Disputed AI-assisted

Disputed claim that Catalan's constant is irrational

A preprint claims a proof that Catalan's constant is irrational, using weighted tails in the Calegari-Dimitrov-Tang framework; a specific technical objection to the argument has been raised publicly and is unresolved.

Lab: Nanjing University
Model: ChatGPT 5.6 Solar, named in the paper as the verifier; the model consulted during the work is named only as AI
Field: Mathematics
Date: 2026-09-03
Human collaborators: Zhi-Wei Sun

Sources

Original work

arXivCatalan's constant is irrational↗

Challenge

Hacker Newstechnical objection to the argument, equation 1.4 against equation 2.12↗arXivWachs, A note on a recent claimed proof of the irrationality of Catalan's constant↗

What was found

The irrationality of Catalan's constant is a long-standing open problem. The preprint adapts the method Calegari, Dimitrov and Tang used for the linear independence of 1, zeta(2) and L(2, chi_-3), replacing part of the arithmetic input with a construction of suitable weights and weighted tails. The acknowledgements set out an unusually specific division of labour: repeated attempts by the model failed, the author then supplied the weighting idea that made the approach work, and the model produced the numerical data. The paper states that 'the whole proof has passed the verification of Chatgpt 5.6 Solar', which is a machine reading rather than a human check.

Novelty check

The irrationality of Catalan's constant is a well-known open problem with no claimed resolution in the four registries before this preprint. The method is explicitly derivative of Calegari, Dimitrov and Tang's 2024 work, which the paper cites, and the claimed novelty is the weighting construction rather than the framework. No competing claim was found. The question of priority is not the issue here; the correctness of the argument is, and it is disputed.

Caveats and known objections

Graded disputed on a specific, unresolved technical objection rather than on general scepticism. The objection, raised publicly and cited here, is that the tail recurrence of equation 1.4 is inconsistent with the factor used in equation 2.12, so the zero count the argument needs in order to force the degree bound fails. The same critique notes that the paper replaces the arithmetic content of the model it follows with a combinatorial minimization plus numerics that no human has checked, and points to internal inconsistencies in the write-up. The preprint remains at version one with no revision or withdrawal at entry. A second, independent critique by Dane Wachs, with published scripts, shows numerically that a quantity the main derivation needs to be negative of order $B^2$ is positive of that order, so the proof of the main theorem is incomplete; it concludes nothing about whether the constant is irrational, which is why the grade is disputed rather than refuted. Autonomy is graded ai-assisted, and the paper's own account is the reason: the model 'made many attempts but we failed again and again' until 'the author realized that we should use suitable weights', so the decisive idea is the author's. The only stated verification of the whole proof is by a language model, which is not an independent check by any human.

Independent checks

Dane Wachs (critical note with computations): Found a slip in the printed proof of the rank theorem that is repairable, and verified that statement exactly at 44 parameter pairs, but showed numerically that a quantity the main derivation needs to be at most $-0.00966B^2 + o(B^2)$ is in fact between $+1.49B^2$ and $+1.79B^2$ , concluding that the published derivation does not justify the estimate and the proof of the main theorem is incomplete. Nothing is concluded about the irrationality of $G$ . · link ↗

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History of this entry

2026-09-24ChallengedCited Dane Wachs's note showing numerically that the main estimate fails as derived, so the proof is incomplete. The grade stays disputed: the note shows a gap, not that the claim is false. source ↗
2026-09-07AddedEntered the registry graded Disputed and AI-assisted.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Graded disputed for verification and ai-assisted for autonomy. What these mean.

Cite this entry

Plain text

whataifound.org. (2026). Disputed claim that Catalan's constant is irrational. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-09-03-catalan-constant-irrationality

BibTeX

@misc{whataifound-nanjinguniversity-2026-irrationality,
  title        = {Disputed claim that Catalan's constant is irrational},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Nanjing University. Verification: Disputed. Autonomy: AI-assisted.},
  url          = {https://whataifound.org/finding/2026-09-03-catalan-constant-irrationality}
}

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