Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates

Will Cook

Long paper

Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates

Erdős #251 By Will Cook September 2026 Authorship, AI use and citation

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Summary. A convergent dyadic series with nonnegative integer coefficients can be made rational by sparse nonnegative corrections of upper Banach density zero, eventually bounded by any f(n)→∞, while preserving every fixed modulus and, for a logarithmic allowance, short-block empirical distributions. For prime gaps the cumulative positions remain asymptotic to n log n; no primality conclusion is asserted. Summation by parts reduces the prime series to the gap series, and the tails $T_{N} = \sum_{j \geq 1} g_{N + j} 2^{- j}$ satisfy $T_{N + 1} = 2 T_{N} - g_{N + 1}$ . That recurrence characterises rationality and yields finite tests with explicit remainder bounds. An exact finite certificate shows that a rational value of the prime series would have denominator at least $2^{39998} > 10^{12040}$ . The bound does not establish irrationality.

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Finite numerical searches

An enclosure must control both the omitted tail and arithmetic error. The scans supply no bound for accumulated double-precision rounding, so their counts remain observations even when the omitted tail is rigorously bounded. By contrast, Proposition 6.6 and Section 7 use exact integer comparisons with linked Lean checks.

Over the $6 841 648$ primes below $1.2 \times 10^{8}$ , the scan tests $h = 1, \dots, 16$ separately. For each offset it records indices satisfying the signed inequalities in Theorem 6.4, with observed proportions between $0.00418$ and $0.008248$ and nearly balanced signs. The reported counts do not require one index to work for all offsets. At $h = 1$ it records $56 427$ hits among $6 841 564$ candidate indices, the last at the prime $119 995 753$ . Multiplying the observed proportion in each band by that band’s mean of $\log p$ gives $0.17671$ in the first band and $0.13148$ in the last. This finite observation establishes no asymptotic counting law and no occurrence beyond every cutoff. The search output is public.

A second scan uses the $1 270 607$ primes below $2 \times 10^{7}$ and pairs of indices in the half-open range $1 143 545 \leq n, m < 1 270 540$ . It searches each residue class modulo each $1 \leq t \leq 20$ for the condition of Theorem 5.5, selecting pairs with a next-gap difference of $2$ and a tail difference in the specified interval. At $t = 20$ , every residue class has at least $90 150$ counted pairs; pairs may share indices. For each class, the program also records the larger index of a selected late witness. The minimum of those recorded indices is $1 270 520$ ; it need not come from the class with the fewest pairs. For the saved sample pairs, floating-point arithmetic gives both small-difference inequalities. The scan supplies neither arbitrary moduli nor witnesses beyond every prescribed cutoff. Its search output is public. The computation guide gives the programs, dependencies and commands to rerun both searches and the $80000$ -bit continued-fraction calculation of Section 7. Appendix D.7 corrects the stated justification of the tail bound in the free-pair program.

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Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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