The Read-Once Factorial-Divisibility Conjecture: Algebraic Proofs Through Six Inputs

Abstract

We formulate the division-free read-once factorial-divisibility conjecture for pairwise distinct integers. Using congruence arguments, signed residue classes, parity decompositions, and disjoint block constructions, we prove the conjecture algebraically for all input sizes 1≤n≤6. We also establish stronger parity-restricted divisibility theorems and record guaranteed-modulus frontiers for the small cases. Computational procedures are used only for independent consistency checks and not as substitutes for mathematical proof. Version 2 update. This version corrects and clarifies several proofs in the original preprint and adds further algebraic/congruence constructions for four inputs. It also adds a parity-pattern summary through six inputs and supplementary consistency checks for the finite signed-class residual tables. The central conjecture, the theorem proving factorial divisibility through six inputs, and the original future-work scope are unchanged. See VERSION_2_CHANGELOG.md in the source package for a detailed list of revisions.