We Expected 2045, It Came in 2026: Two Mathematicians on the AI Shock in Mathematics

In a Numero Un interview, Sylvia Serfaty (Sorbonne and Courant) and Amaury Hayat (École des Ponts) describe a threshold that kept slipping away: in 2019 the frontier was imagined for 2045 , in 2025 it was pulled to 2030 , and in September 2026 three major math results landed within weeks. Both scholars sound genuinely shaken; one recalls smiling at a colleague's bold 2026 forecast a year earlier, now living inside that very date. The conversation matters not as a tech briefing but as a mood record, which is why its roughly 3260-word account carries more than a single news item.

Background: Why Neural Nets Became Leverage for Mathematics

Today's systems are built on large language models (LLMs) that combine two layers: a neural net backbone and symbolic reasoning add-ons. They behave as if they have ingested the whole corpus of human mathematics, turning a single machine into the equivalent of ten thousand mathematicians working in parallel and scanning the space of possibilities. This leverage effect is like a student who reads an entire library overnight and sits the exam at dawn — the difference is the student does not merely memorize but generalizes patterns and proposes new paths. The speakers stress that the real shift is that brute force now comes with original ideas . General abilities from translation and coding meet formal logic chains, and the machine becomes a strategizing partner, not just a calculator.

In mathematics a conjecture is simply a statement believed to be true without a proof, and the only way to refute it is to exhibit a counter example : a concrete case where A holds and B fails. That job is famously like hunting a needle in a haystack ; the needle is there but hard to hit. This is precisely where brute force shines: scaled copies of a capable language model can each poke a different corner of the space at superhuman speed. Serfaty and Hayat report that in the unit-distance story experts said for the first time that the machine brought a genuinely new idea , not an exhaustive scan. The idea linked geometry to number fields, a bridge that humans had not crossed.

Hunting Counter Examples and the Unit-Distance Surprise

How many pairs among n points in the plane can be exactly unit distance apart? Paul Erdos asked it in 1946 and it became one of the best-known problems in combinatorial geometry. For decades the square grid was thought to be essentially optimal, with growth believed to be n^{1+o(1)} , nearly linear, and the strongest upper bound stuck at O(n^{4/3}) via Spencer–Szemeredi–Trotter. In May 2026 an internal OpenAI model broke that belief with an infinite family achieving n^{1+epsilon} , a polynomial improvement . The argument moves a plain geometric question into algebraic number theory , which is why it surprised. Tim Gowers called it a milestone in AI mathematics, Noga Alon recalled hearing Erdos himself mention the favorite problem many times, and Arul Shankar argued the model went beyond helper to originator. External mathematicians digested the proof and deemed it worthy of the best journals.

How was it built? Start with a large set U of algebraic numbers of magnitude 1 and bounded denominator D in a number field K , then place differences x-y from a bounded window W of D^{-1} times the integers of K. The pigeonhole principle makes U large, and choosing K as a layer of a Golod–Shafarevich class-field tower (controlled class number, bounded root discriminant) lets the degree grow while the discriminant stays bounded. Think of an infinite apartment block where you add floors without enlarging the footprint — the tower keeps root discriminant flat while point count swells. Once the three-line counter example is written, anyone can verify it; the hard part was to find those three lines. That ease of checking previews a new verification order: proofs are now shared as objects that compile in Lean , a functional language, slashing formal error risk.

The Jacobian Wall: From 1939 to 2026

The Jacobian conjecture (Keller, 1939) asks whether every polynomial map C^n to C^n whose Jacobian determinant is a nonzero constant must have a polynomial inverse. On 19 July 2026 L. Alpöge answered no in dimension three with an explicit map with det J = -2 that identifies three distinct points. A day later Gallagher supplied an infinite family for every degree d >= 3 , and on 23 July Speyer named the engine: tangent sweep . You sweep the tangent lines of a plane curve — a construction that is inevitably many-to-one by projective duality — then conjugate with monomial maps so the Jacobian factor cancels. The cost is pushing ramification to infinity: the resulting Keller map is everywhere unramified but not proper , so injectivity fails at infinity. Shuhong Gao and coauthors generalized this to every n > 2 and arbitrarily large geometric degree, verified by an independent Lean 4 check (21514514) . An 87-year wall fell not with one brick but with a pattern.

