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.

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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In this paper

The finite truncation criterion

For the actual gaps, we enclose each complete weighted difference by a finite sum and a bound on the omitted terms. A nonnegative summable majorant gives such an enclosure, but need not provide a computable remainder. Proposition 6.5 supplies the rational bound used in the finite tests.

Proposition D.3 (finite truncation). Let $M : N \to R$ satisfy $M (n) \geq g_{n}$ for every $n$ and $\sum_{n \geq 0} M (n) 2^{- n} < \infty$ , and put

S_{h, N, L} = \sum_{j = 1}^{L} \frac{g_{N + h + j} - g_{N + j}}{2^{j}}, R_{h, N, L} (M) = \sum_{j > L} \frac{M (N + h + j) + M (N + j)}{2^{j}} .

If for every $h \geq 1$ and every $N_{0}$ there are $N \geq N_{0}$ and $L \geq 1$ with $dist (S_{h, N, L}, Z) > R_{h, N, L} (M)$ , then the nonintegrality condition in Problem 10.1 holds.

Proof. The part of $\sum_{j \geq 1} (g_{N + h + j} - g_{N + j}) 2^{- j}$ omitted from $S_{h, N, L}$ has absolute value at most $R_{h, N, L} (M)$ , so under the displayed inequality the full sum lies at positive distance from every integer. ◻

With $M (n) = 1250 (n + 2)^{4}$ , the quantities $S_{h, N, L}$ and $R_{h, N, L} (M)$ are exactly $F_{h, N, L}$ and $E_{h, N, L}$ from Proposition 6.5. The same enclosure therefore tests separation from the integers, without requiring either of the two consecutive differences to be small. To compute the distance from the truncated sum to $Z$ , write the weighted block sum as

D_{h, N, L} = \sum_{j = 1}^{L} 2^{L - j} (g_{N + h + j} - g_{N + j}), S_{h, N, L} = \frac{D_{h, N, L}}{2^{L}} .

Then

dist (S_{h, N, L}, Z) = 2^{- L} min {D_{h, N, L} mod 2^{L}, 2^{L} - (D_{h, N, L} mod 2^{L})},

where $D_{h, N, L} mod 2^{L}$ is the least nonnegative residue, including when $D_{h, N, L} < 0$ . The criterion requires this residue to be more than $2^{L} R_{h, N, L} (M)$ from both $0$ and $2^{L}$ . One successful comparison certifies nonintegrality at that index; a failed comparison is inconclusive. Irrationality requires such certificates arbitrarily late for every positive $h$ . At prescribed logarithmic depth, the needed residue must be farther from both endpoints than the error bound. With the polynomial majorant in Proposition 6.5, the remainder bound is rational and explicit, so the whole test reduces to integer comparisons. The modular rewriting is an ordinary deduction, not a separately checked Lean declaration.

Proposition D.4 (completeness of finite separation). Suppose $D \in R$ , $S_{L} \in R$ and $R_{L} \geq 0$ satisfy $| D - S_{L} | \leq R_{L}$ and $R_{L} \to 0$ . Then

D \notin Z ⟺ there exists L with dist (S_{L}, Z) > R_{L} .

Proof. The reverse implication follows from the triangle inequality. For the forward implication put $δ = dist (D, Z) > 0$ . For all sufficiently large $L$ , one has $R_{L} < δ / 2$ , and the $1$ -Lipschitz property of distance gives $dist (S_{L}, Z) \geq δ - R_{L} > R_{L}$ . No monotonicity of the errors is required. ◻

For fixed $h, N$ , increasing the depth eventually certifies every nonintegral $σ_{h} (N)$ . This is not a decision procedure for integrality: finitely many failed tests prove nothing, although a proof of failure at every depth implies integrality. It gives neither a prescribed depth as $N$ varies nor arbitrarily late successful indices. For example, $[1 - 2^{- N}, 1 + 2^{- N}]$ contains both the noninteger $1 + 2^{- 2 N}$ and the integer $1$ for every $N \geq 1$ . The radii shrink, but the values also change; no enclosure certifies nonintegrality. Completeness keeps the value fixed while refining its enclosure.

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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