Directed Quadratic Sieve: Eliminating Non-Smooth Polynomial Paths via Orthogonal Body–Tail Super-Sieves

We introduce a directed optimization framework for the Quadratic Sieve (QS) prime factorization algorithm that eliminates the massive computational overhead caused by evaluating non-smooth polynomial outcomes. Classical QS implementations blindly evaluate consecutive polynomial values Q(x) = x^2 - N within a baseline interval, relying heavily on trial division or index-shifting across the entire factor base to locate smooth relations. By applying the author's multi-component "body–tail" super-sieve operating over the complex orthogonal ring Z_90090, we implement a rigorous predictive filter directly onto the polynomial initialization phase. Experimental verification on N = 4,335,869 proves that our framework isolates a high-density footprint of exactly 540 valid polynomial residue corridors, suppressing 99.40% of the initial arithmetic noise and transforming chaotic relationship assembly into a highly directed algebraic stream.