A Prescribed-Factor Sieve and a Prime-Gap Bound of 216

Abstract

Let H_1 = liminf_{n→∞}(p_{n+1} − p_n). This preprint develops a prescribed-factor equidistribution theorem and combines it with a certified Maynard sieve weight to derive the bound H_1 ≤ 216. The analytic component treats Type-I convolutions in arithmetic progressions to squarefree moduli possessing an explicit chain of divisors in prescribed scale windows. The finite component uses a layered support region and a 208-parameter rational test function for k = 46, certified by rational support arithmetic and 1024-bit outward-rounded Arb integration. The deposit includes the manuscript, LaTeX source, finite certificates, hashes, and verification programs.