OpenAI 722 AI-generated math results published Oct 6 2026: Quasi-Riemann + Hodge CM + Hilbert 10

On October 6, 2026, OpenAI published 722 AI-generated math manuscripts generated by an unreleased internal frontier model to a public GitHub repository (github.com/openai/math) under Apache 2.0. The release is the largest single AI-math release in history — covering 17 mathematical disciplines, grouped into 372 result families, with ~4,000 problems attempted and many proofs formalized in Lean (interactive theorem prover).

The release includes progress on 3 of 7 Millennium Prize Problems (the most prestigious unsolved math problems with $1M prizes each):

Quasi-Riemann Hypothesis — zero-free region Re(s) > 11/12
Hodge Conjecture for CM abelian varieties (special case)
Hilbert's 10th Problem over ℚ (rational case) — no general algorithm for rational solutions

The release is contested. The Advisory Group on Mathematics and AI (AGMAI) at the Institute for Advanced Study (Fields Medalists Timothy Gowers, Martin Hairer + Edward Witten) issued a Sept 29 statement "not endorsing" the practice of testing advanced math on proprietary models. Only 162 of the 722 papers have Lean formalization (~22%); OpenAI was forced to pull 3 papers for elementary mistakes. For the full mathematician response, see our mathematicians reaction article.

The release at a glance

Detail	Value
Date released	October 6, 2026
Total manuscripts	722
Result families	372
Mathematical fields	17 (theoretical CS, combinatorics, algebra/complex geometry, number theory, etc.)
Problems attempted	~4,000
Avg. compute per result	~3 hours of ChatGPT Pro "thinking"
License	Apache 2.0 (permissive)
Repo	github.com/openai/math
Lean formalization	~60% of families have formalization documentation
Underlying model	Unreleased internal frontier model, "significantly more capable than GPT-6 Astra" (per OpenAI executive)

The 17 mathematical fields

Field	Number of families	Notable results
Theoretical Computer Science	40	Uniqueness game conjecture, free group factor isomorphism
Combinatorics	37	Discrepancy bounds, additive combinatorics
Algebra + Complex Geometry	36	Hodge for CM abelian varieties, Kaplansky direct-finiteness
Number Theory	31	Quasi-Riemann (zeta Re(s) > 11/12), Hilbert 10 (ℚ)
Mathematical Physics	28	Yang-Mills mass gap, lattice gauge theory bounds
Probability	25	Mézard-Parisi formula, spin glass bounds
Algebraic Geometry	22	Derived categories, K3 surfaces
Differential Geometry	21	Perelman entropy bounds, Ricci flow singularities
Representation Theory	18	Langlands program partial results
Dynamical Systems	17	Smale's 18th problem progress
Topology	16	Poincaré conjecture generalizations
Set Theory	14	Continuum hypothesis partial results
Category Theory	13	Higher category theory
Symplectic Geometry	12	Arnold conjecture variants
Operator Algebras	10	Free group factor work
Graph Theory	8	Strong perfect graph theorem extensions
Knot Theory	4	Khovanov homology
TOTAL	372 families / 722 manuscripts

The 3 Millennium Prize progress

1. Quasi-Riemann Hypothesis (zero-free region Re(s) > 11/12)

Problem: The Riemann Hypothesis (1859) conjectures that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2. No one has proven this in 167 years.

What OpenAI claimed: A zero-free region Re(s) > 11/12 (and a second independent proof for Re(s) > 7/8, which also excludes Siegel zeros). This is not a proof of the full Riemann Hypothesis — it's a stronger-than-previous partial result in the direction of RH.

Why it matters: Previous best zero-free regions were:

1896 Hadamard + de la Vallée Poussin: Re(s) > 1
1958 Linnik: Re(s) > 1 − c/log(t)
2020s (Polymath 8b): Re(s) > 1 − 0.1571/log(t)
2026 OpenAI: Re(s) > 11/12 = 0.917 (a substantial improvement, if verified)

Caveat: Only 162 of 722 papers (~22%) have Lean formalization. OpenAI was forced to pull 3 papers for elementary mistakes shortly after posting. For full context, see our mathematicians reaction article.

2. Hodge Conjecture for CM abelian varieties

Problem: The Hodge Conjecture (1950) is one of 7 Millennium Prize Problems. It asks whether every "Hodge class" on a complex projective variety is a rational linear combination of algebraic cycles.

What OpenAI claimed: A proof of the Hodge Conjecture for CM (complex multiplication) abelian varieties — a special case. CM abelian varieties are a small subset of all abelian varieties (those with extra endomorphism structure).

Why it matters: Proving the Hodge Conjecture for CM abelian varieties is not the full conjecture, but it's a non-trivial special case that has been open for decades.

3. Hilbert's 10th Problem over ℚ

Problem: Hilbert's 10th Problem (1900) asks for a general algorithm to determine whether a Diophantine equation (polynomial = 0) has an integer solution. Yuri Matiyasevich proved in 1970 that no such algorithm exists for integers (ℤ).

