Singleton safety: $\{r\}$ is $\ell$-sum-free iff $(\ell-1)r \not\equiv 0$

This is the singleton safety criterion, the elementary test that drives every bound in the mission.

Let $m$ be a modulus, let $\ell \ge 1$, and let $r \in \mathbb{Z}/m$ be a residue. The one-element class $\{r\}$ is $\ell$-sum-free modulo $m$ exactly when

The reason is immediate: the only $\ell$-fold sum available inside $\{r\}$ is $\ell r$, and the only possible target is $r$ itself, so the class fails precisely when $\ell r = r$.

Singletons are the cheapest possible colour classes, so this criterion is what decides how far an all-singletons colouring can reach. It is used in both directions throughout the mission: to certify that small residues are safe, and to exhibit the one residue that is not.

Formalization Note The cast $(\ell : \mathbb{Z}/m)$ is the image of the natural number $\ell$, so the left-hand factor is written $(\ell : \mathbb{Z}/m) - 1$ rather than as a cast of $\ell - 1$; the two agree because $\ell \ge 1$.