#503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory

Reports that AI systems had recently produced results involving the Riemann hypothesis, Navier–Stokes, the Unique Games conjecture and other major problems prompted both excitement and concern. The episode repeatedly cautioned that reports are not the same as independently validated breakthroughs: some claims were rumors, and the speakers had not reviewed every proof. The full Riemann hypothesis remains unsolved, and the Navier–Stokes result discussed does not establish whether ordinary solutions can blow up in finite time.

The reported Riemann-related work proves a zero-free region for the zeta function and Dirichlet L-functions, excluding zeros with real parts between 7/8 and 1; another proof reportedly reaches about 11/12. These results improve a boundary that one speaker said had not moved in 70 years, but they do not reach the critical line at real part 1/2, as the full hypothesis requires. The proof combines established analytic number theory techniques, including the Möbius function and sextic characters. One speaker viewed it as a long chain of known methods rather than a new conceptual idea, while acknowledging that AI may be combining ideas in ways people have not explored. The technical proof still needs review, and the speakers could not say whether AI can close the remaining gap.

The Navier–Stokes discussion distinguished mathematical existence questions from practical fluid prediction. The equations describe momentum in fluids, but turbulence involves interacting scales that make useful predictions difficult. The result described uses specially arranged, smooth external forcing to produce a three-dimensional vortex whose vorticity becomes infinite while its energy remains finite; it lies in a highly nonphysical regime and does not resolve the central question of whether viscosity always prevents blowup. Better turbulence models could have substantial engineering value, from reducing aircraft drag to improving turbines and artificial hearts, but the speakers emphasized that the mathematical million-dollar problem is not the same as the much larger engineering challenge. They suggested that AI could help identify useful ways to represent turbulent flows, while noting that the right coordinates and scale for understanding complex systems remain open questions.

The episode also examined AI’s impact on complexity theory. An Anthropic three-sum result, written up by Virgie Vasilevskaya Williams and Josh Alman, reportedly gave an asymptotically faster algorithm than researchers expected. The speaker said this challenges the practice of using shared hardness assumptions to organize many problems and could make separate study of those problems more valuable. OpenAI’s reported Unique Games proof follows substantial prior human work, but its difficult-to-read arguments still need researchers to identify the key ideas and assess the consequences. By contrast, P versus NP remains unresolved and appears especially difficult: proving that a proposed SAT assignment works is easy, but whether SAT itself can be solved in polynomial time is unknown. The discussion also noted a gap between worst-case theory and practical performance, where structure in real-world data can make apparently hard problems manageable.

Formal verification can establish that a proof follows from its encoded assumptions without establishing that people understand it. Lean checks formalized steps, and Mathlib offers a shared library of formalized mathematics; the speakers said Lean was used in part to check proofs in an OpenAI release. A reported Lean formalization of Fermat’s Last Theorem ran to 13 million lines and was described as needing substantial refactoring before it would be a clean, reusable contribution. The speakers argued that future work may pair an explanation of a proof’s ideas with a machine-checkable formalization. That approach matters because journals were already struggling with verification, and even Perelman’s Poincaré Conjecture proof required years of human scrutiny. Collaborative review systems that record revisions and reuse of results could help, but checking validity alone does not make a proof illuminating.

The speakers disagreed over whether this is a crisis or an opportunity, but agreed that the value of mathematics extends beyond collecting proofs: understanding, generality, beauty and the ideas developed along the way remain important. AI may make more research accessible and lower the cost of solving some problems, yet a flood of results could disrupt the career paths of graduate students and postdocs or make exploration feel less rewarding. The speakers urged researchers to choose deliberately what to pursue and what to delegate, while acknowledging that the right strategy and the future of academic institutions are uncertain. They also stressed that AI output should be checked in proportion to the stakes: errors in low-risk, testable optimization differ from errors in high-impact decisions.