bpowSumTarget

This module turns the O1 yardstick discussion into a finite combinatorial search: four candidate $B_{\mathrm{pow}}$ values and four candidate $r_0$ values are assigned to sectors by permutation, then filtered by structural constraints. Valid choice sets collapse to singletons under those filters.

The constant $A$ is the active-edge count per tick. In the foundation gap derivation it is defined as $A:=1$, and the mass-anchor layer records the same fact as active edges per tick equal to 1. Related identities include $\eta_B\cdot\Theta_{\mathrm{crit}}=\varphi^A=\varphi$.

Here the sum target is simply that integer $A$, so any assignment $(B_{\ell},B_{u},B_{d},B_{\mathrm{ew}})$ is required to satisfy $B_{\ell}+B_{u}+B_{d}+B_{\mathrm{ew}}=A$ when the structural sum filter is applied.

One-line definition: cast the active-edge constant $A$ to $\mathbb{Z}$. No lemmas or tactics.

The sum filter is the backbone of the Boolean structural constraints on $B_{\mathrm{pow}}$ assignments. Downstream theorems such as the anchor structural identities, the derivation of Boolean constraints from edge roles, and the equivalences that force the electroweak magnitude to equal $A$ all take the hypothesis that the four sector exponents sum to this target.

That filter is what collapses the enumerated choice set to the anchor assignment in the O1 progress narrative. In the broader Recognition picture the same $A=1$ enters the active-edge bookkeeping that ties $\varphi$-powers on the mass ladder to the eight-tick and gap structure; fixing the sum target at $A$ keeps the combinatorial search aligned with that bookkeeping.