Andrew Curran

False? Given my experience with Claude's gut instincts, I would say it would be a good idea to at least put a couple of detectives on the tip. Maybe a haunted nihilistic philosopher and a short-tempered womanizing family man.

Over two weeks, Prime Agent orchestrated a swarm of over 2,000 agents to rewrite itself in Rust.

With 10,000+ sandboxes, 200B+ GLM-5.3 tokens, and 16,000 agent-to-agent messages, this was our largest test yet of Prime Agent’s multi-agent capabilities and our infrastructure powering these agent swarms.

Prime Agent now reaches usable input ~13× faster and uses 83% less startup memory than before.

With Ben Chow, Yuan Liao and Ziyang Qin, we have completed a full Lean formalization of the Hamilton-Perelman proof of the Poincaré conjecture!

The proof is around 4.7 million lines of code, written in roughly two weeks. Grateful to the @DARPA expMath program for its support!

Most of the recent math results were not reached by a swarm, the way Navier–Stokes was. I think that part got lost in the excitement around the release. OpenAI's unnamed internal model, which I'm going to call Aeon, reached most of them in one shot, from a single prompt, with no interruptions. This model did not even exist before the end of August, and it is still training. Notice how the returns are not dropping off? That chart is from a month ago. It is a log scale. What does Aeon look like now?

On average, each result used three hours of thinking. Aeon was given 4000 problems to work on by OpenAI. How many more have been solved in the last three days? That number is not zero.

It's like hearing notes in a song that is slowly rising. I don't think Pacing the Frontier was just about Hugging Face or hacking. They've seen how close we are to closing the loop, and as the hour draws near, their resolve begins to quaver. The last piece was model creativity, and I think that threshold was crossed internally by Anthropic and OpenAI in September. You see it in Opus 5.5, which gets it from Fable 5.5. You see it in the math results from Aeon.

All that was needed to start the event was the ability for models to think of novel ways to improve themselves. That was the last piece. This is directly analogous to the ability to think of strange, alien ways to solve math problems: solutions so inhuman that they are difficult to express in existing human terms, so the proof winds up incomprehensible. I think these same kinds of alien solutions are now being applied to model improvements internally. That's what all this recent consternation is really about. They see the invisible frontier. And they see what is about to happen.

Introducing Microsoft-Decision-1, our new model for fast decision-making.

It delivers top performance on structured decision tasks, outperforming both LLMs and other decision models in latency and quality.

We’re already testing it across Microsoft for everything from incident response and quality control to scientific discovery.

Working with AI models, I've arrived at what I consider the most significant pure-theory breakthrough of my research (in theoretical neuroscience) so far:

arxiv.org/abs/2610.12426

Chaotic systems show what is colloquially called the butterfly effect, namely, a tiny nudge to the initial state grows exponentially in time. How fast the nudge grows depends on its direction. Each direction has its own growth rate, called a Lyapunov exponent, and the full set of rates is the Lyapunov spectrum.

The Lyapunov spectrum is a rich (and diffeomorphism-invariant) characterization of chaos, telling us how many directions are chaotic, how many dimensions the attractor occupies, and how fast the dynamics generate information.

In 1988, Sompolinsky, Crisanti, and Sommers (Physical Review Letters) introduced a nonlinear recurrent-network model of N randomly coupled neurons that becomes chaotic when the coupling strength exceeds a critical value. This network has since become a cornerstone of theoretical neuroscience, both as a model of the spontaneous activity of cortex and as the starting point for training recurrent networks on tasks. I have worked on this model for much of my research.

In the same paper, the authors obtained the largest Lyapunov exponent from the ground-state energy of a Schrödinger equation and named the full Lyapunov spectrum as an open problem. In 2023, beautiful work by Engelken et al. (Physical Review Research) computed the full Lyapunov spectrum numerically. However, an analytical calculation of the full Lyapunov spectrum has remained out of reach.

Four decades after the problem was posed, this paper derives the full Lyapunov spectrum at large N from a simple self-consistent single-site problem.

This calculation has three steps.

First, we lower every exponent by a constant s, which turns the question "how many exponents lie below s?" into the question "how many tangent directions decay?" We show that, if we drive the shifted tangent dynamics with a source and take the minimum-norm solution, the one-step response is the projector onto the decaying directions. Thus, the trace of this projector counts the decaying directions. This step is exact at finite N.

Second, to work with the minimum-norm solution, we cast it as the vanishing-regularization limit of the solution of a least-squares problem. This regularized solution in turn solves a dynamical system with one field running forward in time and another running backward.

