AI Math in 2026: What Is Solved, What Is Not, and Why

The first-principles guide to what AI has actually proved, disproved, formalized and failed at in mathematics, with the evidence status of every claim.

On 20 May 2026 an unreleased OpenAI model disproved the Erdős unit distance conjecture in one shot, and nine mathematicians including Tim Gowers wrote that they would have recommended the paper for the Annals of Mathematics "without any hesitation." Eleven weeks later, an Anthropic model formalized the entire proof of Fermat's Last Theorem in Lean in eleven days. Four days after that, two competing groups claimed finite-time blowup for the Navier-Stokes equations, and Quanta ran the headline "AI has solved one of math's $1 million Millennium Prize problems" - Quanta Magazine.

But here is the problem: almost nobody reading those headlines can tell which of them are true, which are true with an asterisk that changes everything, and which are lab marketing. The same eighteen months that produced the unit distance disproof also produced the October 2025 episode in which OpenAI executives announced that GPT-5 had "solved" ten open Erdős problems, and the maintainer of the Erdős database replied within a day that the model had found existing papers, not new mathematics. On Epoch AI's new benchmark of 68 hard open Erdős problems, the best model in the world solved two. Riemann, P versus NP, Goldbach and the twin prime conjecture have not moved a millimeter.

This guide explains what the words mean (problem, theorem, proof, formal, solve, from their Greek and Latin roots, because the roots tell you what is being claimed), how machines came to do mathematics from Leibniz to Lean to reinforcement learning, the exact ledger of what has been solved with its verification status, what remains untouched and why, and how to read the next headline without being fooled. It draws on primary sources: the arXiv papers, the Lean repositories, Terence Tao's digestion posts, Kevin Buzzard's and Tim Gowers's blogs, the Epoch AI methodology pages, and the public statements of the people who checked the proofs.

Contents

1. The Answer First: What Is Solved, What Is Not, as of September 2026
2. The Words: What "Problem", "Proof", "Formal" and "Solve" Actually Mean
3. How Machines Learned Mathematics: Leibniz to Lean to Reasoning Models
4. The Watershed: The Erdős Unit Distance Disproof
5. Counterexamples Everywhere: Jacobian, HRT, Grothendieck and Why Constructions Fall First
6. OpenAI's Ten Results, GPT-6 Astra and the Cost of a Theorem
7. Navier-Stokes: What Was Claimed, What Was Proved, and the Forcing-Term Loophole
8. Formalization at Industrial Scale: Fermat's Last Theorem in Eleven Days
9. The Benchmarks: FrontierMath, FrontierMath Erdős, First Proof and the IMO
10. The Erdős Ladder: What the Numbers Actually Say
11. What Remains Unsolved and Why the Millennium Problems Are Different
12. How to Read an "AI Solved X" Claim
13. What the Mathematicians Say and Why They Disagree
14. What This Means for Anyone Building With AI
15. What to Watch Next
16. Conclusion: A Decision Framework for the Next Headline

The Systems, Scored

Ten organizations produced the results discussed in this guide. The table scores each on four criteria that matter to a reader trying to decide how much weight a claim deserves. Verified research results (35%) measures new mathematics that independent mathematicians or a proof checker confirmed. Formal verification (25%) measures whether the system produces machine-checkable Lean output. Independent benchmarks (25%) measures standing on evaluations the lab did not grade itself. Transparency (15%) measures whether the model, prompts and process are public enough for anyone to check.

The weights reflect what a careful reader is really asking. New verified mathematics is the point of the whole exercise, so it carries the most weight. Formal verification is the only mechanism that removes human trust from the loop, so it comes second. Independent benchmarks matter because self-graded competition scores have been the single largest source of inflation. Transparency is weighted lowest not because it is unimportant but because the labs with the strongest results are also the least transparent, and the table should show that tension rather than hide it. OpenAI ranks first on results and last on transparency, and both facts are true at once.

1. The Answer First: What Is Solved, What Is Not, as of September 2026

The cleanest way to describe the state of AI mathematics is that the machines have crossed from competition mathematics into research mathematics, but only into the part of research mathematics where a solution is an object you can exhibit and check. Counterexamples, constructions and bounds are falling. Universal statements that need a new theory are not. Every result in this guide fits that pattern, and the pattern is the most useful thing to carry away.

The table below is the ledger. The status column is the important one. "Lean-checked with human inspection" means a proof assistant verified the proof and a named mathematician confirmed the formal statement matches the intended theorem. "Human-digested" means a strong mathematician rewrote and confirmed the argument. "Lab claim" means the organization published it and nobody independent has confirmed it yet. "Contested" means qualified people disagree about what it shows.

What is not on that table matters as much as what is. No Millennium Prize problem in its central form has been solved. The Navier-Stokes claim concerns a version with a forcing term that many experts consider a loophole, and Section 7 walks through exactly why. Goldbach, the twin prime conjecture, Collatz, the Riemann Hypothesis, P versus NP, the Hodge conjecture, Birch and Swinnerton-Dyer, and the Yang-Mills mass gap have no credible AI claim against them. The reason is structural, not a matter of more compute, and Section 11 builds it from first principles.

Why this matters: every headline you will read in the next year sits somewhere on that table, and the status column is what determines whether it changes anything. How to apply it: before reacting to a claim, find its row or its nearest analogue, and ask which status it has earned. That single habit filters out most of the noise.

2. The Words: What "Problem", "Proof", "Formal" and "Solve" Actually Mean

Almost every word in a headline like "AI solves open math problem" is two to three thousand years old, and each one encodes a specific idea about what mathematics is. This is not decoration. The etymology tells you what kind of thing is being claimed, and the places where the words have drifted from their roots are exactly where hype lives. Understanding five roots is enough to read the whole field correctly.

Mathematics comes from Greek máthēma, "that which is learned," from manthánein, "to learn." The Pythagorean school split into the akousmatikoi, the hearers who followed oral precepts, and the mathēmatikoi, the learners who pursued the mathēmata: arithmetic, geometry, music and astronomy. The name of the discipline is literally the part of knowledge you can reconstruct yourself from stated premises, rather than take on authority. A machine that follows rules is doing exactly what the mathēmatikoi valued, and every argument about whether an AI "understands" its proof is a very old argument about whether reconstruction is understanding.

Problem is Greek próblēma, "a thing thrown forward," from pro and bállein, "to throw." Its first senses were physical: a headland, a bulwark, a shield held out in front. In Euclid's Elements, as Proclus explained in his commentary, propositions come in two kinds. A problem asks you to construct something, and ends "which was to be done." A theorem asks you to demonstrate a property, and ends "which was to be shown." This ancient distinction is the single most predictive fact about AI mathematics in 2026: the results that fall are overwhelmingly problems in Euclid's sense, where the output is an object you can exhibit and check, and Tim Gowers observed exactly this in August, noting that the famous AI solutions "have almost all been with counterexamples rather than proofs" - Gowers's Weblog.

