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We’re releasing a broad range of new mathematical results produced by an internal frontier model.

We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study, and we have drawn on their advice and public recommendations to inform how we release these results.

Read the blog

The public openai/math repository currently contains 722 manuscripts organized into 372 related result families. It includes papers, supporting proof artifacts, Lean formalizations for many results, and 10 abridged summaries of the model’s reasoning.

What’s included

  • Results from an evaluation involving approximately 4,000 problems
  • An average compute expenditure per result equivalent to roughly three hours of ChatGPT Pro thinking with the internal model
  • Computer-checkable Lean proofs for many, but not all, manuscripts
  • Revision and citation protocols, with previous versions remaining accessible
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Independent re-check of the quasi-Riemann result (family 003): it passes

Thanks for publishing the Lean code with comparator configs. It made this straightforward to check.

I re-ran the Lean check for OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re (ζ(s) ≠ 0 for Re(s) > 7/8) from the public repo at commit adc7f12:

  • With your challenge and config, Lean FRO’s comparator printed “Lean default kernel accepts the solution” and “Your solution is okay!”, and exited 0.
  • I ran it again against a challenge statement I wrote myself, with the independent nanoda kernel turned on. Both kernels accepted it.
  • #print axioms lists exactly propext, Classical.choice and Quot.sound.

All 2,924 modules in the proof compiled with no errors and no warnings. The report, the raw logs and a container recipe to reproduce it are on GitHub at davegoldblatt/openai-zeta-proof-check (I can’t post links yet as a new forum member).

Caveats: this checks the Lean proof, not the paper. It’s one run on one machine, and an AI agent (Claude Code) carried it out at my request, so independent reproductions would help.

One suggestion: 404 of the 405 comparator configs in the repo leave the second kernel off. Turning it on by default would make these checks stronger.

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One of the results in the release is a new matrix multiplication upper bound showing that ω ≤ 2.25 over the complex numbers. I was surprised that the underlying field mattered, so I tried to understand why the results is restricted to ℂ.

Long story short, Codex extended the proof to arbitrary fields, producing a Lean proof in a fork of the repo. The resulting theorem states that ω(F) ≤ 2.25 for every field F, using the same definition of ω.

Lean verification passes, as do Comparator checks. From reading the formalized statement, I am pretty convinced it is correct, and so are the agents.

Cannot include links, but the proof is on GitHub under selanavot/matrix-multiplication-all-fields

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The Snaky game solution is very nice. Since this strategy is a finished program, you should post this as a playable game, for example on Android.

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Let me suggest: Open AI needs to set up a WIKI so that humans can comment on their giant output of math alleged-problem-solutions. The rate is so high that it is drinking from a firehose. Humans cannot even digest this rate. With a wiki, humans will be able to collaboratively digest, increasing the digestion rate; and also feedback from the wiki might be usable by the AI to enable it, in future, to rewrite its manuscripts with better writing.

“Better” meaning, more useful for human readers.

I am NOT in favor of holding back machine discoveries so some self-interested committee of human anuses can restrict progress. And I do not like the human arXiv and science journal systems as they presently work - I consider them corrupt and intentionally dysfunctional. But I AM in favor of figuring out how to digest the AI flood and how humankind should make use of it, and we need a wiki as part of that.

Let me elaborate on what science needs. (I’ve been saying this for about 30 years, but the science community refused to give a crap about what I’ve been saying, which is because I’m right and they are wrong and they are corrupt.)

The arXiv is corrupt because it only permits those with arXiv-approved employers to publish on arXiv. You can publish utterly false crap there immediately, no questions asked, IF you are a lifelong academic fraud. FOR EXAMPLE, OpenAI in their “irrationality exponent of pi is 2” paper cite work of Nelson A. Carella. Unfortunately OpenAI neglects to mention that all or nearly all work that Nelson A. Carella has ever published, is fraudulent, including this one, and his “proof” of the Riemann hyporthesis, and his “proof” that INTEGER FACTIORNG is in polynomial time, and his “proof” that there are no odd perfect numbers, and on and on. The reason Carella is allowed to keep publishing his garbage is that he is employed by academic institutions who simply do not care that he is a lifelong fraud. If arXiv had commenting, Carella would have been exposed many years ago and hopefully fired from his jobs. I know I personally tried to point out this fraudulence is this and other cases, years ago, and never received any satisfaction.

