18.6. Small lattices as checked proposals🔗

On eligible large, dense inputs, the total factorizer also uses CLD data as a cheap source of proposals. It prepares sixteen leading coefficient columns once and tries nested lattices with four, eight, twelve, and sixteen columns. These lattices have dimension close to the number of remaining lifted factors, instead of adding every coefficient of the polynomial.

No theorem trusts the proposed partition. Hex reconstructs the proposed pieces in the original integer coordinates, checks their exact product, and runs the unrestricted proved classical factorizer on every piece. Only those classical results supply the final irreducibility proof. A failed proposal continues to the full proved CLD method and then, if necessary, trial division.