Multiplicative Two-Step Recurrence and Complete Factorization in Generalized Pascal Triangles: Classification for m ∈ −1, −2

We study generalized Pascal triangles T (m) (n, k) defined by the recurrence T (m) (n, k) = (m (n − k) + 1) T (m) (n − 1, k − 1) + (m (k − 1) + 1) T (m) (n − 1, k). For the critical parameterm = −1, we prove a Complete Factorization Theorem: the row polynomialRn (y) = Pn k=1 T (−1) (n, k) yk−1 (degree n − 1, constant term 1) satisfies, for n ≥ 4, Rn (y) = (1 − y) dn · Qn (y), where dn = n − 2 (n even) or n − 3 (n odd), and Qn (y) = 1 + y (n even) or 1 + y2 (n odd). The key step is a Multiplicative Lemma: Rn+2 (y) = (1 − y) 2Rn (y). We prove thisby showing that the “excess” E (n, k) satisfies the same recurrence as T (m) (n, k), so thatE (n, k) ≡ 0 follows from E (4, k) = 0 by induction. We also prove a Classification Theorem: among integer parameters m, the multiplicativeidentity holds if and only if m ∈ −1, −2. While m = −2 also satisfies the multiplicativelaw, its odd rows make Qn (1) = 0, so universal cyclotomic quotients Qn ∈ Φ2, Φ4 withconstant boundary residue Qn (1) = 2 occur only at m = −1.