Meanwhile if somebody not part of the club tries to publish on arXiv, no matter how important and correct, his/her work is rejected. In particular, the 1905 works of Albert Einstein would today be insta-rejected. Ditto Srinivasa Ramanujan. As a result virtually the entire continent of Africa is near 100% excluded from doing science. Ever. Indeed 99% of humanity is excluded. The cost to human progress is enormous.

Now suppose somebody on arXiv publishes some utterly false fraudulent bullshit. Then I am not allowed to point that out. I repeat, I am not allowed even to provide a COMMENT pointing out an error in somebody’s work. No comments or questions of any kind are allowed by arXiv. This is absurd. Repositories like arXiv offer the ability to distribute science far faster and cheaper than in the pre-internet age. Why not also provide far superior refereeing than attainable in the old days? But no: arXiv chose not only to have zero refereeing, but in fact to FORBID even pointing out an error.

If each arXiv paper had a wiki attached allowing commenting by anybody - not just a member of the “club” but anybody - the result would be effectively infinite refereeing, far faster, far cheaper, ultimately far better than ever before in the whole history of science. But no. They chose to forbid that.

And I want a numerical rating system: readers can RATE the papers they read. So there are good papers rated 100 and bad papers rated 0. And it should be tied into searching. So I can search for papers on topic X using keywords Y, but only if ratings>Z. Places like stackoverflow and allrecipes have already solved the problem of providing ratings and ratings-of-raters, etc.

viXra is better than arXiv in the sense they provide a commenting system and they do not have a “club” of people who are allowed to publish, with everybody else arbitrarily excluded. BUT they have no rating system, and no searching-tied-to-ratings capability, and their paper collection sucks. I estimate 80% of vixra papers are garbage, the opposite of arXiv with 20% garbage. I would not care about 80% garbage fraction on vixra if there were a search+ratings capability enabling me to find the few good papers. Then it would be a useful science repository. But since no such capability, that just makes viXra a pile of garbage.

And now with AI providing a flood of papers, you have the opportunity to SET UP such a system as a place to put your AI-generated papers; and you could let everybody else put science papers there too including those partly-generated using AI. As a side effect you will revolutionize science in a way that will vastly benefit humankind, democratize science, and overcome decades of corruption. Also the potential now exists for AIs to act as referees of human papers with comments, questions, alleged refutations, plagiarism finding, lit-search cite-adds, etc etc. That also might cause dramatic improvements both in the human papers and in the AIs.

Please - I’m begging you - set this science-paper system up. Doing so will be far cheaper and far more useful than any conference you set up where a bunch of blowhards pontificate about how to mishandle AI science.

In the OpenAI papers “Universal computation in forced Navier–Stokes flows,” “Universal computation with eventually stationary Navier Stokes forcing,” etc you do the same thing I already did in my 2005 paper that you did not cite (vixra 2503.0157) except better in these senses: (a) OPENAI claims to be fully rigorous and (b) they unlike me did not need a weird rigid container. But worse in the sense (c) my 2005 thing solved halting problem in bounded time, think OpenAI’s takes longer. I have doubts OpenAI really accomplished a+b because as I said in my paper there are fundamental obstacles to rigorization, and if OpenAI really overcame those obstacles, that is bigger news than this.

Anyhow, that is just one example of the useful discussion that could happen on a wiki. Also note, probably the reason OpenAI failed to cite my paper was it was not on arXiv. Proving I am correct that asrXiv is corrupt and dysfunctional since you are a victim of exactly that, as this and the Carella case both prove.

Want more such proof? Another: in your OpenAI paper “One rational hitting point for noncommutative formulas” fails to cite my 2004 work (vixra 2607.0013), chapter 11 of which shows results very related to yours. Are my results better than yours? Worse? Related exactly how? (Mine seem simpler and better explained anyhow.) Those answers are not immediately obvious to me, but in any case it is clear you would have been better off knowing about my work and citing it.

--Warren D. Smith (PhD), 7 Oct 2026.

