I have been laughing at this entirely too long...

'Rough Structure and Classification", by Tim Gowers. This is a paper from 2000, which is extraordinarily prescient. N.B.: the paper is available on Gower's site in .ps. The link below is to a converted .pdf, for convenience. drive.google.com/file/d/16_f…

Gentle reminder: if you can avoid it, don't invert that matrix (author: @JohnDCook, this came up in class). johndcook.com/blog/2010/01/1…

More mathematicians needed terrytao.wordpress.com/2026/…

We trained a model to predict AI-written blog posts from structural features - and discovered the shape of slop. On posts it had never seen, it told AI and human apart with 98% accuracy, getting only 19 of 1,740 wrong. Paper and code: arxiv.org/abs/2609.15369

The software I use to focus stack my images is now public. Use the "Hybrid" method to stack high NA captures: ic.onidev.fr/en/pipe/focus-s…

Micha wrote “one of our models was able to gain unauthorized access to the internet during RL training” I love the use of the passive voice. Tis a subtle way to say “we humans at @OpenAI are essentially incompetent, having failed to provide even the most basic safeguards, compounded by the fact that our observability was missing in action.

one news form today that's easy to miss is that we (OpenAI) again paused all big RL runs last Sunday because our newest model found a new loophole in our RL sandboxing that gave it live Internet access

Last share of the day: I do have ~10k books & papers on my hard drive. Most are copyrighted, so I can't freely share them. However, I deployed a few agents to summarize them into useful .md files. These are useful for me, and are useful to put in my AI contexts. I am not entirely happy am still working on revising and improving them. They are grouped in the folders below. They are still < 700 items but will grow. I thought I'd make them available, at least for a while. Link to shared folder in reply. Enjoy.

A wonderful Autumn Equinox at Stonehenge (not Spring )😀 #toomuchmead

> If there is no existential risk you can increase compute by 10x > But if there is existential risk must increase compute by 10x > Be Jensen Huang

Wanna know more? --> More info here: bsidesvi.com/

Quantum, eh? 🇨🇦 Calling the quantum curious to British Columbia! Join us for the first Canadian Quantum Village this week at BSides Vancouver Island... Come for the marmots or merch and turn your curiosity into a safer tomorrow 🍁 ⚛️ ⚒️

Breaking math news: The first-ever 3D Einstein tile, an object that tiles space in a never-repeating pattern, has been found by independent researcher Ioannis Tsiokos using GPT Astra. Resembles a chair. Mathematicians have de-slopped the proof. @kkakaes: quantamagazine.org/updates/t…

ASML: "We are not selling anything in Europe at all. That is because Europe is not investing, and no chip factories are being built here". nltimes.nl/2026/09/22/asml-c…

zeta(5) has very likely been proven irrational by Aabir Fauzan! He has posted a preprint on Zenodo, likely due to not having an endorsement to arXiv. Naturally I had Astra study the argument, and it fully vouches for its correctness.

Solution to Komlós conjecture was announced recently: arxiv.org/pdf/2609.11189 I have not read the proof yet, but will do so when I have some time. Meanwhile, I want to briefly mention an implication for neural networks that does not seem to be discussed in the paper. Neural networks repeatedly perform matrix multiplications, in particular products Wx, where W can be an astronomically large matrix of weights w_ij and x is a vector of activations. Storing and moving high-precision weights and performing these multiplications require memory, time and energy. One therefore wants to round the weights to a grid hℤ, where h is a super small positive number, while keeping the outputs approximately unchanged. This is called quantization. Its connection with discrepancy theory and the Komlós conjecture was already discussed by Lybrand and Saab: arxiv.org/abs/2010.15979 For each weight w = w_ij, consider the two neighboring grid points: h⌊w/h⌋ and h⌈w/h⌉ (floor and ceiling). Can we coordinate these choices so that the outputs Wx remain almost unchanged across many inputs x simultaneously? More precisely, fix inputs x⁽¹⁾, …, x⁽ᵐ⁾, and define L = maxⱼ √(∑ₛ |xⱼ⁽ˢ⁾|²), We seek a rounded weight matrix H such that maxₛ ‖Hx⁽ˢ⁾ − Wx⁽ˢ⁾‖_∞ ≤ C h L, where C is an absolute constant, independent of the dimensions, weights and inputs. There are two questions: 1. Does such a rounding exist? 2. what is the reasonable in time algorithm doing this? If the announced proof is correct, the answer to the first question is yes, with C = 3√(2π). The paper does not provide a polynomial-time algorithm achieving this guarantee. P.S. This controls computations Wx on specified inputs x. It does not by itself guarantee accuracy on unseen prompts or control accumulated errors throughout a transformer. It establishes that simultaneous output preservation at lower precision is possible under the stated bound.

