4,000 Problems, 3 Hours Each: The AI Math Flood Nobody Can Check Fast Enough

What happened The numbers The pace Fields Famous problems What's a conjecture? Verification Problems & criticism Myths vs facts Verdict Sources

AI × Mathematics · October 2026

The 722 Manuscripts: What OpenAI's AI Actually Claims to Have Proved

On October 6, 2026, OpenAI dumped hundreds of AI-written math papers onto GitHub, touching the Riemann Hypothesis, the Hodge conjecture, Kakeya, Mahler and more. Headlines screamed "722 problems solved." That is not what happened. This explainer covers what was released, which fields it spans, which famous conjectures are involved, and why mathematicians remain cautious.

Reading time ≈ 14 min · All figures sourced at the bottom

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manuscripts released on Oct 6

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problem "families"

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subject areas covered

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top-line results formalized in Lean

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manuscripts withdrawn in 24 hours

01 · The event

What actually happened?

OpenAI published a public repository, github.com/openai/math, containing mathematical manuscripts written by an unreleased internal model. Alongside the papers it posted Lean proof files (machine-checkable proofs) for some results and abridged "reasoning summaries" for ten of them.

According to the repository README, OpenAI was testing its models on open research problems because its existing math benchmarks had become too easy for them. Over the evaluation, the model was posed roughly 4,000 problems. The outputs were then grouped into "families" and filtered for significance, which produced the catalogue.

On average, each result used about three hours of ChatGPT Pro-level thinking compute. That detail is what alarmed many observers: it is a small budget, not a supercomputer grinding for months.

⚠️

Correction: "722" counts manuscripts, not solved problems. One family can hold a main result plus companion papers, consequences and alternative proofs. The real unit is the family (372), and even those are a mix of full resolutions, partial results, special cases and counterexamples.

Two exceptions OpenAI itself flags

The README says nearly everything came from the same automated procedure, with two named exceptions: work on a zero-free region for the Riemann zeta function, and the proof of the Hodge conjecture for CM abelian varieties. It also says the write-up of one zeta result (a zero-free region for $\operatorname{Re}(s) > \tfrac{11}{12}$) was edited by humans for readability. So "fully autonomous" is not accurate for the two most headline-grabbing items.

02 · By the numbers

From 4,000 questions to 300 checked results

The clearest way to understand the release is as a funnel. Each stage throws something away. Hover the bars.

The filtering funnel

How a large evaluation became a published catalogue

Sources: OpenAI README and history log (Oct 7, 2026). "Formalized" = top-line results with a Lean proof: 300 of 719.

Going from ~4,000 problems to 372 families means most attempts did not produce something OpenAI considered publishable. That is normal in research, but it means the hit rate is far lower than "722 solved" implies. The last bar matters most for trust: only 300 of 719 top-line results currently have a Lean formalization. The rest are, for now, unverified claims.

How much is machine-verified?

Top-line results with vs. without a Lean formal proof (after withdrawals)

300 · formalized in Lean (~42%)
A computer has checked every logical step of the formal statement.

419 · not formalized (~58%)
Human-readable proofs only. OpenAI itself warns some may have issues.

Some coverage claimed "every result comes with a Lean certificate," and another outlet estimated ~60% of families. The repository's own count is $\tfrac{300}{719} \approx 41.7\%$ of top-line results. Trust the primary source.

03 · The pace

A curve that went nearly vertical

This was not OpenAI's first math release in 2026. It was the third major one in about nine weeks. Toggle between linear and log scale: on a log scale, roughly 10× growth per month looks like a straight line.

Publicly released AI math results, 2026

Approximate counts reported for each month's release

August ≈ 10 results (Astra), September 100+ (including the Navier–Stokes claim), October 722 manuscripts. Figures are approximate as reported by MindStudio and Startup Fortune; August and September counts are results, October is manuscripts, so this is not a perfectly like-for-like comparison.

🧭

Note: a steep curve of outputs is not a steep curve of accepted mathematics. Publication speed went up 70×; verification capacity (the number of experts who can referee a Kakeya or Hodge paper) did not. That gap matters more than the curve itself.

Timeline of the year

MAY 2026

OpenAI disproves the 1946 Erdős unit-distance conjecture, published with a human-verified companion paper. Widely seen as the first historically significant AI-generated proof.