Third, we apply the cavity method, that is, we add one neuron to the network, along with its forward and backward fields. The effect of the rest of the network on this neuron then reduces to the classical dynamical mean-field description of the activity of this neuron, plus two kernels that couple the forward and backward fields of this neuron to each other across time. Thus, the full Lyapunov spectrum follows from a single-site problem, which boils down to the self-consistent calculation of these kernels.

The resulting theory agrees closely with simulations for the Lyapunov spectrum, the attractor dimension, and the entropy rate, and recovers known results, including the Schrödinger equation of Sompolinsky, Crisanti, and Sommers for the largest Lyapunov exponent.

As a byproduct, the theory establishes analytically that the chaos is extensive, i.e., the attractor dimension and the entropy rate grow in proportion to N.

I had assumed that this problem was analytically intractable. After working independently for 100 minutes with Max thinking, GPT-6 Astra produced the initial derivation, yielding a self-consistent problem whose solution matched simulations of large networks.

I found this derivation intriguing, albeit complicated and somewhat mysterious (see AI Methodology in the paper). After investigating the structure of the proposed solution more deeply, together with AI models, I arrived at a physically transparent derivation.

I am excited to apply this approach to other network models, trained networks, and, more broadly, to high-dimensional systems to which Lyapunov analysis is relevant, including AI models themselves. Of course, as with any paper, I took care to write up the paper clearly.

A few broader thoughts...

Neural circuits are high-dimensional nonlinear dynamical systems, and such systems are notoriously hard to understand. I would frame the central problem of theoretical neuroscience as understanding how such systems function, process information, learn, and generate behavior.

Many people in the theoretical sciences are anxious about AI. However, in my field at least, we are nowhere close to fully understanding the high-dimensional computations unfolding in our brains. With AI, we could reach a level of understanding that so far we have only gestured toward. So, I am excited.

A note from our research leaders:

Last week we parted ways with Jasmine, Mikita, and Tomek after a thorough investigation found they violated clear policies on handling sensitive information. Our internal investigation uncovered a significant breach of trust beyond what’s outlined in the letter they published and we stand by the decision to not continue their employment. We generally keep individual employment matters private and don't believe a back and forth would be productive or lead to a resolution, but we want to address the points they raised in their letter directly.

- We want to be very clear that these decisions were not about raising safety concerns or speaking out. Safety and research debates happen every day at OpenAI, often spirited and highly critical. We actively encourage these discussions and consider them essential to making the right decisions. We cannot do the work in front of us without a high degree of trust. We will continue to be extremely forgiving of our team making good-faith mistakes. We have not and do not terminate any of our employees for raising concerns.

- We are actively finalizing contracts with third-party safety assessors and will announce details in the coming weeks. People across the company have been working really hard on getting these partnerships up and running. We are committed to embedding external assessors and continue to make close collaboration with independent safety organizations a core part of our safety work. Many of our researchers already work with 3p safety organizations productively.

- We agree with the letter that preserving the monitorability of frontier models requires an industry-wide commitment, including from OpenAI. Monitorability has long been a core piece of our research program, and something we continue to invest significant resources in (see our publications on Monitoring Monitorability and the subsequent open sourcing of monitorability evals, our system card for GPT-6 Astra, Jakub’s blog and post on X, and the numerous blog posts on our Alignment blog on the topic).

We are deeply sad about this outcome. We appreciated Jasmine, Mikita, and Tomek’s contributions to AI safety at OpenAI and their willingness to speak up and challenge ideas. We championed their voices, supported their work, and placed enormous trust in them. These decisions were not about them raising safety concerns. We have always encouraged that and always will.

Axios is reporting that the top labs are wargaming scenarios for the aftermath of a catastrophic event in the next 6 - 12 months, that some executives now see as being inevitable.

To become a member of the Association for Human Mathematics you must make three vows:

1. Members commit to not working with commercial AI companies by providing technical labor, mathematical knowledge, consultation, or publicity.

2. Members commit to not publishing mathematical texts (including papers, reviews, referee reports, lecture notes, teaching materials, and books) generated by AI models.

3. Members of the AI-free caucus do not use AI models in a research capacity.

We announced an agreement to acquire a nationwide low-band spectrum license portfolio that will pave the way for @Starlink to become a major mobile carrier in the US.

With this new spectrum and our Gen2 constellation, Starlink Mobile can ensure Americans have access to high-speed mobile broadband no matter where they are → spacex.com/updates#starli…