one_fails

The module derives the charge-to-band map $Z(\tilde Q)$ from recognition boundaries on the 3-cube, without mass anchors. Stage 1 asks for a positive integer scale $k$ such that the ledger entries $kQ$ are integers for every SM electric charge, matching the T8 requirement that $\delta$-units live in $\mathbb{Z}$.

The three charges are the fixed list $\mathrm{sm_charges}={-1,2/3,-1/3}$. A natural number $k$ integerizes all of them when $\forall Q$ in that list there exists $n\in\mathbb{Z}$ with $kQ=n$ (as rationals). The face count $F=2D$ at $D=3$ supplies the candidate $k=6$; the surrounding lemmas check which small $k$ work.

Upstream, integerizes_all is exactly that universal quantification, and sm_charges is the concrete list used as the domain.

Assume for contradiction that $k=1$ integerizes every SM charge. Specialize to $Q=2/3$ (membership is by simp on the charge list) to obtain $n\in\mathbb{Z}$ with $1\cdot(2/3)=n$. Clear the trivial factor to get $2/3=n$ in $\mathbb{Q}$, then multiply by 3 and cast to $\mathbb{Z}$ to reach $2=3n$. omega discharges the impossible integer equation.

Feeds directly into integerization_results, which packages the full pattern: $k\in{1,2,4,5}$ fail and $k\in{3,6}$ succeed. That package is the arithmetic half of Stage 1 of the topological $Z$-map derivation: among positive even integers, $F=6$ is the minimal scale that integerizes all three SM charges (matching $F=2D$ at $D=3$ from the forcing chain T8).

Without the $k=1$ failure, the minimality claim for the face-count scale would be empty. The result is pure rational arithmetic; it does not yet encode the even-polynomial band form or the color offset $2^{D-1}=4$, which appear in Stages 2–3.