Closed form at residue level: $\mathrm{schurModResidue}(m,k,\ell) = n - 1$

This is the closed form for the residue-level modular Schur number 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 $k \ge n - 1$,

The value is obtained by combining the uniform upper bound with the singleton-colouring lower bound, which meet exactly.

This is the residue-level form of the mission's headline identity. It is stated separately from the integer-level version because the residue side is where the argument lives, and because a consumer already working in the cyclic group can use it without passing through the reduction theorem.