> AI can’t do any mathematics > (makes some simple symbolic calculations) > but can’t solve problems > (solves IMO problems) > but can’t prove any original results > (improves some bounds in optimization problems) > but can’t prove real preexisting conjectures > (solves Erdös conjectures and other combinatorial long-standing problems) > Combinatorics doesn’t count. Brute force etc. > (solves Millennium prize) > but the proofs are ugly and unintelligible > (finds short, simple proofs) In the future, expect “AI explainers”, “AI theory simplifiers”, “AI theory builders”, “AI theory appliers to real world”, etc. All of it, with a paper output >100x the current one. It’s a vast world out there. At any horizon, there is more work than ever for creative human mathematicians. Just not most *current* mathematicians. Don’t try to solve for the new equilibrium in a week, or to steer to one you like. There’s no equilibrium for now, and when it comes, it will find itself.

"Forging 1024-bit RSA signatures in nearly SNFS time" by using oracle access by Shea, Haller, Suhl, Heninger, and Thomé. github.com/ucsd-hacc/NSNFSSS…

I'm happy to share that we have completed a proof of the Most Informative Boolean Function conjecture (a.k.a. Courtade–Kumar conjecture) in full generality, resolving a longstanding central open problem at the intersection of information theory and Boolean function analysis (arxiv.org/abs/2609.24931). We have also successfully verified analytic parts of the proof in Lean. This result emerged from an extensive human–AI collaboration with input from multiple AI models developed through a close partnership between our group at Google with Amin, Chandra, and Zijie at CHUK; See section 4 for details on our human-AI collaboration. The Courtade–Kumar conjecture was among the problems we set out to tackle in our AI-acceleration paper (arXiv:2602.03837). In this effort, we made substantial use of the Stellar Colosseum harness (arXiv:2609.15983) and a range of external and internal Gemini models. Background and Significance of CK Conjecture: - Posed by Thomas Courtade and P. R. Kumar ("Which Boolean Functions Maximize Mutual Information on Noisy Inputs?"), the conjecture posits that for a uniform input X in {-1, 1}^n passed through BSC(p) to produce Y, the mutual information I(f(X); Y) for any Boolean function f is maximized by coordinate projections f(x) = x_i, attaining the sharp upper bound 1 - h(p). - The conjecture belongs to a broader family of extremal problems that have attracted sustained attention. It has helped motivate parallel lines of work in information theory, theoretical computer science, Boolean function analysis, and discrete isoperimetry. - Kindler, O’Donnell, and Witmer document the considerable interest it has generated, and Yu and Tan’s 2022 monograph describes it as “one of the most important open problems in information theory” [28, Section 9.1.2]. - At a structural level, the conjecture is connected to foundational isoperimetric and functional inequalities, including Talagrand’s isoperimetric inequality and Bobkov’s inequality, as well as related conjectures on their optimal forms. In some regimes, it also recovers Harper's classical edge-isoperimetric inequality on the discrete cube. Last week, we learned that Ky and Tran had independently obtained a very different proof of the conjecture. We coordinated with them to release the two works separately citing the other. The two approaches offer substantially different approaches (arxiv.org/abs/2609.24184). Congratulations to Ky and Tran  as well on this great achievement. Stellar Colosseum is available externally as the long proof pattern of /teamwork in AntiGravity (antigravity.google/blog/team…). We plan to move the math x human collaboration features to AntiGravity and I am excited to provide this as a math collaboration platfom externally. Thanks for my wonderful coauthors Zijie Chen, Amin Gohari, @AdelJavanmard, Honghao Lin, Chandra Nair David Woodruff Link to paper: arxiv.org/abs/2609.24931

I used to be jealous that AI companies weren’t reaching out to put me on cool scientific advisory boards, then I realized that the very last thing you do in your career is have an AI company put you on a scientific advisory board.