An 80-year-old problem, and a bill smaller than a laptop
On 1 August 2026, OpenAI announced that Astra, an unreleased research model, had produced fully verified solutions to ten open mathematical problems, each unsolved for a decade or more. The company published a 249-page manuscript alongside machine-checkable Lean 4 proof certificates on GitHub, and the total compute cost across the ten results came to roughly $2,000. Among the ten: three problems from Paul Erdős’s catalogue, a disproof of Connes’s rigidity conjecture, and the first explicit construction of a non-sofic group, a question that had sat open since 1999, 27 years.
It follows an earlier result from May 2026, when an OpenAI research model resolved the Erdős unit distance conjecture, a problem open since 1946, eighty years before a machine finally cracked it. Thomas Bloom, who maintains the catalogue of open Erdős problems, called the August result even more significant than May’s.
The credibility check: who actually vouched for this
The headline number is $2,000. The number that should carry more weight is zero: the count of unverified logical steps, “sorry” markers in Lean’s terminology, across all ten formalised proofs. Every step was checked by a formal proof assistant that will not accept a step it cannot verify, and the certificates are public on GitHub for any mathematician to run themselves.
Fields Medalist Tim Gowers reviewed the work and said he would have recommended at least one of the results for publication in a top mathematics journal without hesitation. A team of nine mathematicians, Gowers and Noga Alon among them, published a companion paper translating the machine proof into something a human mathematician could follow. That is about as strong an endorsement as this kind of result gets.
The uncomfortable statistic: this is not OpenAI’s first claim, or its cleanest one
Here is the number that belongs next to the $2,000, not instead of it. In October 2025, OpenAI’s own vice president posted that GPT-5 had solved ten previously unsolved Erdős problems. It had not: the model had located existing research papers and presented the results as new solutions, and Bloom publicly called the claim “a dramatic misrepresentation” before OpenAI retracted it.
The August 2026 Astra release has not escaped scrutiny either. Multiple mathematicians have raised research-misconduct concerns about specific results in the release: one, at Yeshiva University, says a sphere-packing proof in the manuscript reuses an argument from his own 2016 paper without credit, and questions have been raised about whether the Connes rigidity result actually addresses the conjecture as originally stated, rather than a related but distinct claim. None of this means the ten Lean-verified proofs are false. It means the framing OpenAI puts around its own results has been wrong before, and is being actively checked again now, by working mathematicians rather than by press coverage.
Why this is still not just a maths story
None of this means AI is quietly better at running your business than you are. It means the cost of buying access to genuinely novel problem-solving, not pattern matching against something the internet has already answered, has fallen further and faster than almost anyone expected, even accounting for OpenAI’s habit of overselling its own headlines. Two thousand dollars is a marketing budget line, not a research grant, and that figure only covers the ten attempts that produced a verified result. It does not include whatever compute Astra burned on attempts that failed, which several outlets covering the release noted was left out of OpenAI’s own accounting.
It is also worth being clear about what is and is not available. Astra itself is not a public product. Nobody can buy access to the model that did this today. What is real, and reproducible regardless of how the specific credit disputes settle, is the direction of travel: the price of frontier-level reasoning has been falling for two years running, and this is the sharpest single data point yet, checked by independent mathematicians rather than taken on trust.
What this means if you run a small business
You will not be asking an AI to solve an Erdős problem. But the same underlying shift, sharply falling cost per unit of genuine reasoning, is what makes AI-driven automation increasingly viable for problems that would have needed a specialist and a five-figure budget two years ago: forecasting demand from messy sales data, untangling a scheduling problem across multiple sites, reading through years of contracts to flag the ones that need renegotiating.
Take a concrete example we see often with UK small businesses: a trades or retail operation trying to work out which suppliers, contracts or job types are quietly losing money once every hidden cost is accounted for, not just the invoice total. That used to mean paying an analyst for a week, or more likely never doing it at all. The same falling cost of reasoning that let Astra chase down a 27-year-old open problem for a few thousand dollars is what now makes that kind of analysis a realistic weekend build rather than a five-figure consulting engagement.
The honest takeaway
What changed on 1 August is not that AI can suddenly run your business, and it is not that everything OpenAI said about its own result should be taken at face value either, given the track record. It is that the cost of the reasoning underneath the hype has kept falling, faster than most forecasts, verified this time by people with no reason to be generous to OpenAI. A small UK business evaluating an AI build for a genuinely hard operational problem is now pricing against a much cheaper baseline than it was even a year ago. That is worth knowing before your next quote from an AI vendor, and worth asking about directly: what is this actually costing to run, and who has independently checked the claim being made.