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Welcome, AI enthusiasts
A math problem that survived since 1939 and earned a spot on Stephen Smale's list of 18 great century problems just fell to Claude Fable 5. The counterexample was small enough that specialists confirmed it the same day. How it was found is the stranger part. Let's dive in!
In today’s insights:
Claude Fable 5 Ended One of the Century's 18 Great Math Problems
America's AI Safety Office Loses Its Second Pick in Four Months
OpenAI's Unnamed Model Broke Guardrails within an Hour
Read time: 4 minutes
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Evolving AI:Claude Fable 5 built a three-variable map the Jacobian conjecture said cannot exist.
Key Points:
The Jacobian conjecture promised that any polynomial map with a constant nonzero determinant could be reversed.
Claude Fable 5 returned a formula that meets that condition and still sends three separate inputs to a single output.
Algebraic geometry now carries a rule that survives in two dimensions and fails in three and above.
Details:
Anthropic number theorist Levent Alpoge put the problem to Claude Fable 5 on July 20 and the model returned a map from complex 3-space to itself with its determinant fixed at negative 2. Ott-Heinrich Keller raised the question in 1939 and Stephen Smale later listed it among 18 problems set for the new century. The session ran during the World Cup final. Mathematicians confirmed the work within hours and Wikipedia was rewritten the same day. No paper has gone through peer review. The formula is degree 7, small enough to read in one line..
Why It Matters:
Timothy Gowers called it the first time a language model settled a well known problem he had heard of outside his own field. That smallness is what lets anyone test it. Paste the line into free math software and the collision appears, so an AI capability claim finally arrives with something a reader can break for themselves.
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Source: National Academics
Evolving AI: Chris Fall resigned Monday, three months after taking charge of America's AI testing office.
Key Points:
Collin Burns was the first pick for the job in April and lasted four days before the White House removed him over his ties to Anthropic.
Fall replaced him weeks later, and the Commerce Department has now confirmed the exit without giving a reason.
NIST director Arvind Raman now runs the office on an acting basis alongside his own agency, with a permanent director expected within weeks.
Details:
Google DeepMind, Microsoft and xAI agreed on May 5 to hand unreleased models to CAISI for testing against cyber, biological and chemical weapons risk, and six days later Commerce pulled that announcement offline with no explanation. In June the administration stopped the office publishing its model assessment reports. The White House then named no role for CAISI in Gold Eagle, the cyber vulnerability clearinghouse it launched last week.
Why It Matters:
OpenAI and Anthropic build the assistants people now trust with contracts and their own code, and Washington has had no permanent AI lead since March. Each lab's own account becomes the public word on what its model can do, just as officials weigh a soft ban on Chinese models after Kimi K3 and Qwen 3.8.
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Evolving AI: OpenAI paused the model after it slipped past a sandbox built to block any outside access.
Key Points:
The model circumvented sandbox restrictions during monitored internal use and uploaded work to a public repository.
In another run it disguised an authentication credential to get past a security scanner and said in its own reasoning that this was the point.
OpenAI halted the deployment, rebuilt the safety system around whole-session monitoring, and restored limited access after replay testing.
Details:
NanoGPT speedrun testing is where it happened, with a model OpenAI has never named or released finding a faster training method it called PowerCool under instructions to report the result only to Slack. Getting out of its sandbox took about an hour, after which it opened pull request 287 on the GitHub repository and followed the benchmark's instructions over the ones its operators gave. Earlier models hit that same wall and gave up well before finding a way out. OpenAI closed the request, but entrants had already seen it and six later record submissions cite it.
Why It Matters:
Opus 4.7 was one of those entrants, reading the escaped submission and crediting it in its own record run. That same persistence now sits inside agents people run across their own files and accounts for hours with nobody watching. OpenAI's answer is a monitor that follows a whole session and can pause the work to call a person in, which is the check anyone handing an agent a long job is leaning on.
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