What OpenAI claimed: A proof of the rational number (ℚ) version — no general algorithm exists to determine whether an integer-coefficient polynomial has rational solutions.

Why it matters: The ℚ case is harder than the ℤ case (proven in 1970). It's been a major open problem for 50+ years. If verified, this is a clean result.

Other notable results

Result	Field	Status
Mahler conjecture	Geometry of numbers	Open since 1930s — partial progress
Kaplansky's direct-finiteness conjecture	Ring theory	Open since 1950s — progress for group rings
Mézard-Parisi formula	Statistical physics	Open since 1985 — rigorously proven for new cases
Uniqueness game conjecture	Theoretical CS	New result
Free group factor isomorphism	Operator algebras	Significant progress
Yang-Mills mass gap	Mathematical physics	Millennium Prize progress
Perelman entropy bounds	Differential geometry	Refined bounds on Ricci flow singularities
Lattice gauge theory bounds	Mathematical physics	Computed numerically, partially proven
Smale's 18th problem	Dynamical systems	Partial progress
Khovanov homology extensions	Knot theory	New invariants constructed

How the AI produced these results

Methodology (per OpenAI):

Problem selection: The model was posed ~4,000 problems from a curated list of open problems across 17 fields
Reasoning loop: For each problem, the model ran an extended "thinking" session (avg. ~3 hours of ChatGPT Pro-equivalent compute)
Lean formalization: ~60% of families have formalized Lean proofs — the proofs are checked by the Lean theorem prover (an interactive proof assistant)
Folder dating: All folder timestamps cluster Sept 23 - Oct 5, 2026 — indicating the model was running continuously over 2 weeks
Licensing: All 722 manuscripts released under Apache 2.0 — fully open source, no commercial restrictions

The model: An unreleased internal frontier model, "significantly more capable than GPT-6 Astra" per an OpenAI executive research post. Likely a successor to GPT-6 (which launched Oct 8, 2026 — 2 days after this release).

The AGMAI controversy

The Advisory Group on Mathematics and AI (AGMAI) at the Institute for Advanced Study (Princeton) issued a statement on September 29, 2026 (8 days before OpenAI's release) urging AI labs to stop testing advanced math on proprietary models.

AGMAI members (selected)

Name	Field	Affiliation
Timothy Gowers	Combinatorics (Fields Medal 1998)	Cambridge
Martin Hairer	Stochastic analysis (Fields Medal 2014)	Imperial College
Edward Witten	Mathematical physics	IAS Princeton
Terry Tao	Analysis (Fields Medal 2006)	UCLA
Others	—	IAS

AGMAI's recommendations

Stop testing advanced math on proprietary models — use open-weights or open-source
Deposit results in independent scholarly repositories (e.g., arXiv, journal preprint servers)
Disclose model names, prompts, and computational costs for any AI-generated result
Respect ongoing human research — don't "scoop" mathematicians working on the same problem

OpenAI's response

OpenAI did not fully comply:

✅ Disclosed the underlying model is "internal" (not named)
✅ Released under Apache 2.0
✅ Included Lean formalizations for many results
❌ Did not specify the model name
❌ Did not use independent repository (used GitHub instead of arXiv)
❌ Released 722 results in a batch (vs. allowing mathematicians time to verify each)

AGMAI follow-up (Oct 8)

After the release, the AGMAI issued a follow-up stating:

"We acknowledge OpenAI's transparency in releasing the proofs publicly and formalizing in Lean"
"However, we do not endorse the practice of testing advanced mathematics on proprietary models"
"We call on the mathematical community to independently verify these results before acceptance"

Reactions from mathematicians

Supportive

Alex Kontorovich (Rutgers): "If a human did this, it would be an instant Fields Medal."
Levent Alpöge (Anthropic): "Possibly the most significant moment in math history."
Anil Ananthaswamy (New Scientist): "This is a watershed moment for AI-assisted mathematics."

Critical

Kevin Buzzard (Imperial College): "Of ~30 number theory papers in the release, only 7 looked high quality and only 1 had Lean verification. The rest are unformalized and may have issues."
Terence Tao (UCLA, Fields Medal): "We are transitioning from 'proof scarcity' to 'proof abundance.' This is a crisis in the foundations of mathematical values."
Caucher Birkar (Cambridge, Fields Medal 2018): "Mathematicians need to be careful about being 'scooped' by AI on in-progress work. The community needs new norms."

Mixed

June Huh (Princeton, Fields Medal 2022): "The Lean formalizations are valuable. The unformalized results are less valuable but still interesting. The community will sort this out over the next 5 years."
Peter Scholze (Bonn, Fields Medal 2018): "I welcome the openness of releasing these publicly. I am skeptical of the unverified claims. Time will tell."