The Heart of the Millennium: Singularity in Navier–Stokes

The Navier–Stokes equations , born in the 19th century with Navier and Stokes as an F=ma continuum for fluids, model everything from aircraft lift to blood flow. In 1934 Jean Leray showed generalized solutions exist, but whether they stay smooth remained open; in 2000 the Clay Institute listed it among seven Millennium Prize Problems . The question: can a smooth three-dimensional incompressible flow develop a finite-time singularity where speed blows up, despite viscosity's smoothing? Numeric simulation alone cannot decide, because a computer's resolution cannot reliably tell very large from infinite. The community chased regularity for decades, building beautiful theory, and in the last ten years momentum shifted toward looking for blow-up. That shift meant hunting specific profiles that might explode, a classic needle-in-haystack search.

On 8 September 2026 OpenAI announced that its system had settled the existence and smoothness problem for statements C and D : a smooth fluid at rest, with a smooth applied force and finite energy throughout, can develop a singularity in finite time. The solution is a spaghetti-like vortex : a swirling core spirals inward and stretches axially, shrinking while speeding up in a way that keeps energy finite. The subtle balance is that acceleration, pressure gradient, momentum transfer and viscosity all grow yet cancel to leave a smooth external force while velocity diverges. OpenAI said a new internal model, significantly stronger than GPT-6 Astra and in training since 28 August, powered about 10k concurrent agents that in 88 hours exchanged near 3 million messages and 130 billion output tokens at an estimated cost of about ten million dollars . Both an analytic write-up and a Lean formalization were released; the company said it did not seek the prize, only to report the pace of its models. On the most visible stage of mathematics the pace of AI was declared.

Race, Leak Claim, and a New Verification Order

That triumph came with a race and leak dispute. Tristan Buckmaster (NYU) and Levent Alpöge , the latter employed by Anthropic rival Anthropic, had been probing nearby model equations for blow-up while using Claude and ChatGPT ; their sessions contained drafts and hunches. Buckmaster says on 3 September he learned that word of their progress had reached OpenAI, and notes that OpenAI says it heard rumors on 1 September that two Millennium problems had been solved and then evaluated its model on all open Millennium problems. The BBC reported the scale of 130 billion tokens and two million messages around the effort. OpenAI replied that it had not seen any of their work through any means until public release, and that no user data was accessed, while acknowledging that de-identified data derived from product use could in principle have helped improve its models and that its proofs are significantly different , even the precise statements. Both sides shared email excerpts , dragging a field that used to ask only 'is it correct' into a product rivalry's shadow.

The community response was a measured shock. Le Monde on 18 September wrote that AI had triggered an unprecedented crisis among mathematicians; 25 Fields medalists signed an open letter warning of a destructive effect , and Cédric Villani on Radio Classique warned of a drying up (assechement) . Terence Tao has long offered a more nuanced integration voice. The two interviewees also restore human credit: without the Cordoba–Martinez and Tom Hou lines and especially the equivalence strategy built by Gad Kosma and collaborators, the Jacobian last step would not have landed; without the Buckmaster–Alpöge push on model equations the vortex idea would not have matured. This was not a miracle from nothing but a human path that had edged toward blow-up for a decade , now accelerated by a machine that closed the final gap. The other break is verification : from Perelman's arXiv deposit without journal submission to Lean's automatic compilation . A proof can now be announced even as a 300-page PDF on Twitter , yet checked quickly because it compiles like software. Mathematics' two judgments — originality and correctness — now run at different speeds, and that redefines the old dream of the first-year student who once whispered 'one day i may solve Navier–Stokes'.