AUG 1, 2026

Internal model "Astra" releases 10 results with Lean certificates, including a sphere-packing bound improvement. Scientific American later reported that two of them leaned on prior published ideas without proper citation.

SEP 8, 2026

OpenAI claims finite-time blow-up solutions related to Navier–Stokes (a 166-page manuscript). Not a Millennium Prize claim, but a major one.

SEP 2026

25 Fields Medalists sign a declaration objecting to how results are being released: no named authors, no attribution trail, no time to absorb. A credit dispute with NYU's Tristan Buckmaster also surfaces; OpenAI disputes it.

SEP 21, 2026

OpenAI forms a nine-member outside advisory group on math and AI, hosted by the Institute for Advanced Study (members include Timothy Gowers, Martin Hairer, Edward Witten, Ravi Vakil).

OCT 6, 2026

The 722-manuscript repository goes public.

OCT 7, 2026

A sign error is found. Three Hodge-related manuscripts are withdrawn; 14 others are revised with proof repairs. Count drops to 719.

04 · How many fields

17 subject areas, but not evenly spread

The 372 families are classified by mathematical discipline into 17 areas. No complete per-field tally has been published, so the chart shows only the reported counts and groups the remaining families together.

Families by field (reported counts only)

Hover a bar for details

Algebraic & complex geometry: 36 families (reported as 3rd largest). Number theory: 31 families. The remaining 305 families are spread across the other 15 areas; their individual counts were not published in the sources used here. 36kr / Synced (Oct 8, 2026).

Because algebraic geometry is described as the third-largest area with 36 families, two other areas must each have more than 36. Which ones has not been reported. The paper titles and reasoning summaries show that these disciplines appear:

Number theory 31 Algebraic & complex geometry 36 Harmonic analysis Convex geometry Additive combinatorics Theoretical computer science Probability / spin glasses Mathematical physics Partial differential equations Operator algebras Ring & group theory Complex analysis Representation theory / Langlands

Proofs vs. disproofs

Families whose abstracts mention "disproof" or "counterexample"

"About 50" is a rough keyword search by 36kr/Synced, not an official count. Treat it as an estimate.

Disproofs matter. Finding a counterexample to a conjecture that stood for decades is just as valuable as a proof, and counterexamples are often easier to check: you can verify the example directly.

05 · The famous names

Which conjectures are involved?

Every "proved" below is OpenAI's claim, not an accepted result. Badges show what kind of claim it is and whether a Lean proof exists. Use the filters, and hover the cards.

Number theoryRiemann · 1859

"Quasi-Riemann Hypothesis" (family 003)

The Riemann Hypothesis says every non-trivial zero lies on the line $\operatorname{Re}(s)=\tfrac12$. Nobody had proved there is any fixed vertical line left of $1$ with no zeros to its right. OpenAI claims:

$$\begin{gathered} \zeta(s)\neq 0 \quad\text{and}\quad L(s,\chi)\neq 0 \\ \text{for } \operatorname{Re}(s) > \tfrac{7}{8} \end{gathered}$$

The README separately mentions a $\tfrac{11}{12}$ zeta result. This is NOT the Riemann Hypothesis.

Partial resultLean (reported)

Number theoryMillennium Prize

Birch & Swinnerton-Dyer formula (families 002, 006)

BSD links the rank of an elliptic curve $E$ to its $L$-function:

$$\operatorname{ord}_{s=1} L(E,s) \;=\; \operatorname{rank} E(\mathbb{Q})$$

OpenAI claims the full BSD formula for "almost all" quadratic twists $E_d$ of every rational elliptic curve. A special family of cases, not the whole conjecture.

Special casesNo Lean

Number theoryGoldfeld · 1979

Goldfeld conjecture

Predicts that across the quadratic twists $E_d$, ranks split half $0$ and half $1$, so the average rank is

$$\lim_{X\to\infty} \operatorname*{avg}_{|d|\le X} \operatorname{rank} E_d(\mathbb{Q}) = \tfrac12$$

Claimed as a by-product of family 006.

