One colour with $m < \ell$: the value is $0$ or $1$ by $\ell \bmod m$

This theorem settles the single-colour value in the regime the paper's one-colour theorem excludes, namely a modulus smaller than the number of summands.

For every $m \ge 2$ and every $\ell$ with $m < \ell$,

When $\ell \equiv 1 \pmod m$ the all-ones $\ell$-tuple sums to $\overline{1}$, so even the interval $\{\overline{1}\}$ fails and no nonempty interval can be coloured; otherwise that singleton interval is safe, and the length cannot be pushed to $2$.

Together with the $2 \le \ell \le m$ case this completes the single-colour picture over all $\ell \ge 2$, leaving no gap in the one-colour regime.