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Documentation note: stale manuscript references in NavierStokesAndEuler

This concerns the separate openai/NavierStokesAndEuler repository and the published Navier–Stokes manuscript referenced in its README.

During an AI-assisted audit of correspondence between the manuscript and its Lean formalization, we identified outdated manuscript references in comments.

How we found and checked them

We compared manuscript statements and formulas with the corresponding Lean declarations and their explanatory comments. For references that did not match the published numbering, we located the corresponding mathematical expressions in the manuscript and checked their surrounding context.

The proposed mappings therefore rely on formula correspondence and the role of each result, rather than simply replacing lemma numbers. They identify the relevant components documented by each file; they do not imply that each file formalizes the entire cited lemma or proposition.

Audited repository commit:
f9e8bc5b38b6e212696e8a30e3e91517af887bbd

All files below are in the NavierStokes/ directory. Page numbers refer to the published PDF referenced in the repository README.

Lean file Existing reference Corresponding reference in the published PDF PDF page
PowerMomentMatrix.lean Lemma 3.6 Lemma 4.7 — matrix nonsingularity component 34
ConeAlgebra.lean Lemma 3.5 Lemma 4.5 — cone criterion and factorization 31
CurlGeometry.lean Lemma 8.8 Lemma 7.7 — algebraic identities in the proof 86
TorusAverages.lean Lemma 8.7 Proposition 7.5, Step 1 / equation (7.27) 82
CurlClassBounds.lean Lemma 9.2 Lemma 7.7 / equation (7.38) — curl remainder 86

ConeAlgebra.lean also contains outdated equation references. Its lower-root definition cites equation (10), which corresponds to equation (4.21) in the published manuscript. Three references to equation/criterion (11) correspond to an unnumbered sufficient inequality or factorization in Lemma 4.5 and its proof. The proposed patch uses descriptive references for these.

Patch and validation

The patch corrects 12 reference occurrences in five files, improving navigation between the manuscript and the formalization.

git apply --check succeeds against the audited commit. Removing comments leaves the non-comment source unchanged in all five files. No declarations or proof terms are changed. The Lean project was not rebuilt for this documentation patch.

The patch and a mapping table with page numbers and hashes are available.

GitHub Issues and Discussions are disabled, and the pull-request page currently states that PR creation is restricted. Could someone direct this report to the repository maintainers or indicate the preferred channel for submitting the documentation patch?

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That’s fantastic! I do hope you’ll put this on Android.

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A 4×4 Obstruction to Polynomial Removal for Ordered Binary Matrices

I would like to share a follow-up to OpenAI’s September 25, 2026 manuscript, Polynomial Removal Fails for Ordered Binary Matrices.

The accompanying paper establishes a counterexample to polynomial induced ordered removal bounds for a fixed 4×4 binary pattern, reducing the obstruction size from 66×66 in the original construction.

Main result

Consider the fixed ordered binary pattern

H_4 = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 \end{pmatrix}.

For every integer h \ge 1, set m=2^h. The construction produces an ordered binary matrix A_h of order

n=(14h+2)m.

The manuscript establishes the following exact identities.

Editing distance:

D_{H_4}(A_h)=m^2.

Induced-copy count:

N_{H_4}(A_h) =2m^6+\frac{m-2}{2}m^5.

Original-copy transversal number:

\tau_{H_4}(A_h)=m.

Consequently, the normalized editing distance is

\frac{D_{H_4}(A_h)}{n^2} =\frac{1}{(14h+2)^2},

while the normalized induced-copy density satisfies

\frac{N_{H_4}(A_h)}{n^8} =\Theta\left(4^{-h}h^{-8}\right).

Together, these estimates establish the failure of polynomial induced ordered removal bounds for the fixed pattern H_4.

Relation to the OpenAI result

The argument develops the tree-propagation approach introduced in the original OpenAI manuscript. Its central improvement is the reduction of the fixed obstruction from a 66×66 pattern to a 4×4 pattern, accompanied by exact enumeration and unrestricted editing-distance analysis.

The release includes a 21-page manuscript, complete LaTeX sources, a Lean 4 formalization, reproducibility scripts, and verification records.

