Finite Differences of the p-adic Valuation of Generalized Multifactorials Across Residue Classes

For a prime q, an integer α ≥ 2, and n = αm + r with 1 ≤ r ≤ α, the generalized-factorial valuation v_q(n!^(α)) was recently expressed as a Legendre-type sum over prime powers by the author. This note studies this valuation as a function of the residue r, with α, m, and q held fixed, and determines how much it can change as r varies. When gcd(α, q) = 1, an exact decomposition v_q(n!^(α)) = v_q(M!) + Σᵢ εᵢ(r) is given, from which a sharp bound on the spread across residues follows, with equality attained. When q | α, the valuation splits into a constant-rate branch and a branch reducing to the coprime case, giving a sharp two-sided bound tying back to classical Legendre's formula. All results are proved elementarily and verified computationally against direct evaluation at large scale.