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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In this paper

Sparse congruence-preserving perturbations

At indices $n, n + 1$ , the pairs $(0, 6), (2, 4), (4, 2), (6, 0)$ all have ordinary sum $6$ , while their dyadic contributions are $6, 8, 10, 12$ divided by $2^{n + 2}$ . If the cumulative correction before the pair is odd, we first add $1$ at $n - 1$ . Choosing the pair then leaves the cumulative residue after the triple unchanged. We repeat this operation with nested moduli eventually divisible by every fixed integer.

Finite changes alone preserve rationality, so we use infinitely many triples. Making their separations tend to infinity gives upper Banach density zero, but also makes each weighted contribution smaller. We compensate by increasing the range of choices inside a pair. The estimate (5) shows that all later ranges together cover the gap between adjacent choices at the current stage. The set containing the supports and the interval of sums are chosen before the value to be attained.

Density and block conventions.

About this paper

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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