Residue-Slice Dynamics and Certified Primorial-Gap Coverings

Let P_k = p_k# and let D(k) denote the set of differences between consecutive integers coprime to P_k. This version 1.0.1 release provides a finite certificate proof that every even integer g with 2 <= g <= 742 occurs in D(50). The frozen archive contains 371 independently verifiable certificates: 349 direct stage-50 certificates and 22 certificates propagated from stages 46 and 47. It also contains canonical JSON and CSV data, a machine-checkable propagation ledger, primary and structurally independent verifiers, mutation tests, regression tests, a manuscript, and one-command reproduction scripts. The search procedure is not trusted as part of the final proof. Each released certificate is checked independently from its residue data. Canonical release SHA-256: 7063106a12fe8012ed9d20f2dd907d18b65d09524aa0de6de7131263e59e4a6b Scope limitation: this release establishes the stated finite certificate theorem. It does not by itself establish historical priority, world-record status, broad structural novelty, implications for prime gaps, or a general theorem beyond the certified finite range. AI and software assistance are disclosed in the manuscript and release documentation. Ari Samuel is the sole human author and accepts responsibility for the mathematical claims, code, data, citations, and released work.