Theorem is theṓrēma, "a thing looked at," from the same root as theatre and spectator. A theorem is a truth you are invited to see. Proof is Latin probāre, "to test," from probus, "upright"; the phrase "the exception proves the rule" preserves the original sense, since Cicero meant the exception proves the existence of the rule. A proof is a test that a claim passes, which is precisely what a formal verifier implements. But probāre also meant "to make good," to vouch, and the gap between the two senses is the gap every careful caveat in 2026 is about: a machine can run the test while nobody has vouched that the statement tested is the statement intended.

Formal is Latin formālis, from forma, "shape." The scholastic sense, form as opposed to matter, is what David Hilbert exploited in the 1920s when he proposed that a formal system is one whose proofs can be checked by the shape of the symbols alone, without reference to meaning. Lean, Coq, Isabelle and HOL Light are formal systems in Hilbert's exact sense. That is why they can check a proof and cannot check whether the proof is of the right theorem. Kevin Buzzard put the consequence plainly: "If your code compiles, the mathematics is correct... if you miss or garble an axiom then your code will still compile, it just won't mean what it is supposed to mean" - Xena Project.

Solve is Latin solvere, "to loosen, untie, release." The mathematical sense, working out the answer to a problem, is only attested from 1737. To solve is to untie a knot, and that requires the knot to already exist. Terence Tao's argument in September 2026 that good open problems are being "mined in a non-renewable fashion" is the etymology made literal: the knots were tied by people over centuries, untying them is now cheap, and tying good new ones is not - Tao's AI views page.

Three more words complete the picture. Conjecture is Latin coniectūra, "a casting together of facts," and its first English sense around 1400 was the interpretation of dreams and omens; only in the 1520s did it come to mean an unverified supposition. A conjecture is a throw, and the Erdős problems are throws by one man, some casual, some deep, ranging over what Tao called "several orders of magnitude" of difficulty. The name "Erdős problem" sounds uniform and is not, which is how the October 2025 episode happened. Benchmark is a surveyor's term from 1838: a horizontal mark chiselled into stone so a levelling staff could be placed at exactly the same height again. A benchmark is repeatable reference against something that does not move, and contamination, the benchmark leaking into training data, means the stone moved. Algorithm is a corruption of the name al-Khwārizmī, the ninth-century Baghdad scholar, and it meant a fixed procedure for calculation. Hilbert asked whether there is one algorithm for all of mathematics; Church and Turing proved in 1936 that there is not. That forbids a procedure that always decides. It does not forbid a procedure that finds many proofs, and the 2026 results live in the vast space that theorem leaves open.

Why this matters: the words draw a map. Throwing (conjecture) and seeing (understanding) remain human. Testing has been mechanized. Taking, finding the lemmas that untie a knot, is what crossed into machine territory in 2025 and 2026. How to apply it: when you read "AI solved X," ask which verb is really being claimed. Usually it is "took" and "tested," and the headline says "saw."

3. How Machines Learned Mathematics: Leibniz to Lean to Reasoning Models

The history is best read as five distinct capabilities, each mechanized in turn: computing, searching, checking a proof, finding a proof, and proposing a conjecture or a definition. Every milestone advanced one of them, and most confusion in 2026 comes from not saying which one a result advanced. A formalization of Fermat's Last Theorem is a triumph of capability three. A disproof of the unit distance conjecture is capability four. Nobody credibly claims capability five, and that is where the Millennium problems live.

Leibniz dreamed the whole thing in the 1660s: a characteristica universalis, a symbolic language for all thought, and a calculus ratiocinator, so that disputes could end with "Calculemus," let us calculate. Boole in 1847 and Frege in 1879 made logic itself calculable. Hilbert in 1928 posed the Entscheidungsproblem, asking for a procedure that decides every statement of first-order logic. Gödel in 1931 and then Church and Turing in 1936 answered no, and Turing did it by defining the machine that now bears his name. The same year that killed the universal algorithm founded computation, and every later system lives inside that boundary.

The first machine proof search was the Logic Theorist of Newell, Simon and Shaw in 1956, which proved 38 of the first 52 theorems of Principia Mathematica and found a shorter proof of one of them; the Journal of Symbolic Logic refused a paper with the program as co-author. The Four Color Theorem in 1976 relied on over a thousand hours of computer time, and Thomas Tymoczko's 1979 paper asked whether an unsurveyable proof is a proof at all. That question is back in 2026 at a thousand times the scale. In 1996 McCune's EQP prover settled the Robbins conjecture, the first genuinely new theorem found rather than merely checked by a machine. Thomas Hales's Kepler proof left Annals referees "99 percent certain" after four years, so he spent eleven more years on Flyspeck, completed in 2014, and the lesson stuck: when a proof outruns human refereeing, formalization is the only exit.

The proof assistants came next: Automath in 1967, the first practical use of the Curry-Howard correspondence (a proof of a proposition is the same kind of object as a program of a type), then Mizar, Isabelle, HOL, Coq, and Lean, released in 2013 with Lean 4 in 2021. Lean won mathematicians over for reasons that were sociological as much as technical: its library mathlib is one coherent, community-maintained body of canonical definitions. Tao said at the ICM in July that "a big reason why AI is so successful in mathematics is because, for centuries, we've been building these canonical definitions" - Simons Foundation.

Then the recipe that changed everything. Train a language model with reinforcement learning against a verifier, so that it is rewarded only when the final answer or the formal proof checks, and let it spend variable compute at inference time: long chains of thought, parallel samples, self-repair. Mathematics was the natural first domain because the verifier is cheap and exact. Dario Amodei described the empirical law behind it in February: "We train the model on math contests... and how well the model does is log-linear in how long we've trained it" - Dwarkesh Podcast. Three generations followed in two years. The o-series of 2024, which Tao compared to advising "a mediocre, but not completely incompetent, graduate student." The IMO gold systems of July 2025. And the 2026 models behind the results in this guide, which we compared for agent work in our GPT-6 Astra versus Claude Fable 5.1 guide.

Why this matters: the ladder tells you what a result is. Anthropic's FLT is rung three at industrial scale. The unit distance disproof is rung four. Tao's "digestion" posts are a human doing rungs five and six by hand on machine output. How to apply it: place any new claim on the ladder before deciding how impressed to be. A rung-three result described as rung five is the most common form of inflation.

4. The Watershed: The Erdős Unit Distance Disproof

If one result marks the crossing from competition mathematics into research mathematics, it is this one. In 1946 Paul Erdős asked how many pairs among n points in the plane can be at distance exactly one, and conjectured the answer was at most n to the power 1 plus a vanishing term. The problem was one of his favorites, it was catalogued as Erdős #90, and mathematicians attacked it for eighty years without moving the exponent. On 20 May 2026 an internal OpenAI model produced, in one shot, a construction with n^(1+c) unit distances for a positive constant c, which is a disproof.

The construction is not elementary. It uses an infinite unramified tower of number fields with bounded root discriminant, a structure from the Golod-Shafarevich theory of the 1960s, together with integral points on the norm-one torus. Nine mathematicians, Noga Alon, Thomas Bloom, Tim Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood, then wrote "Remarks on the disproof of the unit distance conjecture," a short human-verified version that credits the ideas "at least in retrospect" to Ellenberg-Venkatesh, Golod-Shafarevich and Hajir-Maire-Ramakrishna - arXiv 2605.20695. Gowers's verdict was the one that traveled: "if a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation" - Scientific American.

