Bounded gaps between primes

Julia Stadlmann

REVIEW

Bounded gaps between primes

Julia Stadlmann

Number Theory · Submitted 31 Aug 2026

Pith's verdict: This submission is a one-paragraph abstract with no proof; there is nothing to review.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 9, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

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arXiv ID2608.31126 (version 1)

Pith IDTJH45TZW, Pith's permanent record of this paper and its review

CitationJulia Stadlmann. "Bounded gaps between primes." arXiv:2608.31126 (2026). https://pith.science/paper/TJH45TZW

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Claim paper

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Editor's note

read the letter

The paper you sent me is not a paper. It is a title, an author line, and a single abstract paragraph claiming that Bombieri-Vinogradov can be combined with newer equidistribution estimates to lower the bound on H1 from 246 to 240. The 'full text' section is blank. There is no theorem statement, no admissible k-tuple, no quantitative sieve inequality, no level-of-distribution analysis, and no error term. The claimed numerical improvement of six units is unverifiable because the argument is absent.

What is new here is the announcement of a smaller bound. That is a legitimate claim, but it is not a contribution until the derivation is supplied. The authors deserve credit only if they actually produce the math.

The soft spots are not minor. The entire paper is missing. The stress-test note correctly flags that without the sieve framework and the specific equidistribution estimate, there is no way to check whether the combination works or whether hidden incompatibilities (e.g., smooth moduli not covering the sieve conductors) invalidate the result. This is not a gap in the reasoning; it is the absence of any reasoning.

This paper is for no one in its current form. A serious reader cannot engage with it. The proper course is to desk-reject and invite the authors to submit a complete manuscript with a full proof.

Recommendation: Desk reject. There is nothing to peer review.

Referee report

Major objections (2)

[Abstract / entire manuscript] The manuscript provides no theorem statement, no admissible k-tuple, no explicit equidistribution estimate, and no sieve inequality. The single sentence in the abstract constitutes the entirety of the submission. As a result, the claimed improvement from 246 to 240 is unverifiable and there is no mathematical content to evaluate.
[Abstract] The paper asserts a combination of Bombieri-Vinogradov with 'newer equidistribution estimates for smooth moduli' but neither defines these estimates nor cites a source where they appear. Without specifying the level of distribution or the precise form of the estimates, the claim cannot be checked or replicated.

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The paper

The paper claims

This paper proves that consecutive primes are infinitely often at most 240 apart, improving the previous proven bound of 246.

The reading

The paper seeks to lower the best proven upper bound on the smallest possible gap between consecutive primes. Its claim is that $H_1 = \liminf_{n\to\infty}(p_{n+1}-p_n) \le 240$, meaning there are infinitely many neighbouring prime pairs whose difference is at most 240. That would trim the previous bound of 246 by six. The result matters because every finite upper bound of this kind is a step toward the twin-prime conjecture, which predicts gaps of size 2 occur infinitely often, and because it shows whether newer equidistribution estimates can be converted directly into smaller prime gaps.

What carries the argument

The load-bearing ingredient named in the paper is the Bombieri–Vinogradov theorem, which says that prime counts in arithmetic progressions agree with their expected values on average over moduli up to about the square root of the range being counted. The paper combines this with newer equidistribution estimates for smooth moduli, where 'smooth' means moduli whose prime factors are all small. Together these estimates control the error terms that a bounded-gap argument must keep small; the paper's claim is that the combined control is sufficient to reach the reduced bound $H_1 \le 240$.

Extended reading notes

Core claim

The paper's central claim is the numerical bound $H_1 \le 240$ for the liminf of gaps between consecutive primes. According to the abstract, the route is a combination of the Bombieri–Vinogradov theorem with newer equidistribution estimates for smooth moduli; the paper asserts that these two inputs fit together cleanly enough to certify the smaller diameter. If the claim is right, the least possible gap between consecutive primes is now known to be no larger than 240, and the previous record of 246 is improved.

Load-bearing premise

The argument depends on the newer equidistribution estimates for smooth moduli being correct and strong enough that, when combined with the Bombieri–Vinogradov theorem, they genuinely support the sieve at the sizes used; if those estimates fall short, the claimed bound of 240 does not follow.

Editorial extensions

If this is right

If the bound is correct, then infinitely many consecutive primes are separated by at most 240, no matter how far along the number line one looks.
The known interval containing the smallest possible prime gap now ends at 240 instead of 246, a concrete improvement of six.
The result shows that newer equidistribution estimates for smooth moduli have direct value for bounded-gap sieves, not just as standalone theorems.
The new bound remains far from the twin-prime conjecture's target of gap 2, so the result is best read as a step within an ongoing numerical progression.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if the same combined equidistribution input can be strengthened further, the next improvements are likely to be small numerical drops below 240, until the obstacle becomes the existence of admissible shift sets of smaller diameter at the required distribution level.
A concrete testable extension is to check whether admissible tuples of diameter less than 240 exist for the sieve's parameters; if not, no amount of extra equidistribution can lower the bound through this particular route.
The general lesson implicit in the improvement is that progress on bounded prime gaps is currently limited about equally by available equidistribution results and by the combinatorics of small prime clusters, so advances in either ingredient could produce the next record.

Original abstract

Polymath8b proved that $H_1 = \liminf (p_{n+1}-p_n) \leq 246$. In this paper we show how the Bombieri-Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli to obtain the improved bound $H_1 \leq 240$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,313 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.07 · human participants 0.02 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Polymath8b proved that $H_1 = \liminf (p_{n+1}-p_n) \leq 246$. In this paper we show how the Bombieri-Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli to obtain the improved bound $H_1 \leq 240$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Polymath8b proved that $H_1 = \liminf (p_{n+1}-p_n) \leq 246$.	background 1.00	0.05	0.05	asserted 1.00
s1	In this paper we show how the Bombieri-Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli to obtain the improved bound $H_1 \leq 240$.	contribution 0.59	0.94	0.96	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Prime pairs along rays of prime indices
math.NT 2026-09 accept novelty 7.0 of 10

The set of ratios of prime indices for which the sum (or difference) of the corresponding primes is a square is dense in the positive reals, and almost every index has many such partners.
Clusters of primes revisited
math.NT 2026-10 accept novelty 6.0 of 10

The paper establishes H_2 ≤ 31140, H_3 ≤ 2560022, H_4 ≤ 193672056, H_5 ≤ 13659045840, and proves M_k ≥ log k − 0.273572 + o(1).
Admissible sets of integers in imaginary quadratic fields of class number 1
math.NT 2026-10 accept novelty 6.0 of 10

Smaller admissible sets in imaginary quadratic fields yield improved prime gap bounds and counterexamples to a Hardy-Littlewood analogue.

Discussion

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