Mod-12 Rigidity Theorem for Consecutive Central Trinomial Multinomial Coefficients

Let M (n) = (3n)! (n!)3 (OEIS A006480), the central trinomial multinomial coefficient. This manuscript isolates a concise, residue-class-closed formula for the greatest common divisor of consecutive terms G(n) = gcd (M (n), M (n + 1)), together with a normalized quantity, the Sakib Index which measures the “shared divisibility exposure” between consecutive central multinomials. The central statement (the Sakib Mod-12 Rigidity Theorem) shows that the co-primeness quotient M (n)/G(n) depends only on (n + 1) mod 12 and is always of the form (n+1)2/d, with d ∈ {1, 2, 3, 4, 6, 12}. We provide an elementary proof (via a gcd-in-recurrence lemma and a 2-adic/3-adic reduction), and a dataset-based validation protocol using exact arithmetic values tabulated from OEIS and deterministic recurrence evaluation.