The reactions from people qualified to judge were unusually uniform. Gil Kalai compared it to the 1976 Four Color Theorem and called it "truly amazing" - Kalai's blog. Daniel Litt called it "the unique interesting result produced autonomously by AI so far." Jacob Tsimerman, who would win a Fields Medal two months later, described the edge in plain terms: "It's not just that they can try all known methods... They can play for longer" - Quanta Magazine. Sébastien Bubeck of OpenAI said the model "just executed like an amazing mathematician." The Lean formalization followed on 26 June, produced by Logical Intelligence using OpenAI's model in about 1.2 million lines over three weeks - Xena Project.

The caveats are as precise as the praise, and a careful reader should hold both. The construction's advantage over the old bound only appears at around 10 to the power 2,000,000 points, so it says nothing about any drawing a human could make. The ideas were in the literature, and Gowers noted in The Conversation that "many of the ideas needed for the proof were present in the literature already" - The Conversation. The model failed to credit them. And the result is a counterexample: it exhibits an object, and the object can be checked. That is a Euclidean problem in the sense of Section 2, not a theorem. Will Sawin later pushed the exponent to about 1.03, and Kalai reports the method's limit near 1.21, so humans immediately improved on the machine's output once the idea was on the table.

Why this matters: this is the result that moved the expert consensus, because the people who checked it were the people best placed to find a flaw, and they found none. How to apply it: when someone says "AI has not done any real mathematics," this is the counterexample to their claim. When someone says "AI is now a mathematician," the caveats above are the answer.

5. Counterexamples Everywhere: Jacobian, HRT, Grothendieck and Why Constructions Fall First

The unit distance disproof was not alone. Within two months, Kevin Buzzard wrote a post titled "Human mathematicians are being outcounterexampled," listing three AI counterexamples in quick succession, and the phrase captures the structural pattern better than any statistic. The reason constructions fall first is not an accident of which problems the labs chose. It follows from what the machines are good at and what a verifier can check.

The Jacobian conjecture, posed by Keller in 1939, says a polynomial map with constant nonzero Jacobian determinant is invertible. On 19 to 21 July 2026, Levent Alpöge of Anthropic used Claude Fable 5 to find a degree-7 polynomial map in three variables with Jacobian determinant minus 2 that sends three distinct points to the same image, which is a counterexample in dimension three and above - Fortune. Paul Lezeau formalized it in Lean by the next morning, as a pull request to DeepMind's formal-conjectures repository - GitHub. Tao's digestion called it "a massive miracle" of cancellations across roughly 1,329 equations with about 360 degrees of freedom, and judged that "finding such a polynomial looks highly unlikely to be located by brute force" - Tao's digestion. The two-dimensional case remains open, the prompting details are not public, and there is no arXiv preprint as of this writing.

The HRT conjecture of Heil, Ramanathan and Topiwala, on the linear independence of time-frequency shifts, was disproved on 5 August 2026 by Faulhuber, Petersen, van Velthoven and Voigtlaender, who exhibit twelve linearly dependent shifts of a Schwartz function with a certified numerical proof and, in a second version, a purely analytic one - arXiv 2608.05044. Here the AI's role was different: Tao reports that the authors used AI for the initial proof strategy, wrote the final argument themselves, and "disclosed their AI use responsibly" - Tao's partial digestion. The arXiv abstract does not mention AI at all, which is itself a data point about how disclosure norms are still forming.

The third example is a question of Grothendieck's about group schemes. On 11 July 2026, Akhil Mathew found, with ChatGPT Sol and Claude Fable, a group scheme of order 4 that is not killed by 4, formalized in 1,076 lines of Lean. Frank Calegari's response to the whole cluster was the sharpest: riffing on Deligne's remark that all problems in mathematics are psychological, he wrote that "AI is an insane psychofreak with no hangups," and dismissed the "obvious in retrospect" objection because "obvious in retrospect is very far from obvious" - Persiflage.

Why do constructions fall first? Gowers's August essay is the best available diagnosis. Language models win through "wide knowledge and the ability to explore many paths," off-the-shelf constructions and probabilistic methods; they struggle where "the mysterious human ability to prune the proof-discovery search tree" matters. A counterexample is a finite object. Once you have it, the check is mechanical, so a model can propose thousands of candidates and let a verifier reject all but one. A proof of a universal statement over an infinite domain needs an argument whose shape nobody has seen, and there is no verifier for an idea that does not exist yet. Gowers did update his view after OpenAI's August results: "LLMs are not just good at finding counterexamples: they can find proofs of difficult statements as well" - Gowers's Weblog. But the proofs in question assembled existing techniques; the difficulty was in the assembly, not in inventing the tools.

Why this matters: it predicts what falls next. Any open problem whose negative answer is a finite object is now at risk, and any positive statement that needs new machinery is not. How to apply it: if you follow a particular conjecture, ask whether a counterexample would be a finite, checkable object. If yes, expect an AI attempt soon. If no, the wait is for something that does not yet exist.

6. OpenAI's Ten Results, GPT-6 Astra and the Cost of a Theorem

On 1 August 2026 OpenAI announced its next model, Astra, not with a demo but with a 249-page manuscript titled "Ten Advances in Mathematics and Theoretical Computer Science," accompanied by Lean 4 certificates for every claim in a public repository with a sorry count of zero - GitHub. The ten results include the first explicit non-sofic group, answering a question of Gromov from 1999; a disproof of Connes's rigidity conjecture; Ehrhart's volume conjecture with a sharp bound in every dimension; the growth rate R_k(3) = k^(Theta(k)), which resolves Erdős #183; bipartite constructions resolving Erdős #146 and #180; the exact asymptotic strength of the Cohn-Elkies sphere-packing linear program; lower bounds for the permanent; exponential parallel repetition for entangled two-player games; and NP-hardness for the closest vector problem - The Decoder.

The reactions split along a line worth understanding. Thomas Bloom, who maintains the Erdős database, called the results "big news." Gowers called the non-sofic group "one of the most important unsolved problems in group theory" and the Ramsey growth result "a major open problem in Ramsey theory that I didn't necessarily expect to see solved in my lifetime." Scott Aaronson wrote that it was "pretty clearly the last year of math and theoretical computer science research in the style we've known it" - Shtetl-Optimized. On the other side, Henry Yuen criticized the writeups as "characteristic of ChatGPT-generated proofs," Gary Marcus called the release "amazing but vastly oversold" - Marcus on AI, and Zvi Mowshowitz supplied the caveat every reader should keep: "There are Lean proofs. That does not mean that all ten results prove the things they assert that they prove. So far it is looking good" - Don't Worry About the Vase.

Two more facts calibrate the results. Levent Alpöge reported that Claude Fable independently solved five of the ten within 24 hours, which bears directly on how hard the problems were for current systems. And the cost was small: Noam Brown stated that generating the proofs for all ten cost under $2,000 at API rates, and added, "Sadly, no Millennium Prize Problems (yet)" - SiliconANGLE. No peer-reviewed publication of any of the ten exists as of 9 September 2026.