Paper, source code, and formal verification:

github：wzn1118/h4-ordered-matrix-removal

I welcome independent scrutiny of the proof and formalization, and would appreciate feedback from the original authors or the repository maintainers.

If verified, I would be grateful for consideration of this work as a follow-up to the corresponding result in the OpenAI mathematical releases.

Zinan Wang

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Regarding the finite Benson-Etingof-Ostrik conjecture (Fiber-functors-…), there is a preprint that appeared before the publication of the openai paper: arxiv 2610.04096.

This should be acknowledged in an updated version of the openai paper.

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Independent Verification of OpenAI’s Exact DFT Preprint (Family 130)

Hello,

I’m Oleksandr Pylypenko, an independent researcher from Ukraine.

Following OpenAI’s recent mathematics release, I conducted an independent computational investigation of the local Fourier compiler described in the September 25, 2026 preprint, “An explicit power saving for the exact discrete Fourier transform” (Family 130).

The investigation resulted in a 31-page research manuscript accompanied by publicly available implementations, experimental data, and reproducibility materials.

Key findings:

  • Exact arithmetic: Verified equality with the DFT matrix for eight selected transform sizes using cyclotomic fields.
  • Finite-field verification: Conducted additional checks at larger sizes, including a streaming experiment at r = 1024.
  • Numerical stability: Observed severe finite-precision instability, with float64 relative error reaching approximately 8.1 × 10²³ at r = 64.
  • Circuit analysis: Examined arithmetic operation counts, fixed computational topology, and parallel depth.
  • Formal verification: Examined the associated Lean formalization and its correspondence to the mathematical claims.

Scope clarification: These findings neither prove nor refute OpenAI’s main theorem. The investigation focuses on the local Fourier compiler, not the full tensor-network construction responsible for the claimed asymptotic improvement.

The research is publicly available:

The manuscript is published on Zenodo under the title:

An independent check of the local Fourier compiler in OpenAI’s preprint “An explicit power saving for the exact discrete Fourier transform”: exact and numerical tests, depth, cost, and limits.

The source code and reproducibility package are available in the GitHub repository cyberdid/fft-power-saving-verification.

I would particularly appreciate technical feedback on the numerical stability analysis, exact verification methodology, and the interpretation of the local construction.

If members of OpenAI’s mathematics research team are following this discussion, I would be grateful if these findings could be brought to their attention.

Thank you for making these mathematical research materials publicly available.

Oleksandr Pylypenko
Independent Researcher, Ukraine

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OpenAI dropped 719 math manuscripts in a single day - all generated by an unreleased internal model that thinks faster than any LLM publicly available today. Problems unsolved for decades. Some for over a century. The model thought for 3 hours per problem and produced more mathematical discoveries than all human mathematicians combined in the past several years.

The results are staggering. They narrowed the zone where Riemann hypothesis zeros can exist by 25% - the biggest breakthrough on the problem in history. They reduced the matrix multiplication exponent from 2.37 to 2.25 - every transformer forward pass and every LLM inference call on earth gets cheaper as that number drops toward 2. They proved Hilbert’s 10th problem impossible even for fractions, establishing a hard ceiling on what any algorithm can ever compute regardless of how powerful AI becomes.

37 minutes. Bookmark & watch today. The week AI rewrote the history of mathematics.

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Independent prime–zeta research (2018–2026): Request for mathematical comparison

I am Kenneth A. Giuffre, M.D., an independent researcher investigating the deterministic mathematical structures associated with prime numbers and the nontrivial zeros of the Riemann zeta function.

Since 2018, I have developed arithmetic and geometric constructions including the P-array, the 6n ± 1 modular prime sleeve, reciprocal-slope boundaries, and logarithmic spectral models.

Of particular relevance, the P-array establishes its slope-1/2 geometric hard deck independently of any nontrivial-zero ordinates or spectral wave summations. The NTZ-derived constructions were introduced subsequently to investigate possible relationships between these independently defined mathematical structures.

My research was publicly documented in January 2024 and subsequently expanded through a four-part ResearchGate series in 2026.