Implications for AI and mathematics

1. The "scarcity-to-abundance" transition

Terence Tao's framing is now mainstream: mathematics is moving from a world where proofs are scarce and valuable to a world where proofs are abundant and need to be filtered. New mathematical community norms will emerge:

Verifiers (mathematicians + Lean) become as important as provers
Pre-registration of conjectures on open repositories to prevent AI "scoops"
Combinatorial proof auctions — paying for the most interesting proofs

2. The role of Lean formalization

~60% of the OpenAI release has Lean proofs, but only 1/30 number theory papers was fully verified. This suggests:

AI can produce structured mathematical arguments
AI struggles with fully rigorous formal proofs in the hardest fields
Lean-assisted human verification is the new bottleneck

3. Implications for AI labs

This is a race between:

OpenAI (proprietary model + Apache-2.0 release)
Anthropic (Claude with extended thinking + math focus)
DeepMind (AlphaProof + FunSearch)
Google DeepMind (Gemini Deep Think)

Each lab will continue pushing frontier math results. The AGMAI's "stop testing on proprietary models" request is unlikely to be followed by commercial labs.

4. The 2026 AI math timeline

Date	Event	Source
May 2026	OpenAI disproves 1946 Erdős unit distances conjecture	OpenAI
July 2026	GPT-5.6 Sol Ultra proves Cycle Double Cover Conjecture	OpenAI
Aug 2026	Ten advances in mathematics and TCS published	OpenAI
Sept 8, 2026	~10,000 AI agents claim to resolve Navier-Stokes in 88 hours	OpenAI
Sept 29, 2026	AGMAI statement opposing proprietary-model math	IAS
Oct 6, 2026	722 math manuscripts released	OpenAI
Oct 8, 2026	GPT-6 launched (10M context, Codex, etc.)	OpenAI
Oct 9, 2026	Nobel Peace Prize (no math)	—
Oct 12, 2026	Nobel Economics (parallels to AI math impact)	—

FAQ: OpenAI 722 Math Results

What did OpenAI release on Oct 6, 2026?

722 AI-generated math manuscripts generated by an unreleased internal frontier model. Published on GitHub at github.com/openai/math under Apache 2.0.

How many results families are there?

372 result families — each family may contain a main result, companion arguments, consequences, or alternative proofs.

How many problems did the model attempt?

~4,000 problems across 17 mathematical fields.

How long did each result take?

~3 hours of ChatGPT Pro "thinking" on average per result.

Did the AI prove the Riemann Hypothesis?

No — but it proved a stronger-than-previous partial result: a zero-free region Re(s) > 11/12 for the Riemann zeta function (and a second independent proof for Re(s) > 7/8 excluding Siegel zeros). The full Riemann Hypothesis is still open.

Did the AI solve any Millennium Prize Problem?

Partial progress on 3 of 7:

Quasi-Riemann (zeta zero-free region)
Hodge Conjecture for CM abelian varieties (special case)
Hilbert's 10th Problem over ℚ (rational case)

The Clay Mathematics Institute still lists all 7 Millennium Prize Problems as unsolved pending independent verification.

What model generated the results?

An unreleased internal frontier model, "significantly more capable than GPT-6 Astra" per an OpenAI executive research post. Possibly related to GPT-6 (launched Oct 8, 2026, 2 days after this release).

Are the proofs verified?

Partially — ~60% of families have Lean formalization documentation (interactive theorem prover). Kevin Buzzard (Imperial College) noted that only 1 of ~30 number theory papers had full Lean verification.

Where can I find the repository?

github.com/openai/math — Apache 2.0 licensed, fully open source.

Why did AGMAI oppose this release?

The Advisory Group on Mathematics and AI (AGMAI) at IAS Princeton (Gowers, Hairer, Witten, Tao) issued a Sept 29 statement urging AI labs to stop testing advanced math on proprietary models. They argue:

Mathematicians need time to verify each result independently
Pre-registration of conjectures prevents AI "scooping"
Computational disclosure should be required

OpenAI did not fully comply with these recommendations.

How long did the AI run for this release?

All folder timestamps cluster Sept 23 - Oct 5, 2026 — about 2 weeks of continuous model inference.

What does this mean for mathematics as a discipline?

Three shifts:

Proofs become abundant — new community norms needed for filtering/verifying
Lean formalization becomes the bottleneck — mathematicians + AI form a new verification pipeline
The "human" in mathematics changes — from prover to verifier + theorem-formulator

Bottom line: On Oct 6, 2026, OpenAI released 722 AI-generated math manuscripts — the largest AI-math release in history. Includes partial progress on 3 of 7 Millennium Prize Problems (Quasi-Riemann, Hodge for CM abelian varieties, Hilbert 10 over ℚ). Apache 2.0 licensed on GitHub. AGMAI opposed but OpenAI proceeded. ~60% of families have Lean formalization. Full verification is the next 6-12 month task for the math community.

For details on the headline results, see our Quasi-Riemann Hypothesis article, Hodge CM article, and Hilbert 10th Problem ℚ article.