Claimed proofNo Lean

Number theory / logicHilbert · 1900

Hilbert's 10th problem over $\mathbb{Q}$ (family 004)

Given any polynomial with integer coefficients, is there an algorithm to decide whether

$$P(x_1,\dots,x_n)=0, \qquad x_i\in\mathbb{Q}$$

has a solution? Over $\mathbb{Z}$ the answer is no (Matiyasevich, 1970). Over $\mathbb{Q}$ it stayed open. OpenAI claims the answer is also "no."

Claimed proofNo Lean

Number theory1860s

Catalan's constant is irrational

$$\begin{aligned} G&=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)^2} \\ &= 1-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots \\ &\approx 0.9160 \end{aligned}$$

No one knew whether $G$ is irrational. OpenAI claims it is. Simple to state, notoriously hard.

Claimed proofLean status unclear

Number theoryfamily 017

Irrationality measure $\mu(\pi)=2$

$\mu(\pi)$ is the smallest exponent such that $\left|\pi-\tfrac{p}{q}\right| < q^{-\mu}$ has only finitely many solutions. The best proven bound was $\mu(\pi)\lesssim 7.1$; OpenAI claims $\mu(\pi)=2$, the value for almost every irrational. That would also settle whether this series converges:

$$\sum_{n=1}^{\infty}\frac{1}{n^{3}\sin^{2} n}$$

Claimed proofLean (reported)

Algebraic geometryMillennium Prize

Hodge conjecture for CM abelian varieties (family 032)

A highly symmetric special class, far from the general conjecture. Not produced by the standard automated procedure, per OpenAI's own README.

Special caseNo Lean

Algebraic geometryfamily 054

Kuznetsov's conjecture on cubic fourfolds

Kuznetsov proposed a category-based test for when a cubic fourfold is rational. OpenAI claims examples that pass the test but are not rational, overturning it.

DisproofLean status unclear

Algebrafamily 008

Deligne–Drinfeld conjecture

About the structure of the Grothendieck–Teichmüller Lie algebra. Claimed solved.

Claimed proofLean (reported)

Representation theoryfamily 069

Quantum geometric Langlands

Builds on the ~1000-page 2024 proof of geometric Langlands. Claims the "quantum" version at irrational parameters.

Claimed proofLean status unclear

Harmonic analysisKakeya · 1917

Kakeya: 3D maximal and 4D dimension (family 074)

Hong Wang and Joshua Zahl proved the 3D Kakeya set conjecture (Wang won the 2026 Fields Medal). OpenAI claims the stronger 3D maximal version and the 4D case. Flagged as urgently needing expert review.

Claimed proofNo Lean

Convex geometryMahler · 1939

Mahler conjecture, all dimensions (family 087)

For a convex body $K\subset\mathbb{R}^n$ and its polar $K^{\circ}$, Mahler conjectured

$$\begin{aligned} |K|\,|K^{\circ}| &\ge \frac{4^n}{n!} \quad \text{(sym.)} \\[4pt] |K|\,|K^{\circ}| &\ge \frac{(n+1)^{n+1}}{(n!)^2} \quad \text{(gen.)} \end{aligned}$$

Only the 3D symmetric case was settled (2020). OpenAI claims both versions in every dimension.

Claimed proofLean (reported)

Complex analysisKoebe · 1908

Koebe's circle-domain conjecture (family 071)

Can every planar domain be conformally mapped to one bounded by circles and points? Claims the existence part.

Partial (existence)Lean (reported)

Ring theoryfamily 197

Kaplansky's direct-finiteness conjecture (char. 2)

A classic about group rings, here in characteristic two only. One of ten results with a published reasoning summary.

Special caseLean status unclear

Operator algebrasfamily 287

Isomorphism of free group factors

One of the most famous open questions in von Neumann algebras. Listed with a reasoning summary. If correct, this alone would be enormous, which is exactly why it needs specialist scrutiny.

Claimed resultLean status unclear

Mathematical physicsfamily 271

Spontaneous magnetization, quantum Heisenberg ferromagnet

A long-standing problem in rigorous statistical mechanics. Reasoning summary released.

Claimed resultLean status unclear

Theoretical CSfamily 102

NP-hardness at the basic SDP threshold

About how hard approximation becomes exactly at the limit of semidefinite programming. Reasoning summary released.