GPT-6 Astra itself shipped on 3 and 4 September. OpenAI reports 97.6 percent on FrontierMath Tier 4 v2, against 87.8 percent for Claude Fable 5.1 and Fable 5, 83.0 percent for GPT-5.6 Sol and 73.2 percent for Claude Opus 5 - Vellum. That number is OpenAI's own; Epoch AI had not run Astra on Tier 4 at launch, and Epoch's independently run ceiling at that date was the 87.8 percent of the two Fable models. We covered Astra's pricing at $10 per million input tokens and $50 per million output in our GPT-6 Astra pricing analysis, and its computer-use results in our OSWorld breakdown.

The gap between the self-reported and independently run numbers is the whole story of benchmark reading in one chart. A Tier 4 score approaching saturation is precisely why Epoch built FrontierMath Erdős, on which the same Astra scored two out of 68. The two numbers are not in contradiction. They measure different things: Tier 4 is hard problems with known answers, and FrontierMath Erdős is open problems with no answer key. Section 9 walks through the difference.

Why this matters: the ten results are the largest single batch of research-level claims any lab has published, and their Lean certificates are real, but Lean certifies the proof, not the choice of statement. How to apply it: treat each of the ten as "very likely correct, awaiting the audit that peer review normally provides," and watch for the first one to be refereed.

7. Navier-Stokes: What Was Claimed, What Was Proved, and the Forcing-Term Loophole

The Navier-Stokes existence and smoothness problem is one of the seven Millennium Prize problems, and the Clay Mathematics Institute's official statement, written by Charles Fefferman, offers four options for what would constitute a solution. Two of them concern the equations as physicists usually state them, with no external force. Two of them allow a smooth forcing term, an external push on the fluid that can be chosen by the prover. That distinction, buried in the fine print for twenty-five years, became the most important sentence in mathematics during the second week of September 2026.

What is solid is this. Building on work of Diego Córdoba and Luis Martínez-Zoroa, Levent Alpöge and Tristan Buckmaster obtained finite-time blowup with a smooth forcing term for the incompressible porous medium equation, the two-dimensional Boussinesq equation and the three-dimensional Euler equation, with a Lean verification completed on 22 August and a public announcement on 7 September. Tao called it "a remarkable achievement" and noted "the arguments here are heavily AI-assisted" - Tao's blog. Jordan Ellenberg added the line that best states the community's ambivalence: "An interesting but incorrect example would surely be of more value than an uninteresting but correct one" - Quomodocumque. Buckmaster's own account is blunt about the raw material: "the first LLM generated proof Levent sent me was the most horrendous I have ever read" - Buckmaster's statement.

What is claimed is larger. OpenAI says that a run of about 10,000 autonomous agents exchanging about 5 million messages over 88 hours produced a Lean-checked finite-time singularity for three-dimensional Navier-Stokes with smooth forcing, at a cost Bubeck put at "several million dollars" - Quanta Magazine. Fefferman, who wrote the Clay statement, told Quanta he "was thrilled that the problem was solved" and that "the heroes of the story are Córdoba and Martínez-Zoroa." Martínez-Zoroa said, "It would have been nice to do this ourselves."

What is contested comes in three parts. First, whether the forcing term is a loophole. Scientific American's report puts it directly: the problem "as many experts imagine it" is the unforced one, and the Clay Institute is "in a bit of a quandary" - Scientific American. Second, priority: Buckmaster and Alpöge announced twelve hours before OpenAI, OpenAI concedes priority on Euler, and Buckmaster's public statement alleges that an OpenAI researcher pressured him over authorship, which OpenAI and Sam Altman deny, with Altman saying the team "acted with integrity and generosity throughout." Third, verification: Quanta notes that the crucial remaining check "must still be done by humans," namely "to guarantee that the statement being shown to be true in Lean is logically equivalent to what mathematicians set out to prove." That is the statement-fidelity gap from Section 2, at Millennium scale.

Tao's assessment of the unforced case is the one to keep. The extension "faces an enormous number of technical difficulties," though "I would not be surprised if one could batter out such an extension by pouring an enormous amount of compute and AI assistance at such a task." His deeper objection is not to the mathematics but to the mode of production, which he described as "an autonomous harness with vast compute running the entire iteration internally while the company keeps the process out of public view" - Tao's AI views page. His September posts warn that a historically productive problem is at risk of becoming "technically solved, with almost no value added to mathematics."

Why this matters: this is the first time a Millennium problem's literal statement has been touched by AI, and the way it was touched (a forcing term, a race, a priority dispute, a proof nobody has read) is a preview of every hard problem's future. How to apply it: when you hear "AI solved Navier-Stokes," the accurate sentence is "two groups, one human-led and one autonomous, produced Lean-checked blowup for a version with an external force that the problem's own author allows but most experts consider a loophole; the unforced case is open." That sentence is long because the truth is.

8. Formalization at Industrial Scale: Fermat's Last Theorem in Eleven Days

On 4 September 2026 Anthropic announced that an internal model roughly comparable to Claude Fable 5.1 had formalized the entire proof of Fermat's Last Theorem in Lean, end to end, in about eleven days: 29,511 theorems in the final dependency tree, about 13 million lines of Lean, roughly 6 billion output tokens, only Lean's three standard axioms, zero sorries, and verification by two external checkers - Anthropic. Humans wrote only the one-line goal statement. The proof follows the Darmon-Diamond-Taylor exposition of the Wiles and Taylor-Wiles argument and proves the needed cases of Mazur's theorem, Langlands-Tunnell, Ribet's level lowering and the R equals T theorem along the way.

The person best placed to judge this is Kevin Buzzard, who has led the human Fermat's Last Theorem formalization project at Imperial College since 2024 on a five-year, one-million-pound EPSRC grant. His post is titled "FLT: Anthropic has beaten me to it." He reports 13.4 million lines, coverage of exponents 17 and above (the small regular primes are handled separately), a compile time about twenty times that of mathlib on a 96-core machine, and, crucially, that he manually inspected every non-definition, non-proof line to rule out soundness exploits and confirmed the theorem statement is the right one - Xena Project. His two conclusions belong together: "mathematically this work tells us essentially nothing," because FLT was proved in 1995, and "in the future we will start to see formalization of modern research being done on the fly."

What this milestone changes is the economics of checking. Formal verification was always the answer to Tymoczko's 1979 question about unsurveyable proofs, but it cost roughly twenty lines of Lean per line of paper, the so-called de Bruijn factor, and years of expert labor. Tao noted in June that autoformalization now "completes most tasks within hours" and has "essentially emptied" the unclaimed formalization queues on at least one project, with the de Bruijn factor dropping and "no fundamental obstacle" to it falling below one - Tao's AI views page. Anthropic itself notes that about 7 percent of the FLT lines came from failed attempts and that the proof is "much longer than it needs to be."

