates: one 0-cell and one 2-cell per boundary component, and $2b_1$ one-cells for total handle rank. It is not an embedded cubical sheet-splitting construction.

Definitional one-liner: the vertex count is set equal to the regular-boundary component count $b_0 + b_2$. No further lemmas or tactics; downstream Euler identities unfold this alias and cancel by ring arithmetic.

Supplies the $V$ term in the canonical CW Euler characteristic $V - E + F$ used by the Phase-28 certificate. Downstream, regularBoundaryCWEuler assembles that combination, and the theorem that the CW Euler equals the doubled region Euler ($2\chi(\mathrm{region})$) unfolds this definition together with the edge and face counts and reduces by ring.

That equality is the cell-count half of the regular-boundary genus bridge: once components and Euler are identified, total desingularized boundary genus equals the region's $b_1$. The module status remains partial through Phase 44 and conditional at Phase 47; the embedded digital-cubical collapse and homeomorphism to the geometric regular-neighborhood boundary stay open. This definition only fixes the combinatorial zero-skeleton budget those later arithmetic wrappers consume.