Claimed resultLean status unclear

Algebraic geometrywithdrawn Oct 7

Rational Hodge conjecture for products of K3 surfaces

Withdrawn with two dependent papers (Weil classes on abelian eightfolds; Kuga–Satake correspondences) after a sign error broke a key cancellation argument.

WithdrawnError found

06 · Back to basics

How is a conjecture formed, and how is it "solved"?

Judging these claims starts with knowing what a conjecture is. A conjecture is a precise statement that mathematicians believe is true but nobody has proved. It usually forms in four stages:

Observe a pattern

Someone computes examples and notices regularity, like Riemann noticing where zeta's zeros fall.

State it precisely

The pattern becomes an exact claim that is either true or false, with no vague words.

Gather evidence

Billions of checks, special cases, heuristics. Evidence builds belief, never proof.

Resolve it

A proof (true for all cases) or a counterexample (one case that breaks it). Then the community checks it.

Most of the 722 manuscripts don't fully resolve a famous conjecture. They do one of four things: prove a special case (Hodge for CM varieties), prove a weaker version (a zero-free line at $\tfrac78$ instead of $\tfrac12$), settle a related question (Hilbert's 10th over $\mathbb{Q}$), or disprove something. These are all real contributions if correct. They are just not "solving the Riemann Hypothesis."

The Riemann Hypothesis, in symbols

Four lines that explain the whole "quasi-Riemann" headline

Definition

$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{p\ \text{prime}}\frac{1}{1-p^{-s}}, \qquad \operatorname{Re}(s)>1$$

RH (1859)

$$\zeta(\rho)=0,\;\; 0<\operatorname{Re}(\rho)<1 \;\Longrightarrow\; \operatorname{Re}(\rho)=\tfrac12$$

Known (classical)

$$\zeta(\sigma+it)\neq 0 \quad \text{for } \sigma > 1-\frac{c}{\log |t|}$$

Claimed (unreviewed)

$$L(s,\chi)\neq 0 \quad \text{for all } \chi \text{ and } \operatorname{Re}(s)>\tfrac78$$

Why this matters: in the classical result, the gap $\frac{c}{\log|t|}$ shrinks to $0$ as $|t|\to\infty$, so the zero-free region squeezes against the line $\sigma=1$. A fixed line at $\sigma=\tfrac78$, if correct, would be the first of its kind. It is still far from $\tfrac12$.

The Riemann picture, visually

Where zeros are allowed to be: what is known vs. what is claimed vs. what RH says

Simplified illustration. Classical zero-free regions hug the line $\operatorname{Re}(s)=1$ and shrink as $|t|$ grows; no fixed line left of $1$ had been proven before. The claimed $\tfrac78$ line would be a major step. RH asks for $\tfrac12$.

07 · Checking the work

What Lean can and can't tell you

Lean is a proof assistant: a program that checks every logical step of a proof written in its formal language. If a Lean proof compiles, the formal statement is proved. That is very strong evidence. But students often misunderstand it in three ways:

1. Lean checks the formal statement, not your intention

If the conjecture is translated into Lean incorrectly ("misformalization"), Lean will happily verify a proof of the wrong thing. An expert still has to confirm the formal statement matches the famous conjecture.

2. Only ~42% of top-line results have Lean proofs

The other 58% are ordinary written proofs. OpenAI's own README says some unformalized results could have issues, and the October 7 withdrawals proved that warning right within a day.

3. A checked proof isn't the same as understood mathematics

UT Austin mathematician Francesco Maggi argued that results nobody understands or builds on risk becoming "dead letters." After the Navier–Stokes claim, some mathematicians said it was hard to extract human insight from the proof even where it was correct.

4. Who referees 700 papers at once?

In many subfields only a few hundred people worldwide can referee frontier work, and one dense paper can take a specialist a week or more. A single drop of this size outruns the field's checking capacity.

Confidence meter, by evidence type

How much weight each kind of claim deserves today

Lean-verified, statement checked by expertsHigh

Lean-verified, statement not yet expert-reviewedModerately high

Counterexample that can be directly checkedModerate–high

Written proof only, no Lean, no reviewLow–moderate

Headline / social-media summaryLow

A rule of thumb based on standard mathematical practice, not measured data.

