The Schimmel Prime-Gap Theorem: The Principle of Mathematical Inevitability

The Schimmel Prime-Gap Theorem: The Principle of Mathematical Inevitability Kevin Mark Schimmel Independent Researcher Oklahoma City, Oklahoma, USA This publication presents the Schimmel Prime-Gap Theorem, an absolute mathematical claim concerning the structural relationship between consecutive prime gaps and the historical absolute-difference construction generated by preceding prime gaps. For n ≥ 3, let p_n denote the nth prime and let g_n = p_(n+1) − p_n. Define the historical gap sequence G_n = (g_2, ..., g_(n−1)), and let R_k(G_n) denote the rightmost entry of the kth iterated absolute-difference row, including row zero. The proposed universal inequality is: g_n ≤ B_n, where B_n = 2 + Σ_(k=0)^(n−3) R_k(G_n). The author's central claim is that this inequality expresses a necessary structural restriction governing prime gaps, rather than merely a statistical prediction. The investigation examines genuine prime histories, reversed and rearranged historical gaps, altered composite controls, near-boundary configurations, and independent computational verification. Recorded experiments include 11,998 genuine-prime cases without a violation, 220,847 valid three- and four-composite control configurations without a violation, and 68,772 effective adjacent-gap rearrangements without a violation. These experiments provide finite computational evidence. They are not presented as a completed universal proof. The author's position is absolute: the mathematical structure governing the relationship does not permit failure. The publication invites independent mathematical examination of that claim under the exact definitions provided. The accompanying evidence archive includes computational materials, results, and reproducibility documentation.