Quadratic Cascade: A Parameterized Extension of Pollard's P-1 Method Complementary to ECM

We introduce the *Quadratic Cascade*, a parameterized extension of Pollard's P-1 factoring method that chains modular exponentiations of the form x^(r·q²+q) mod n with different ratio parameters r. Unlike the standard P-1 method, which fails when p-1 has a large prime factor F > B, the Quadratic Cascade can incorporate F into its cumulative exponent whenever F divides r·q+1 for some prime q ≤ B and some ratio r in the formula set. We provide experimental evidence that an arsenal of Quadratic Cascade formulas, optimized by simulated annealing on general P-1-resistant corpora, factorizes 50-digit semiprimes that GMP-ECM (Zimmermann et al.) at its recommended parameters (214 curves, B1=50,000, Stage 2 automatic with B2 ≈ 13,000,000) does not factorize within the tested budget. Evidence of out-of-sample generalization is observed across two independent test corpora with verified GMP-ECM return codes. A theoretical mechanism is proposed and verified on a specific case.