The problem
The planar Tusnády problem asks: given n points in the plane colored red or blue, how evenly can one color them so that every axis-aligned rectangle contains nearly equal numbers of red and blue points? The objective is to minimize the maximum imbalance over all rectangles. It is a classic of combinatorial discrepancy theory, and the gap between upper and lower bounds in the plane had stood for over forty years—Nikhil Bansal's 2022 ICM survey (Section 1.2.4) still listed closing it as an important open problem.
The result
On October 6, 2026, the gap was closed. The paper is *Tight Bounds for Tusnády's Problem in the Plane* by Zhehui Wei (professor and vice dean, Gaoling School of Artificial Intelligence, Renmin University of China), arXiv:2610.08130.
- Previously known upper bound: (log n)^{3/2} — one can always keep the maximum imbalance at this order.
- Previously known lower bound: log n — some point sets force at least this much imbalance under any coloring.
- New result: the two bounds are matched.
- Sept 8: OpenAI claimed an internal model produced a finite-time blow-up solution for 3D Navier–Stokes with smooth forcing—a disputed claim.
- Sept 11: 25 Fields Medalists signed a statement criticizing hasty AI-company announcements.
- Sept 12: The "Human Mathematicians Association" was founded, pledging not to provide technical labor to commercial AI companies.
- Oct 6: OpenAI released 722 model-generated mathematical manuscripts in one batch, drawn from about 4,000 submitted open problems, each result reportedly costing roughly the compute of three hours of ChatGPT Pro reasoning. Some manuscripts address special cases of the Hodge Conjecture—general Hodge remains unsolved, with no formalized proofs among the set.
- The arXiv paper's full text, theorem statements, and proofs were not independently examined; the mathematical content relies on Wei's own essay.
- The "forty-year gap" claim comes from Wei himself; Bansal's 2022 ICM survey text was not checked directly.
- Peer-review and journal submission status of the paper are unknown.
- Figures about the OpenAI release, Fields Medalist statement, and compute costs are as relayed in Wei's essay, not re-verified against original announcements.
- Zhehui Wei, *Tight Bounds for Tusnády's Problem in the Plane*, arXiv:2610.08130, 2026-10-06.
- Zhehui Wei, *The Golden Age of Mathematics and the Lost Age of Mathematicians*, Gaoling School of Artificial Intelligence, Renmin University of China, published 2026-10-09.
For intuition: with log n, doubling the number of points increases the allowed imbalance by only about one unit. At n = 10^6, log n ≈ 20 while (log n)^{3/2} ≈ 89—a gap of more than 4x that grows like √(log n) without bound.
A companion finding: to build the hardest-to-color point sets, no intricate construction is needed—uniformly random points achieve the optimal order with high probability.
The proof and the AI role
Wei describes the proof as unexpectedly simple, using only elementary combinatorial and probabilistic arguments: design a quantity measuring local amplitude, reveal point positions layer by layer, accumulate gains in conditional expectation, and control random fluctuations.
His own timeline: he had thought about the problem since 2013; starting in July 2026 he interacted continuously with GPT and Claude, exploring ideas and hunting for flaws, simplifying an initially more complicated proof, and using AI to complete a Lean formalization of the main lower-bound theorem. He writes: "The core idea emerged from this interaction process; I was responsible for posing the question, judging direction, and checking arguments. This can hardly still be called 'assistance.'"
The broader 2026 context
Wei's companion essay, *The Golden Age of Mathematics and the Lost Age of Mathematicians* (published October 9, 2026), situates the result amid recent events:
Wei cautions against direct cost comparisons with a researcher's annual funding, noting unclear accounting for failed attempts, training, and human curation.
Three distinctions he draws
1. Formal verification. A Lean-checked proof validates the *translated* proposition; translation errors (extra assumptions, weakened conclusions, altered definitions) can silently change the theorem. This step still relies on human checking, and novelty remains a concern—*Scientific American* reported two of ten OpenAI results from August were questioned for insufficient credit to prior work. 2. "AI proves, humans pose problems." He agrees conjectures shape fields more than proofs, but warns against treating this as a permanent division of labor, citing chess: for over a decade after Deep Blue, human+engine teams beat pure engines—until that window closed. 3. Usable vs. understandable. A formalized theorem can be cited and engineered even if unread (usable); a genuinely understood proof changes how we think (understandable); a result with neither is "not knowledge, just text awaiting verification."