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

Will Cook

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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.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #251, which remains open.

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Bounded residue-preserving perturbations

For one fixed positive modulus $M$ , we may add either $0$ or $M$ after a prescribed prefix. The weighted choices form a binary expansion, so we can choose a rational sum while preserving every coefficient modulo $M$ . This construction permits changes at every index after the prefix; it imposes neither density-zero support nor eventual preservation of all moduli.

Theorem 8.1 (bounded perturbations preserving congruences). Let $a_{n}$ be natural numbers with $\sum_{n \geq 0} a_{n} 2^{- (n + 1)}$ convergent. For every integer $M \geq 1$ and every cutoff $K$ there are digits $ε_{n} \in {0, 1}$ , zero for $n < K$ , such that $\sum_{n \geq 0} (a_{n} + M ε_{n}) 2^{- (n + 1)}$ is rational.

Proof. Write $A = \sum_{n \geq 0} a_{n} 2^{- (n + 1)}$ and choose a rational $r \in (A, A + M 2^{- K})$ . A binary expansion of $(r - A) / M$ has the form $\sum_{n \geq K} ε_{n} 2^{- (n + 1)}$ with $ε_{n} \in {0, 1}$ ; set $ε_{n} = 0$ for $n < K$ . The perturbed series converges and equals $r$ . ◻

Hence a property shared by every allowed perturbation cannot by itself force irrationality, since one such perturbation has a rational sum; this consequence is not stated separately in Lean. Lean also checks the construction at the actual consecutive prime gaps, for every $M \geq 1$ and every $K$ : the perturbed prime-gap series is rational, and each perturbed gap lies in $[g_{n}, g_{n} + M]$ and is congruent to $g_{n}$ modulo $M$ . For each fixed $k \geq 1$ , the sum of $k$ consecutive gaps increases by at most $k M$ . Thus infinitely many bounded gaps and bounded clusters of each fixed size survive, but their original numerical bounds need not. Nonnegative corrections also preserve lower bounds for large gaps. Taking $M = 2 q$ preserves parity and the cumulative positions modulo a prescribed $q$ , hence any limiting residue frequencies of those positions. These conclusions concern the size and residue information in the results discussed in Section 10; they establish neither primality of the new positions nor preservation of every quantitative assertion of those theorems.

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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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