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

Two corrections

We distinguish a bound for the actual tail from the polynomial series used to compute it, and then correct an insufficient offset condition.

The tail bound in the free-pair program.

The $4096$ -bit free-pair program linked through Section 9 bounds $T_{s}$ by summing the first $400$ terms of $1250 (s + j + 2)^{4} 2^{- j}$ after rounding each down, then adding $1$ . This does not necessarily bound that polynomial series: at the recorded $s = 1 270 306$ , the returned value is $40$ below the exact sum $1250 P (s + 2)$ , with $P$ as in Proposition 6.5.

The returned bound nevertheless has a valid justification. The stronger estimate $p_{n} \leq 2 (n + 5)^{4}$ proved in Appendix A gives

T_{s} \leq 2 P (s + 6) < 625 (s + 3)^{4} (s \geq 0) .

The last expression is already the first summand of the program, and all its remaining summands are nonnegative. The strict inequality follows from

625 (s + 3)^{4} - 2 P (s + 6) = 623 s^{4} + 7436 s^{3} + 32958 s^{2} + 62972 s + 40437 > 0.

Thus the returned bound exceeds the actual tail, even though its stated comparison with the larger polynomial series fails. This validates the analytic bound only; it neither controls floating-point errors in other scans nor reruns the denominator-exclusion search.

An insufficient condition.

Theorem 5.5 allows the offset to vary. For each modulus $t$ , one must still find pairs of congruent indices beyond every prescribed cutoff whose tail difference is nonintegral. Two even coefficient values differing by $2$ , each occurring infinitely often in the same residue class of indices, do not give the required tail difference for an integer-coefficient recurrence: the sequence of Theorem 8.8 has both properties and only integral tail shifts.

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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