January 2024 publication

January 2024 publication: DOI: 10.13140/RG.2.2.32559.92326

Part IV (2026): DOI: 10.13140/RG.2.2.19444.51846

Part IV — Proofs, Models, and Provenance (2026):

I am interested in whether the mathematical mechanisms underlying these constructions have a substantive relationship to those employed in OpenAI’s recently released 7/8 zero-free-region result.

I am not asserting that my framework proves the same theorem. I would welcome an independent mathematical comparison and guidance on submitting my accompanying two-page research notice to the researchers involved.

I would also appreciate clarification as to whether publicly indexed mathematical preprints were accessible to the research agents.

Kenneth A. Giuffre, M.D.
Trilobot Institute for Applied Cognitive Research

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Independent Research Inspired by OpenAI’s Critical Percolation Work

OpenAI’s recent mathematics release, particularly Result Family #213 on critical percolation, inspired us at Nutelligence Research to investigate related questions in site percolation.

Our work led to three formal theorem developments:

  • Theorem P: Critical site percolation on periodic graphs.
  • Theorem P′: A broader result for quasi-transitive graphs under specified growth conditions and an explicit structural hypothesis.
  • Theorem S: Critical site percolation on specialized lattice graphs.

All three formal propositions have been verified using Lean 4, with exact proof-verification certificates obtained through our MathVerify framework. Theorem P′ retains an additional mathematical assumption.

These results form part of Nutelligence’s independent follow-on research. Their novelty relative to existing literature remains under review.

We have prepared a detailed technical research page under Nutelligence Research, including theorem statements, proof explanations, assumptions, and verification evidence.

We welcome feedback from researchers in probability theory, graph theory, and formal mathematics.

Nutelligence Research

Independent research; not affiliated with or endorsed by OpenAI.

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Where is Proposition 2.1 of OpenAI’s paper on the 7/8 Quasi-Riemann Hypothesis formalized in lean 4 ?

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The proposal described is discussed further in the update of Part IV as Part IVa…DOI:

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I think Proposition 2.1 is replaced in the lean 4 formalization by theorem nonzero_of_common_signal in OAI/NumberTheory/DirichletL/Continuation.lean .

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A numerical observation on off-critical-line zeros of a Riemann–Siegel approximation

I would like to share a numerical observation that may be relevant to the discussion of quasi-RH and the difficulty of narrowing the zero-containing strip toward the critical line.

Consider the symmetric Riemann–Siegel main-sum approximation:

ζ_N^dual(s) = P_N(s) + χ(s)P_N(1−s),

where P_N(s) = Σ(n=1 to N) n^(−s), χ(s) is the factor in the Riemann functional equation, and N = floor(√(t/2π)).

The exact relation is ζ(s) = ζ_N^dual(s) + E_N(s). On the critical line, the Hardy-normalized remainder is O(t^(−1/4)), so the approximation becomes more accurate in absolute terms at large heights.

Near the classical Lehmer pair #6709–6710, numerical calculations with N = 33 give two off-critical-line zeros of ζ_N^dual:

s₊ ≈ 0.576484125614720 + 7005.084141009866i

s₋ ≈ 0.423515874385280 + 7005.084141009866i

These zeros are symmetric about Re(s) = 1/2 and can be independently checked numerically. They are zeros of the finite approximation, not of ζ(s).

Both lie within the strips 1/8 ≤ Re(s) ≤ 7/8 and 1/12 ≤ Re(s) ≤ 11/12.

My question is whether the persistence of such off-critical-line zeros as t increases, together with the decreasing Riemann–Siegel remainder, could reveal something about the limitations of progressively narrowing the zero-containing strip.

Of course, a small remainder alone does not guarantee that zeros of the approximation correspond to zeros of ζ(s). Their stability under the remainder correction would need to be established.

I would be interested in any mathematical insights or independent verification of this phenomenon.

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  1. Humans make more mistakes in maths than AI: All papers · Math Referee

  2. More conjectures proved (some open since 30 years, with only partial cases settled): https://github.com/sashakolpakov/math/tree/contribute/sarnak-trace-gap/preprints/Trace-Sparsity-and-Arithmeticity-for-Compact-Hyperbolic-Surfaces-October-23-2026)

    It would be really great if OpenAI allowed those academics who do not take the defensive stance towards AI to contribute.