Improved deterministic integer factorization by quadratic-image sieving

Let $N=pq$ be a semiprime and let N be an odd prime. For a≠0 modulo , put Hₐ(N;)=+Nx⁻¹:x∈_^*\. For a primitive root $γ$, the size of H_γ(N;)∪ Hγ²(N;) is determined by the Frobenius trace of Y²=(X²-1)(X²-γ). We study the resulting trace-selection problem through finite certificates, a congruence modulo $8$, and a GRH-conditional positive-density family from the CM specialization $γ=2$. Independently of this conditional result, we obtain a deterministic factorization algorithm from the residue-known coefficient lattice of Harvey and Hittmeir (Math. Comp., 2022). The actual babystep index is a nondegenerate quadratic image modulo each selected prime divisor of the auxiliary modulus. After a computable translation, every such image avoids a fixed set of (-χ(-1))/4 residues. This forbidden set is optimal among translated sets that must handle both square classes of the leading coefficient. Gauss reduction, a nearby lattice-vector selection, arbitrary-point multipoint evaluation, and the unrestricted large-order algorithm of Harvey and Hittmeir (arXiv, 2026) give the bit-complexity bound N1/5exp(-(log(4/3)/50+o(1))log N/loglog N). Its secondary factor is asymptotically smaller than every fixed negative power of log N, improving the general bound of Harvey and Hittmeir (Math. Comp., 2022) and the balanced-semiprime bound of Gao et al.\ (arXiv, 2025).