two_fails

This module derives the charge-to-band polynomial $Z(\tilde Q)$ from recognition boundaries on the 3-cube, without anchor constraints or empirical masses. Stage 1 fixes the integerization scale: a boundary of charge $Q$ couples to the $F=2D$ faces of the cube, and the ledger demands integer entries (T8: $\delta$-units $\simeq\mathbb{Z}$). The three SM charges are $Q\in{-1,,2/3,,-1/3}$.

A natural number $k$ integerizes all SM charges when $kQ\in\mathbb{Z}$ for each of those three values. The target theorem of the stage is that the face count $F=6$ (at $D=3$) is the least positive even such $k$. Checking small $k$ one by one is the bookkeeping that makes that minimality claim hold.

Sibling facts already record that $k=1$ fails and that $k=3$ and $k=6$ succeed; this declaration handles the even candidate $k=2$.

Assume toward contradiction that $k=2$ integerizes every SM charge. Specialize to $Q=-1/3$ (membership in the SM charge set is discharged by simp). Obtain an integer $n$ with $2\cdot(-1/3)=n$, i.e. $-2/3=n$ in $\mathbb{Q}$. Clearing the remaining denominator yields $-2=3n$ in $\mathbb{Z}$. Omega closes the contradiction: no integer $n$ satisfies that equation.

Stage 1 of the topological Z-map derivation needs a complete pass/fail table for small multipliers before it can assert that $F=6$ is the minimal positive even integerizer. This theorem supplies the $k=2$ failure half of that table.

It is packaged into integerization_results, which states that $k\in{1,2,4,5}$ fail while $k\in{3,6}$ succeed, and it is consumed by six_smallest_positive_even_integerizer, whose second conjunct is the minimality claim among positive even $k$. Together with T8 ($D=3$ spatial dimensions, eight-tick octave) this pins the face-count scale $F=2D=6$ as the canonical integerization of SM charges before Stage 2 forces the even polynomial $Z=a\tilde Q^2+b\tilde Q^4$ with $a=b=1$.