Complete one-colour formula for all $m \ge 2$ and $\ell \ge 2$

This theorem states the single-colour modular Schur number for every admissible pair, with no restriction relating the modulus and the number of summands.

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

The first branch is the regime treated in the literature; the remaining two cover a modulus smaller than $\ell$, where the answer degenerates to whether the all-ones tuple is already fatal.

This is the single statement a consumer can quote for the one-colour case without checking which regime a given pair falls into, and it closes the single-colour question completely.