The same capability is spreading. Axiom Math formalized the bounded prime gaps theorem with gap 246, the Polymath8b result that Ken Ono described as "the threshold of human knowledge about prime numbers" - IEEE Spectrum. Mistral released Leanstral 1.5 on 2 July under an Apache-2.0 license, claiming 100 percent on miniF2F and 587 of 672 on PutnamBench - Mistral AI. Buzzard launched the Annals Challenge on 13 August, releasing 50 Lean statements of theorems from recent Annals of Mathematics papers, and in formalizing the statements alone found errors in the human literature, "four of which were in papers written by Fields Medallists" - Xena Project. Tao launched Palomar, a registry of Lean-verified mathematics, on 18 August - Tao's blog.

Why this matters: Lean is the only mechanism that removes human trust from the loop, and it just became cheap. Every result in this guide that carries the label "Lean-checked" is trustworthy in a way that no natural-language AI proof is. How to apply it: the residual human job is now statement fidelity. When a formalization is announced, the question is not "did it compile" but "who read the statement," and Buzzard reading every line of the FLT statement is the model of what that looks like.

9. The Benchmarks: FrontierMath, FrontierMath Erdős, First Proof and the IMO

Benchmarks are the instruments through which most people experience AI progress, and in mathematics they have been systematically misread. There are three levels of grading, and only one of them is evidence in the mathematical sense. Self-reported means a lab graded its own model. Independent means a third party graded it. Official means the body that owns the competition graded it, or a proof checker plus a human who read the statement did. The gap between the first and the third has been the largest source of inflation in the field.

FrontierMath was built by Epoch AI with problems contributed by mathematicians including Tao, Gowers, Borcherds and Green. Tiers 1 to 3 run from advanced undergraduate to early-researcher difficulty, and a separate Tier 4 holds research-level problems. OpenAI funded the benchmark and had access to much of it, a fact disclosed only in January 2025, which is why a holdout set exists. In June 2026 Epoch released version 2 after finding errors in 42 percent of problems: 123 Tier 1 to 3 problems and 12 Tier 4 problems were corrected, others removed, leaving 295 problems in Tiers 1 to 3 and 43 in Tier 4 - Epoch AI. Greg Burnham of Epoch had warned in 2025 that the benchmark rewards "background knowledge" over creativity, and Elliot Glazer said Tier 4 was designed "not to get saturated until AI has genuinely mastered the main ideas of most of the major fields of mathematics" - Lemmata. With Astra reporting 97.6 percent, that design goal has been reached or breached, depending on whether you trust the self-report.

FrontierMath Erdős, announced on 1 September 2026, is the corrective. Thomas Bloom curated 68 significant open Erdős problems, about 10 percent of the 652 unsolved problems in his catalog, all formalized in Lean 4 with Mathlib; 50 already existed in DeepMind's formal-conjectures repository and 18 were formalized with AI. The run gave each model $300 and 72 hours per problem, one attempt each, with bash, Lean 4, SageMath, Python and 476,000 arXiv papers available. GPT-6 Astra solved two, a disproof of Erdős #74 and a proof of #126. GPT-5.6 Sol, GPT-5.5, Claude Fable 5.1 and Claude Fable 5 each solved zero - Epoch AI. Bloom estimated that only three to five problems of this caliber had been solved by AI as of August.

First Proof is the most honest instrument in the field because it cannot be contaminated. Ten research mathematicians, including Mohammed Abouzaid, Martin Hairer, Lauren Williams and Daniel Litt, took unpublished problems from their own work in progress and graded the AI solutions double-blind to the standard of "accept with minor revisions." In the first batch, released 5 February 2026, only two of ten problems were solved correctly, one was contaminated, and most submissions were "very convincing nonsense" - Scientific American. In the second batch, run 28 May to 1 June and graded by about 30 referees at Harvard's CMSA, seven of ten problems received at least one passing grade across four systems, the best harness being an ETH Zurich and Aarhus ensemble; the recurring complaints were glossed-over hard steps, citations to papers that did not contain the claimed results, and "line-by-line reuse of an author's earlier paper without citation" - arXiv 2606.18119. Scientific American's headline grade was "C minus" - Scientific American.

The IMO is finished as a discriminating test, and the way it finished is instructive. In 2024 DeepMind's AlphaProof reached silver level at 28 of 42. In July 2025 Gemini Deep Think scored 35 of 42 with official grading by IMO coordinators, while OpenAI announced the same score from an experimental model it graded itself and released before the IMO's requested waiting period; Buzzard wrote that the self-graded claims were "completely unable to be independently verified... one wonders if one is even allowed to call it science" - Xena Project. At IMO 2026 in Shanghai, an independent post-hoc evaluation using Claude-based graders found four systems at 42 of 42: Claude Fable 5 on the first pass, GPT-5.6 Sol and Kimi K3 after repair rounds, and AxiomProver with machine-checked Lean proofs; the evaluator's own repository says to "treat scores as strong but not authoritative" - 36kr. Two Chinese systems, Huawei's Celia and RedNote's dots-note-3.0, were reported as scoring 42 of 42 under official grading, but the IMO's own site records nothing about AI participation, so that remains contested - SCMP. We rank Kimi K3 among open-weight models in our Kimi K3 benchmarks guide.

Why this matters: the four instruments measure four different things, and the honest summary is that competition mathematics is saturated, hard problems with known answers are nearly saturated, unpublished research problems are at "C minus and rising," and serious open problems are at two out of 68. How to apply it: never let a score from one instrument stand in for another. Our 50-benchmark ledger and our analysis of why coding benchmarks lie apply the same logic outside mathematics.

10. The Erdős Ladder: What the Numbers Actually Say

Paul Erdős, born in Budapest in 1913, wrote more than 1,500 papers, lived out of a suitcase, and attached cash prizes from $25 to $10,000 to problems he cared about. Thomas Bloom's erdosproblems.com catalogs them, and by August 2026 the database held 1,217 problems, with 565 solved and 652 open - Quanta Magazine. The problems became AI's proving ground for structural reasons: the statements are short and self-contained, the database is public, many problems were never seriously attacked, and Bloom is one person who can be asked whether a problem was really open. Jared Duker Lichtman's summary in Quanta is the whole story in a sentence: "labs realized that this could effectively be a benchmark."

The October 2025 episode is the template for how "solved" gets inflated, and it deserves to be told exactly. On 17 October, Mark Sellke of OpenAI posted that "using thousands of GPT5 queries, we found solutions to 10 Erdős problems that were listed as open"; Kevin Weil amplified it as "GPT-5 just found solutions to 10 (!) previously unsolved Erdős problems"; Bubeck wrote that "science acceleration via AI has officially begun." Bloom replied within a day that this was "a dramatic misrepresentation": "GPT-5 found references, which solved these problems, that I personally was unaware of," and added, "Its literature searching ability is already useful and impressive enough, no need to describe it as something it's not!" - Fortune. Demis Hassabis replied "this is embarrassing," and the posts were deleted - The Decoder. Two lessons survive. The claim was true as literature search and false as mathematics, and only the maintainer of the list could tell the difference in a day. And literature search is genuinely valuable: Tao's own regular use of these tools includes turning "weeks to minutes" of literature search.

