Main closed form: $S_m(k,\ell) = m/\gcd(m,\ell-1) - 1$ for $k \ge n-1$

This is the mission's headline result: a single closed form for the modular Schur number, valid at every modulus in the many-colours regime.

Throughout, $m \ge 2$ is the modulus, $\ell \ge 2$ the number of summands, $k \ge 1$ the number of colour classes, $d = \gcd(m, \ell - 1)$ and $n = m/d$.

For every $m \ge 2$, every $\ell \ge 2$, and every $k \ge n - 1$,

Here the number counts colourings of an actual integer interval: it is the largest $N$ for which $[1,N]$ splits into at most $k$ classes, none of which contains $\ell$ elements summing to a member of the same class modulo $m$.

The formula is closed in a strong sense: it produces the value from $m$ and $\ell$ in a fixed number of elementary steps, one gcd, one division and one subtraction, with no search over colourings, no recursion, and no case split on $\ell \bmod m$. Earlier work in the literature settled individual small moduli by case analysis; this single identity covers every modulus at once in the stated range of $k$.