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.

This paper covers the complete problem-specific reasoning surface for Erdős #251, including all registered result families and their boundaries.

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In this paper

Coefficient conditions and counterexamples

The examples test different proposed substitutes for information about the actual tails. Bounded changes preserve a fixed modulus and transfer polynomial nonconcentration through finitely many translates. Sparse changes preserve the larger collection of properties in the main theorem because most sampled blocks are untouched. Later examples prescribe integer tails first and derive coefficients with recurring values or growth conditions. These mechanisms have different conclusions; none constructs consecutive primes.

We also show that the actual small-pair witnesses, if they occur arbitrarily late, must have density zero. This does not rule them out: infinitely many witnesses suffice, and positive density is not required.

Bounded residue-preserving perturbations

Algebraic nonconcentration survives the rationalising perturbation

Density of indices with two small tail differences

Recurring gap values differing by two do not suffice

A rational series with recurring values in every residue class

Polynomial coefficients and telescoping sums

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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