The genuine solutions followed. Erdős #728 was resolved on 4 January 2026 by GPT-5.2 Pro with Harmonic's Aristotle, Lean-verified, in what the write-up calls "the first Erdős problem... regarded as fully resolved autonomously by an AI system" - arXiv 2601.07421. Tao's response set the tone for the year: the writeup was "within ballpark of an acceptable standard for a research paper," but the win "says more about speed than difficulty," because Erdős problems vary "several orders of magnitude" in difficulty and only one or two percent of the open ones were simple enough for AI with minimal help - The Decoder. Erdős #1196 in May was different in kind: Tao wrote that its Markov-chain approach "reveals a previously undescribed connection between the anatomy of integers and Markov process theory" and "would be a meaningful contribution... that goes well beyond the solution of this particular Erdős problem" - Tao's blog.

The tally shows where the machine sits on the ladder. The community wiki, frozen on 30 June 2026, counted roughly 56 standalone-AI solutions, 25 alongside literature, 37 building on literature, 130 or more human-AI collaborations and 70 or more literature-search finds - Erdős problems wiki. Quanta reports about 100 problems moved to "solved" with AI help since October 2025. Tao's verdict on the whole set, from his consolidated AI views page, is that the true success rate is "only a few points," concentrated at the easy end, with "no evidence the median problem is in reach." The false positives are real too: Kevin Barreto and Liam Price's claimed solution to #333 in December 2025 turned out to have been solved by Erdős himself in 1977, and Barreto told Quanta, "As someone who has fallen for this twice now, it's quite gut-wrenching." Bloom's other complaint is about volume: non-mathematicians now submit "100- to 200-page papers" that "no human has read."

The upper rungs are the point. FrontierMath Erdős, curated to contain only problems a mathematician would call serious, put the best model at two of 68. Noga Alon, whose name is on the unit distance digestion, gave Quanta the darkest reading: "Once AI started to solve them, there is no point anymore." Tao's September reframing is more constructive: he proposes that companies race to be "the first to announce a new mathematical insight" rather than a solution, and warns that current incentives "point toward not sharing promising directions at all, which would reverse centuries of traditions of open science" - Tao's AI views page.

Why this matters: the Erdős database is the only place where AI's research-level hit rate can be measured against a catalog, and the honest number is a few percent, rising, concentrated at the easy end. How to apply it: when a lab announces an Erdős count, ask which rung of the ladder the problems sat on, and whether Bloom or Tao has commented. If neither has, wait.

11. What Remains Unsolved and Why the Millennium Problems Are Different

The list of what AI has not done is short to state and long to explain. The Riemann Hypothesis has no credible AI claim. Neither do P versus NP, the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, or the Yang-Mills mass gap. Navier-Stokes is touched only in the forced version discussed in Section 7. Poincaré was proved by Perelman in 2003 without machines. Goldbach's conjecture, the twin prime conjecture and the Collatz conjecture have not moved. The abc conjecture remains in the limbo it has occupied since Mochizuki's claimed proof. Kevin Buzzard's list of the remaining challenges on Freek Wiedijk's famous list of 100 theorems is now empty, because FLT was the last one, but the open problems are untouched.

The first-principles reason is the difference between recombination and invention. The problems that fall are ones where the answer is a combination of known tools nobody had combined, found by exploring more paths for longer than a human can. Tsimerman's "they can play for longer" is exactly this. Gowers's caveat, that apparently original ideas "might merely reflect training data," is the pessimistic reading of the same fact. Either way, the search space is the existing literature, and the method is search. The Riemann Hypothesis has resisted every known method for 167 years, and the people who work on it believe it needs an object or a theory, something like the Weil cohomology that proved the function-field analogue, that nobody has yet defined. Inventing the definition is rung five of the ladder in Section 3, and no system has demonstrated it.

There is a second reason, which is that there is no verifier. Reinforcement learning against a Lean kernel works because the kernel gives an exact reward: the proof checks or it does not. For a problem that needs a new theory, the intermediate steps are new definitions and new conjectures, and there is no kernel that rewards a good definition. A definition is good if it is fruitful, and fruitfulness is only known years later. Tao's ICM essay describes this as an "impedance mismatch" across the five stages of the mathematical pipeline, generation, verification, exposition, publication and canonicalization, where AI accelerates each stage less than the one before - arXiv 2608.16753. The stage that cannot be accelerated by a verifier is the one the Millennium problems require.

A third reason is subtler and concerns what counts as solved. Ravi Vakil, president of the American Mathematical Society, told the ICM panel: "We don't just prove random stuff. We prove it because it's interesting and it's interesting because it has a story" - Quanta Magazine. Tao's September posts make the same point about Navier-Stokes: a problem can be "technically solved, with almost no value added to mathematics," if the solution is a hundred pages of machine-generated estimates that no human has understood. His proposed publication rule is the "talk test": if the authors "cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, that is correct and properly attributed, then the result should not be published." By that standard, several of the results in this guide are not yet solved even though they are Lean-checked.

A final word on the arXiv flood. Every open problem now attracts machine-generated proof attempts, and the Erdős database "already contains dozens of AI-generated proof submissions," in Tao's words. Joel David Hamkins described "this ocean of slop that is overwhelming our journal systems" in Quanta's April feature - Quanta Magazine. Daniel Litt, in the same piece, warned of "pollution of the commons by AI-generated nonsense" while also saying "it's very likely that this technology is bigger than the computer." Both are true, and the second does not cancel the first.

Why this matters: the boundary between what AI does and does not do in mathematics is not "easy versus hard." It is "assemble versus invent," and that boundary will not move by adding compute to the same recipe. How to apply it: the accurate statement, as of September 2026, is that "AI solved an open problem" reliably means a frontier model, with substantial compute and some human framing, produced a construction or an assembly-of-known-techniques proof for a problem whose statement fits in a paragraph, and a strong human or a proof checker confirmed it within days. It does not mean the model chose the problem, invented a concept, wrote a paper a journal accepted without rewriting, or touched a problem that needs a theory nobody has.

12. How to Read an "AI Solved X" Claim

Every claim in this guide passed through the same seven questions, and the questions are more durable than any particular answer. They are ordered by how often they catch an inflated claim, and each one has a case behind it.

Was the statement actually open, and who says so? The October 2025 Erdős episode failed here. Ten problems listed as open had solutions in the literature, and only Bloom could say so quickly. The same question caught the #333 retraction in December. The person who maintains the list is the only reliable oracle, and if they have not commented, the claim is unconfirmed.

Who graded it, and would they grade a human the same way? OpenAI's IMO 2025 gold was self-graded and released against the IMO's request to wait; DeepMind's was officially graded; both scored 35. The evaluator of IMO 2026 used Claude-based graders and said so. Astra's 97.6 percent on Tier 4 is OpenAI's number. The rule is that a lab's own grade of its own model is a press release, not a measurement.

Is there a formal certificate, and did a human read the statement? Lean checks that a formal statement follows from the axioms. It cannot check that the formal statement is the theorem. The FLT formalization passed this test because Buzzard read every statement line. The Navier-Stokes claim has not yet passed it, by Quanta's own account. A Lean certificate without a named human who confirmed the statement is a compiled program, not a confirmed theorem.

