The Fractal Body–Tail Sieve Prunes Pollard's rho Factorization Trajectories: Converting Probabilistic Walks into Directed Search

We introduce a directed probabilistic factorization framework that eliminates the computational blindness inherent to Pollard's classical rho-algorithm. While Pollard's method effectively exploits the birthday paradox to locate non-trivial factors via pseudo-random walks, it performs costly greatest common divisor (gcd) extractions across the entire residue field Z_N without any geometric constraints. By integrating the author's multi-component "body–tail" residue filter operating modulo 900, we restrict the domain of valid factor candidates to a minute, precomputed algebraic subset based on quadratic residue tracks. Experimental verification on the 7-digit integer N = 4,335,869 demonstrates that the proposed sieve successfully prunes 98.08% of the cold computational branches, reducing the number of multi-precision gcd extractions from 52 total loop iterations down to a single, deterministic check.