B_pow_sum

This module treats Open Problem O1: why each particle sector receives a specific binary-power exponent and rung offset from the counting layer. The principle is that each sector couples to a distinct level of the 3-cube hierarchy (passive edges, active edge, total edges).

The binary power $B_{\mathrm{pow}}$ is the integer exponent attached to each sector. From the anchor definitions it is fixed by cube edge counts: lepton gets $-2E_{\mathrm{passive}} = -22$, up quark gets $-A = -1$, down quark gets $2E_{\mathrm{total}}-1 = 23$, and electroweak gets $+A = +1$. Here $A = 1$ is the active-edge count per tick, and the edge totals come from the $D=3$ cube.

The module records the structural observation that these four integers partition into magnitude pairs ($22+1=23$, and $|\pm 1|=A$), reflecting how much each sector's recognition boundary borrows from the cube edge network.

Term-mode proof by simplification and numeric evaluation. Rewrite each summand via the four closed equalities (lepton $=-22$, up $=-1$, down $=23$, electroweak $=1$), then norm_num checks $-22+(-1)+23+1=1$. No induction or external arithmetic lemmas beyond those equalities.

This is the raw sum identity behind the yardstick assignment principle. It is rephrased immediately as equality to the active-edge count $A$ (since $A=1$), and it is the target matched by the choice-set verification theorem that the declared $B_{\mathrm{pow}}$ sum equals the principle's sum target of 1.

In the Recognition framework the identity is not decorative: the binary shifts encode sector coupling to the eight-tick / 3-cube edge network (T7–T8 landmarks). Leptons borrow heavily (factor $2^{-22}$), quarks sit at $\pm A$, and down quarks amplify via total-edge doubling. Closing the sum to 1 confirms the four sector exponents form a complete partition of the counting-layer budget, which is a necessary check for O1 before the mass yardstick $\phi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$ can be assigned sector by sector.