08 · The problems

The criticisms, and the replies

The main concerns raised by mathematicians, followed by the strongest points in OpenAI's favour.

Concerns

🧾

No peer review. The papers went straight to GitHub. "Proved" and "solved" are OpenAI's own words throughout.

✍️

Attribution and credit. Fields Medalists objected to results with no named authors and no attribution trail. A credit dispute over the Navier–Stokes work (Buckmaster and Anthropic's Levent Alpöge) remains contested; OpenAI denies using their private work. Two August results were reported to under-cite prior work.

🔁

Errors already found. Within a day: 3 manuscripts withdrawn, 14 repaired, 13 more updated to cite revisions. That is responsible correcting, but it shows the unformalized part is not yet reliable.

📏

Unclear denominator. A widely shared "90 of the 500 hardest problems" framing came from outside trackers, not a published OpenAI breakdown. Open-problem lists mix brutal problems with near-folklore, so "solved X of Y" can mislead.

🔒

Unreleased model. Nobody outside OpenAI can rerun the model to reproduce how results were found, only check the outputs.

Counter-points in OpenAI's favour

✅

It's checkable. Everything is public under an Apache-2.0 licence, with Lean files, reasoning summaries and a versioned history that keeps withdrawn papers visible. That is more transparent than a press release.

🏛️

Serious mathematicians take parts of it seriously. Timothy Gowers reportedly said he would recommend at least one earlier AI proof to a top journal. The May unit-distance disproof is widely accepted.

🔧

Errors were fixed fast and in public. Withdrawing papers within 24 hours is how science should handle mistakes.

09 · Correcting the headlines

Myths vs. facts

Several viral headlines got this wrong:

MYTH

"AI solved the Riemann Hypothesis."

FACT

It claims a zero-free line at $\operatorname{Re}(s)=\tfrac78$ (and $\tfrac{11}{12}$ in another write-up). RH needs $\tfrac12$. Partial, unreviewed.

MYTH

"722 problems solved overnight."

FACT

722 manuscripts (now 719) in 372 families, from ~4,000 attempts over a long evaluation.

MYTH

"Every result is Lean-verified."

FACT

$300$ of $719$ top-line results: $\tfrac{300}{719}\approx 41.7\%$.

MYTH

"Millennium Prize problems were solved."

FACT

None were fully solved. BSD, Hodge and RH appear only as special cases or weaker versions.

MYTH

"It was all fully autonomous."

FACT

Most results used a fixed automated procedure, but the zeta and CM-Hodge work are named exceptions, and one write-up was human-edited.

10 · Verdict

So, is this a big deal?

Yes, probably, but not in the way the headlines say.

Even if only the Lean-verified 42% hold up and their statements are correctly formalized, that is hundreds of new results across 17 fields from a model using a few hours of compute each. Several touch problems that had resisted specialists for decades. That changes what research mathematics looks like.

The status today is: a large set of claims, a minority machine-checked, almost none peer-reviewed, and a few already withdrawn. The most famous items (Hodge, BSD, Kakeya 4D) are exactly the ones without Lean proofs. The responsible position is neither "AI has solved mathematics" nor "this is hype." It is: wait for the experts, and watch which results survive.

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What to watch next: expert reviews of the Kakeya, Hodge-CM and BSD families; whether the Lean share rises above 42%; how many more withdrawals follow; and whether the IAS advisory group publishes an assessment.

Sources

Where these numbers come from

• OpenAI — openai/math repository README (primary source: 719 manuscripts, 372 families, ~4,000 problems, ~3 h compute, ~42% formalized, exceptions)
• OpenAI — history.md, Oct 7, 2026 (3 withdrawals, 14 revisions, 300/719 formalized)
• 36kr / Synced — "Top 30 problems from the 722 manuscripts" (17 areas, field counts, ~50 disproof families, famous problems)
• MindStudio — What its secret model actually found (monthly pace, verification bottleneck)
• Startup Fortune — the "90 of 500" framing (advisory group, Maggi's criticism)
• Startup Fortune — mathematicians not impressed (timeline, Fields Medalist declaration, attribution dispute)

Figures accurate as of October 9, 2026. This repository is actively being revised, so numbers may change.

Published October 2026 · Figures current as of October 9, 2026