How much human framing preceded the model's contribution, and is the transcript public? Tao's ICM recommendation is that authors disclose actual chat logs. The Jacobian counterexample's prompts are not public. The unit distance construction is described as one shot, and Buckmaster describes his Euler proof as heavily iterated. These are different degrees of autonomy, and the headline usually erases the difference.

Is the improvement marginal, and was the argument in the literature? AlphaEvolve improved 23 of 67 problems, several by tiny margins, and Tao warned that "blindly trusting the AE values can be risky as they may be a consequence of verifier exploits" - Tao's blog. Even the unit distance disproof drew on ideas "present in the literature already." Neither fact makes the result worthless. Both change what it means.

Has anyone given a talk on it? Tao's talk test is the last filter, and it is the one most 2026 results have not yet passed. Sendov's conjecture took Tao "several days (with heavy AI assistance)" to digest, and he reduced the Lean proof from about 90,000 lines to about 15,000 in the process. A result nobody can explain is a result that has not yet entered mathematics.

Why this matters: these seven questions separate the six or seven results in this guide that changed the expert consensus from the dozens that produced a news cycle. How to apply it: keep them in the order given. The first two catch most inflation in a minute; the last two take an expert and weeks. We use the same discipline for agent benchmarks in our guide to AI agent evals.

13. What the Mathematicians Say and Why They Disagree

The public record of what leading mathematicians think has moved faster in 2026 than in any previous year, and the movement is not in one direction. The most useful way to read it is to notice that people are answering different questions. Those looking at competition mathematics and Erdős counts say the problem is solved. Those looking at Riemann say nothing has happened. Those looking at First Proof say "C minus, improving fast." All three are describing the same field accurately.

Terence Tao has moved on capability and hardened on values, and both movements are documented on his own consolidated page. In September 2024 he compared o1 to "a mediocre, but not completely incompetent, graduate student." In February 2026 he wrote that his 2023 forecast of a trustworthy AI co-author by 2026 was met "almost exactly on schedule." By July he said, "I'm not sure anyone is capable of any reliable forecasting beyond a year at best." On values he has become the field's organizer: the talk test, disclosure of chat logs, priority to "first to explain" rather than "first to generate," the Palomar registry, a new video-talk journal called Mathematical Discourse, and, on 9 September 2026, a blog comment that "the most pressing issue is for the entire mathematical community to unite around our core values and objectives, and reject irresponsible and unsustainable usages of AI technology," in which he also distanced himself from OpenAI's use of an interview with him as what he called "that infamous advertisement" - Tao's AI views page.

Tao's ICM 2026 public lecture in Philadelphia, "Mathematics in the age of AI," delivered on 24 July, is the single best hour on the subject, and it is the primary source for the five-stage pipeline argument in Section 11. The Simons Foundation recorded it.

The lecture's central line, quoted by the Simons Foundation, is the one that separates a checked proof from a mathematical result: "We can have these 100,000-line proofs that we have to verify, but no one understands them... The proofs need to be explained well enough that they can be communicated and understood by the community" - Simons Foundation.

Tim Gowers went from skeptic to shaken in three months, and said so. In May, after ChatGPT 5.5 Pro produced what he called "a piece of PhD-level research in an hour or so" on sumset problems, he wrote, "We are all having to keep revising upwards our assessments of the mathematical capabilities of large language models" - Gowers's Weblog. In July, declining to sign the Leiden Declaration on AI and mathematics, he admitted, "It felt very strange and not particularly pleasant to have the rug pulled out from under my feet like that," and disputed the declaration's demand to "affirm the humanity of authorship" - Gowers's Weblog. The Leiden Declaration itself, drafted after a September 2025 Lorentz Centre workshop, had 3,807 signatories by 9 September 2026 and an endorsement from the International Mathematical Union - Leiden Declaration.

Kevin Buzzard is the formalist conscience, and his position has been consistent: Lean or nothing. "I am not reading AI-generated informal mathematics," he wrote in July, while also saying of the Jacobian result, "It is a big day. I think it's a great time to be alive, personally." His practical advice to students was that "any PhD student who was not paying $200 per month to access these tools was crazy" - Xena Project. Scott Aaronson moved from "an AI that can merely fill in the insights that should've been obvious to you is a really huge freaking deal" in September 2025, when GPT-5 supplied a key step in his QMA amplification paper, to "the last year of math and theoretical computer science research in the style we've known it" in August 2026 - Shtetl-Optimized.

Jacob Tsimerman, who won a Fields Medal on 23 July for his work on the André-Oort conjecture, is the most striking case, because he co-authored the unit distance digestion and then said in public that AI "will be better than mathematicians at doing math within two years" and that "some version of maths as it exists today will be lost, and it's ok to grieve that" - Quanta Magazine. He has taken leave from Toronto to work at an AI lab; the ICM report from Plus Magazine names Google DeepMind - Plus Magazine. Akhil Mathew called the change "very rapid and very unsettling... especially for junior mathematicians."

The labs and the critics frame the same facts differently. Dario Amodei's January essay states that "AI models are beginning to make progress in solving unsolved mathematical problems" and puts a "country of geniuses in a datacenter" possibly one to two years away - Dario Amodei. Sam Altman's 2025 essay predicted that "2026 will likely see the arrival of systems that can figure out novel insights" - Sam Altman. Gary Marcus and Ernest Davis wrote in April 2025 that "none of the AIs scored higher than 5 percent" on the USAMO and that "all evaluated LLMs consistently claimed to have solved the problems" - Marcus on AI; by September 2026 Marcus called Astra's symbolic capabilities "extraordinarily vindicating" while insisting that "what we don't know is how robust that capability is." Ravi Vakil's phrase for what everyone is waiting for is "a phase change, not a slow climb," with the warning that "the predictions will be even more wrong this time" - Epoch AI.

Why this matters: the disagreement is structural, not factual. People differ on reference class, on time horizon, and on the one open question, which is whether recombination at scale eventually produces the new definitions that hard problems need. How to apply it: when reading any expert's verdict, ask which instrument they are looking at. Tao and Gowers looking at First Proof and Erdős say "real and rising." Anyone looking at Riemann says "nothing yet." Both are right.

14. What This Means for Anyone Building With AI

The mathematics story is a preview of every domain where AI does work that matters, because mathematics has the one thing other domains lack: a cheap, exact verifier. Everything that happened between the IMO gold of July 2025 and the unit distance disproof of May 2026 happened because a Lean kernel or a numeric answer could reward a model for being right. The transferable lesson is not "AI is good at math." It is that capability follows the verifier, and wherever you can build one, the same curve will follow.

That is why mathematics and code moved first and moved together. The Curry-Howard correspondence, that a proof is a program and a proposition is a type, is the reason Lean is simultaneously a proof assistant and a programming language, and the reason coding agents and proving agents have converged on the same architecture. The same models that ran Buckmaster's Codex sessions wrote the Euler blowup proof. Amodei's line, "I'm very confident on tasks that can be verified," is the strategic version of the same point. When we compared the best coding CLIs and the sandboxes where agents run code, the winners were the tools that closed the loop between generation and a test, which is the software equivalent of a Lean kernel.

