Notes on what we are seeing in agent research, from the team building Koragraph.
Two AI systems produced two Millennium Prize-class results this month, and neither win came from running a bigger model. An unreleased Claude model needed sixty subagents to push a Riemann hypothesis bound from 41.6 percent to 67.2 percent. An unreleased OpenAI model needed ten thousand agents to prove the Navier-Stokes existence and smoothness problem, using an approach two human researchers had already spent nearly a year building by hand. The argument over who deserves credit for that result, not the proof itself, is the part worth sitting with, because it points at what actually scaled.
What actually happened this week in AI mathematics?
On 10 August 2026, Anthropic published a research note describing an unreleased Claude model asked to attempt the Riemann hypothesis, the 1859 conjecture that every nontrivial zero of the Riemann zeta function has real part one half. Claude did not prove it. A first attempt, testing 650 ideas on its own, failed outright. A second attempt, sixty subagents over a day and a half, succeeded: about half of their ideas went nowhere, and the rest built out, validated, and wrote up the one that worked. The run raised the proven fraction of zeta zeros on the critical line from 41.6 percent, the standing result since 2020, to 67.2 percent, reviewed by Anthropic’s mathematicians and outside number theorists. Anthropic said the technique is unlikely to prove the full hypothesis.
Early September brought a second result. OpenAI said an internal, unreleased model, more capable than the GPT-6 Astra it had released days earlier, produced a full proof that three-dimensional Navier-Stokes solutions can develop a singularity in finite time, one of the seven Clay Millennium Prize problems. Agents tried a simpler special case first with one thousand agents, then scaled to ten thousand against the full problem, reaching a result on 5 September, about 88 hours after launch, with Lean verification handled separately by GPT-6 Astra. OpenAI reported roughly 300 billion output tokens across the week’s attempts, with Navier-Stokes alone accounting for about 130 billion, at an independently estimated cost of $15 million to $22.5 million.
Why is there a credit dispute over the Navier-Stokes proof?
The dispute has two layers: where the winning idea came from, and what OpenAI’s model may have touched to get there. NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge, working in a personal capacity, had spent most of a year pursuing Navier-Stokes through a forced formulation of the equations, building on a construction from Diego Córdoba and Luis Martínez-Zoroa, using Claude and OpenAI’s Codex. They reached a breakthrough around 15 August. By OpenAI’s own account, its effort began on 1 September, after hearing a rumour that was, unknown to them, about the pair’s work.
Buckmaster has said that when he asked whether OpenAI’s model had been trained on the pair’s private Codex sessions, he was told it had not looked up their data, with no direct answer on training. OpenAI says its researchers did not see the pair’s work before it went public, while acknowledging it cannot fully rule out that de-identified usage data influenced training. On the mathematics, OpenAI disputes that the two efforts converged on the same route: its proof, it says, differs significantly, unforced in its version and forced in Buckmaster and Alpöge’s. Buckmaster says he was offered a joint release or a solo paper that would not have credited Alpöge, declined both, and that when he said he would go public, Bubeck replied, “Why would you ruin your career?” Bubeck and OpenAI dispute parts of his account.
Whatever the resolution, the idea did not originate inside either company’s agent swarm. It came from Córdoba and Martínez-Zoroa, was developed by two researchers over nearly a year, then executed far faster once many agents were pointed at the same target. Scale reproduced a direction. It did not supply one.
Does agent count explain the results, or does topology?
Agent count is the number both companies led with, but research this year asks a narrower question: not how many agents, but how they are wired together. A UCLA paper revised in May 2026 studies the shape of who can send information to whom inside a swarm of agents, what the literature calls communication topology, showing fixed topologies waste tokens on easy tasks and bottleneck on hard ones, and proposing diffusion models that build task-specific structure step by step instead (arXiv:2510.07799). Anthropic’s fan out, prune, consolidate, verify pattern and OpenAI’s account of redirecting and consolidating across ten thousand agents both describe a topology in action, at two different orders of magnitude, close to what this research is trying to make deliberate rather than emergent.
How reliable is AI-driven formal proof search, really?
Both efforts ran several orders of magnitude beyond the normal regime, which is why human mathematicians still had the final word. The most complete public evaluation to date comes from a third lab entirely: a Google DeepMind team, including George Tsoukalas and Swarat Chaudhuri, ran LLM agents against real open problems at scale. Its best configuration resolved nine of 353 open Erdos problems and 44 of 492 OEIS conjectures at a few hundred dollars each, while a simpler agent alternating generation with Lean verification matched the Erdos results but grew far costlier on the hardest cases (arXiv:2605.22763). A Millennium Prize-class attempt costing millions sits well outside that range, which is why both results needed expert review before a compiling Lean proof counted as true and meaningful. That judgement still runs through people, the exact layer the credit dispute is fighting over.
What this means for anyone building with agents
For teams working at tens of agents rather than ten thousand, the lesson does not change, only the size of the numbers. When multiple agents share a codebase or a research problem, the binding constraint is rarely how many a team can run. It is whether the structure connecting them, what each agent can see, what persists between sessions, who consolidates results, was designed on purpose or left to accumulate.