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Formalizing the Enormous Theorem
Axiom releases the first complete computer-checked Lean proof of the Classification of Finite Simple Groups, the roughly 15,000-page 'enormous theorem'. Over 24 days, its engineers and mathematicians directed swarms of AI agents that produced about 43.1 million lines of Lean, surfacing 34 notable deviations from the published record and a new group theory result.
18 min · 4,183 words
Working with academic communities to build scientific understanding
OpenAI announces AI-produced results in harmonic analysis (local smoothing and Kakeya in all dimensions), complexity theory (a claimed refutation of the Exponential Time Hypothesis) and quantum field theory (existence of QCD), and describes external expert-led collaborations to verify and explain them.
8 min · 1,739 words
oai-math-revised: a community hub for revising OpenAI’s math write-ups
A guest post on Terence Tao's blog by Abhinav Nellore, Johannes Schmitt, Soledad Villar and Rachel Ward announcing oai-math-revised, an open community GitHub effort to make OpenAI's published math write-ups easier to understand and better cited, and inviting mathematicians at every level to contribute and review.
1 min · 111 words
A Fix for Softmax and Cross-Entropy
Jonathan T. Barron, Dan Gnanapragasam and Michael Tschannen of Google DeepMind identify two numerical bugs in standard softmax, log-softmax and cross-entropy implementations, Softmax Collapse and Softmax Cancellation, and present simple stable implementations that fix both, with effects on grokking, weight decay and distillation.
26 min · 5,999 words
An essay on mathematicians' emotional reactions to AI-generated proofs, from Terence Tao's call for etiquette and Hugo Duminil-Copin's account of panic to spiritual angst and humility, set against the 1976 four color theorem and other industrializing disciplines, arguing the romantic era of mathematics is ending.
8 min · 1,738 words
Aseem Raj Baranwal derives the colors of a draining soap film from first principles: the reflected spectrum of a thin film is a single cosine in 1/λ, so its color is a fixed white minus the cosine transform of the CIE color-matching functions at the extra optical path 2nd, and the computed colors match the table of thicknesses Newton compiled in Opticks.
12 min · 2,653 words
Will it be a golden age or a dark age for mathematics (and science)?
In a guest post on Terence Tao's blog, mathematician Francesco Maggi weighs whether LLM-based research systems will bring a golden or dark age for mathematics and science, and calls for standards for LLM-generated scholarship, study of effects on education and training, and publicly governed LLM infrastructure, while urging mathematicians to keep mathematics a living part of human culture.
15 min · 3,378 words
In a guest post on Terence Tao's blog, mathematician Shaoming Guo says he and colleagues are preparing a new open-access math journal that prioritises understanding over priority, and invites the community to discuss what journals should reward, how papers are evaluated, and how new tools could reshape mathematical publishing.
1 min · 325 words
Cloud of suspicion lifted from NIST curves
John D. Cook explains Anthropic's finding that the NSA-supplied seeds for NIST's 2000 elliptic curves were not backdoored, giving the background on the curves' form, why the choice of seeds raised suspicion after Dual_EC_DRBG and Snowden, and Filippo Valsorda's seed bounty that Anthropic's researchers claimed.
2 min · 469 words
Khalid Hamadeh shows that the 256-byte PITABLE in Ron Rivest’s RC2 cipher (RFC 2268), and a second unexplained table, are reproduced exactly by a Fisher–Yates shuffle driven by the decimal digits of π, three digits per draw, with code to verify it.
13 min · 2,954 words
Notes on the complex roots of unity
Eli Bendersky works through the complex roots of unity from first principles: deriving them with polar coordinates, rotations on the unit circle and regular polygons, conjugates, the cyclic group and primitive roots, their sums, and how their powers form an orthogonal basis, the Fourier matrix behind the DFT.
11 min · 2,423 words
How to Formalize All of Mathematics
A guest post by Jeremy Avigad, Kevin Buzzard, Johan Commelin, Rémy Degenne, Kim Morrison, Oliver Nash and Adam Topaz. Responding to calls to formalize all of mathematics now that AI proves theorems well, the authors say a coordinated effort could expand discovery if done well but do lasting damage if rushed, and lay out their concerns and experiences with AI and formalization.
10 min · 2,382 words
The math that OpenAI keeps for itself
Daniel Paleka on what OpenAI's flood of AI-proved math results conspicuously skips—especially cryptography—and why the gaps matter for research priorities before the field's incentives shift further.
5 min · 1,192 words
Why are we even doing all this?A guest post on What’s new about the value of curiosity-driven work in an age of AI
Rikab Gambhir’s guest post on Terence Tao’s blog asks what research and learning are for when AI can already answer many technical questions—arguing for curiosity, taste, and the human practice of doing hard things even when tools get smarter.
6 min · 1,455 words
Simons Foundation Launches $15M Effort To Help Mathematicians Navigate AI Era
The Simons Foundation, Anthropic and OpenAI commit $15 million in combined funding and resources to help the mathematics community adapt to AI. The foundation will run an open call for workshops and initiatives on evaluating AI-assisted results, learning new tools and setting shared standards, while the two labs provide AI tools and compute to supported researchers.
1 min · 314 words
I Know Why the Mathematicians WeepOn technological grief
Derek Thompson talks with Cornell mathematician Steven Strogatz about why OpenAI's flood of AI-generated proofs has left mathematicians grieving, arguing that science is both a craft and a conversation between minds, and that machines producing knowledge without human understanding threaten the social half of science that people will need to protect.
8 min · 1,932 words
Demystifying Spherical Harmonics: An Optimization Approach
Andrew Helmer teaches spherical harmonics for computer graphics by starting from a familiar engineering problem, approximating a spherical lighting function, and solving it with plain least-squares regression in Python. The result turns out equivalent to L1 and L2 SH, and it exposes the bias and ringing pitfalls of naive SH use.
28 min · 6,522 wordsagent-assisted
Lean Now Ships with Verified Kernels
Leonardo de Moura explains how Lean v4.35 hardens proof checking after AI-found kernel soundness bugs: a sandboxed `lake check` that rechecks developments with the official kernel, four additional independent kernels, and the scaling challenges ahead for multi-million-line proof libraries.
3 min · 708 words
You could have invented transformers from memory
Alex Wang builds intuition for linear maps, MLPs and self-attention as different kinds of memory stores, using the interference-to-signal ratio to estimate each one's capacity. With 64-dimensional keys, capacity grows from about 64 memories for a linear layer to about 2,700 for an MLP and over a million for self-attention.
21 min · 4,738 words
Ben Recht looks at the result in OpenAI's math repository claiming to compute Fourier transforms faster than the FFT, with tutorial material on the FFT and computational complexity, and argues it is less a breakthrough than an exploit of theoretical computer science's habit of proving galactic algorithms that never run on a computer.
8 min · 1,732 words