quartic_only_separated

This module derives the charge-to-band map $Z(\tilde{Q})$ from recognition boundaries on the 3-cube, without anchors or empirical masses. Stage 1 fixes the integerization scale: face count $F=2D=6$ is the least positive even $k$ with $kQ\in\mathbb{Z}$ for SM charges $Q\in{-1,2/3,-1/3}$. The integerized charges are $\tilde{Q}{\ell}=\pm 6$, $\tilde{Q}{u}=\pm 4$, $\tilde{Q}_{d}=\pm 2$.

Stage 2 requires $Z$ even in $\tilde{Q}$, nonnegative, and $Z(0)=0$, so the minimal form is $Z=a\tilde{Q}^2+b\tilde{Q}^4$ with $a\ge 0$, $b>0$. Family separation (distinct ordered $Z$ on the three SM $\tilde{Q}$) is the next filter. The polynomial evaluator is $Z_{\mathrm{poly}}(a,b,q)=a q^2+b q^4$; lepton/up/down specializations plug in the three $\tilde{Q}$.

ordered_hierarchy a b packages the strict ordering of those three family values at coefficients $(a,b)$.

Term-mode proof by simplification and linear integer arithmetic. Unfold ordered_hierarchy together with $Z_{\ell},Z_u,Z_d$, the polynomial $Z_{\mathrm{poly}}$, and the three $\tilde{Q}$ constants. The goal reduces to concrete inequalities among $0\cdot q^2+1\cdot q^4$ at $|q|\in{6,4,2}$, i.e. $1296>256>16$, which omega discharges.

In the Stage-2 uniqueness story, family separation alone does not pin $(a,b)=(1,1)$: the pure quartic $(0,1)$ still separates the three SM bands. The module therefore needs an extra minimality principle (stated immediately below): among nonnegative $(a,b)\neq(0,0)$ with ordered hierarchy, minimize $a+b>0$, which selects $(1,1)$.

That pairs with the module claim that distinct SM family $Z$ values force $a=b=1$ once minimality is imposed, and with the color-offset stage $Z_{\mathrm{quark}}=2^{D-1}+\tilde{Q}^2+\tilde{Q}^4=4+\tilde{Q}^2+\tilde{Q}^4$ at $D=3$ (T8). No downstream consumers are wired yet; the lemma is local casework inside the topological Z-map derivation.