Three practical consequences follow for anyone deploying agents. First, build the verifier before scaling the generator. First Proof's second batch found that models "gloss over the hardest steps" and cite papers that do not contain the claimed results; an agent writing code or filing documents does the same, and the only defense is a check it cannot argue with. Second, statement fidelity is your job. Lean cannot tell you whether the theorem is the right theorem, and a test suite cannot tell you whether the specification is the right specification; Buzzard reading every line of the FLT statement is what responsible deployment looks like. Third, cost is no longer the constraint. A perfect IMO score cost about $20 of API usage in 2026, ten research results cost under $2,000, and an attempt at a serious open problem is budgeted at $300. We track the underlying prices in our LLM price table and the routing tactics in our model routing guide.

The same logic drives the platforms that run autonomous agent workforces. Whether you assemble your own harness from the pieces we describe in our subagent fleet guide and our context engineering guide, or use a managed platform such as O-mega, which runs agents against verifiable outcomes rather than open-ended chat, the question that decides whether the system works is the same one the mathematicians are asking: what is the kernel that rejects a wrong answer, and who reads the statement it checked.

The mathematics community is also modeling the sociology that every field will need. Tao's proof-parenting norms, that whoever claims a proof should commit to developing it to publication stage, and that priority attaches to the first public explanation rather than the first generated artifact, are answers to a problem every organization will face when generation becomes free and understanding does not. His toy model is worth quoting: if generation collapses from six months to one day while exposition falls only from one month to three weeks, then "90 percent of literature now poorly written." Replace "literature" with "codebase" or "policy" and the model still holds.

Why this matters: the mathematics results are the cleanest available evidence about what AI does when a verifier exists, and the failure modes (skipped steps, invented citations, statement drift) are the same ones every agent deployment sees. How to apply it: treat every agent task the way Tao treats an AI proof. Demand the verifier, read the statement, and require that a human can give the talk.

15. What to Watch Next

Predictions in this field have a poor record, and the people making them say so. Tao's line that nobody can forecast "beyond a year at best" is the frame for everything below. What follows is not a forecast but a list of instruments, with the specific reading that would change the picture, and who has staked a prediction on each.

FrontierMath Erdős is the cleanest measure of research-level capability, because the problems are open and Lean-formalized. The current reading is two of 68 for GPT-6 Astra and zero for everyone else. A move into the tens would mean the recombination frontier has advanced into problems Bloom considers serious. Epoch's methodology page is the place to watch - Epoch AI. Alongside it, Epoch's independent run of Astra on Tier 4 v2 will either confirm or deflate OpenAI's 97.6 percent, and the difference between those numbers is the most useful single fact about self-reporting that 2026 will produce.

Navier-Stokes has three open threads. Whether the Clay Mathematics Institute rules on the forcing term. Whether an unforced result appears, which Tao expects to face "an enormous number of technical difficulties." And whether OpenAI publishes its Lean file and its formal statement for audit, without which the claim remains a compiled program rather than a confirmed theorem. Buckmaster's Mastodon post quoted OpenAI as saying that since 28 August it has been training "a new internal model that has exhibited unprecedented performance in our benchmarks, including mathematics," so the next model is already in the room.

The ten Astra results will start entering peer review, and the first referee report on any of them will be more informative than the announcement. Gil Kalai's 3 September note that a Claude document with Lean verification claims the Kozma-Nitzan conjecture, implying no percolation at the critical probability in all dimensions, is the next candidate in the same category: "If verified, this is a remarkable breakthrough," with "quite a few details which are still unprovided" - Kalai's blog. First Proof's third batch, called for on 25 August, will show whether "C minus" becomes a B - First Proof.

The human literature under audit is the sleeper story. Buzzard's Annals Challenge released 50 formalized statements from recent Annals papers, found errors in four papers by Fields Medallists just by formalizing the statements, and carries his prediction of "a 50-50 chance that one of these papers is unformalizable." A machine that formalizes FLT in eleven days will formalize the recent literature next, and it will find things. DARPA's expMath program, with thirteen university teams and a stated goal of an AI co-author that "proposes and proves useful abstractions," is the institutional bet on rung five of the ladder - DARPA. The Renaissance Philanthropy and XTX Markets AI for Math Fund, at $31.5 million, financed the Annals Challenge dataset - Renaissance Philanthropy.

The named predictions, for the record. Tsimerman: AI better than mathematicians at doing math within two years, said in July 2026. Amodei: the country of geniuses possibly one to two years away, said in January 2026. The Besiroglu-Litt bet, that AI autonomously produces an Annals-quality number theory paper for $100,000 of compute by March 2030, on which Litt now says, "It's clear I was wrong about what capabilities were necessary to produce one, and it's just a matter of time." And Tao, on 5 September, on the rumors around Navier-Stokes: "hypothetical, but not completely implausible."

Why this matters: each instrument above has a specific reading that would change the answer to "what has AI solved," and none of them is a press release. How to apply it: bookmark the four pages named in this section, and ignore any headline that does not move one of them.

16. Conclusion: A Decision Framework for the Next Headline

The state of AI mathematics on 9 September 2026 can be stated in three sentences. AI systems now produce research-level results that the best mathematicians accept as real, concentrated in counterexamples, constructions and proofs that assemble known techniques. Formal verification has become cheap enough that the trustworthy results are the Lean-checked ones, and the residual human job is confirming the statement. No problem that needs a theory nobody has yet invented has moved, and the Millennium problems, in the form mathematicians mean, are untouched.

When the next headline arrives, run it through four decisions. If the result is a counterexample or a construction, expect it to be real, check whether Bloom, Tao, Buzzard or Gowers has commented, and read their caveats as part of the result. If the result is a proof of a universal statement, ask whether the techniques existed; if they did, it is the assembly frontier advancing, which is significant but not new mathematics in the sense of new concepts. If the result is Lean-checked, ask who read the statement; a certificate without a named human is a compiled program. If the result is a Millennium problem, read the Clay statement's fine print before believing the word "solved," because the first such claim turned on a forcing term.

For builders, the mathematics results are the clearest evidence that capability follows the verifier. Every domain that can build a kernel that rejects wrong answers will see the same curve, and every domain that cannot will see the same failure modes: skipped steps, invented citations, and statements that drifted from what was meant. The tools for the first case are the ones in our September 2026 model ranking; the discipline for the second case is the seven questions in Section 12, and platforms such as O-mega exist because the second case is where most real work lives.

The last word belongs to the people doing the work. Kontorovich at the ICM: "you give it 1,000 problems, and maybe it has a 1 percent hit rate, and that's 10 good papers a year, which is a fantastic career in mathematics." Tao, the same week: "It is not enough to generate and verify the proofs. The proofs need to be explained well enough that they can be communicated and understood by the community." Both are true, and the space between them is where mathematics, and everything else that AI touches, will be decided.

This guide reflects the state of AI and mathematics as of 9 September 2026. Several results discussed here (OpenAI's ten Astra results, the Navier-Stokes blowup claims, the Kozma-Nitzan claim) are days or weeks old and have not been peer reviewed; their status may change. Benchmark scores, model names and pricing change frequently, so verify current details before relying on them.