Pruning the Base-Sampling Space of Williams' p+1 and Pollard's p-1 Factoring Algorithms via Orthogonal Body–Tail Super-Sieves

We introduce an integrated algebraic selection framework for Pollard's p-1 and Williams' p+1 factorization algorithms that eliminates the massive computational overhead associated with blind base initialization. While classical implementations of these dual algebraic filters depend on randomly choosing baseline parameters to initiate multi-precision exponentiation, they frequently lock the execution routine into dead modular orbits inside unaligned local fields. By adapting the author's multi-component "body–tail" super-sieve to function over the orthogonal super-modulo ring Z_90090, we establish a predictive gatekeeper directly on the base-generation phase. Experimental simulation on the semi-prime N = 4,335,869 isolates exactly 540 valid base-residue tunnels, compressing the initial search space by 99.40% and ensuring that sub-exponentiation resources are targeted exclusively at active algebraic pathways.