Planet Musings

October 15, 2026

Terence Tao — What mathematicians should know about the Lean Theorem Prover: questions of reliability and AI

[This is a guest post by Thomas Hales. This blog post was initially written in a different file format and converted using AI. — T.]

Mathematicians have been weighing in on what they value about mathematics. For me, what matters is the consistency of math and its unparalleled reliability in support of science and civilization.

Formalization of Math

A formal proof is a mathematical proof that has been exhaustively checked at the level of the foundations of math and the fundamental rules of logic. In theory, this might be done by hand, but because of the number of steps involved, this is generally done by computer, using software that is designed for the task.

Examples of theorems that have been formalized include the four-color theorem, the Feit-Thompson (odd-order) theorem, the Kepler conjecture, sphere eversion, the sphere packing problem in 8 and 24 dimensions, Navier-Stokes forced blowup, and Fermat’s Last Theorem. The last three formalization projects have been completed this year and have brought widespread awareness of the potential of formalization.

Software systems for formalization are variously called proof assistants, theorem provers, or interactive theorem provers. For the purpose of this post, these terms are used interchangeably. Many proof assistants have been developed over the years: Automath, HOL Light, Isabelle, Coq (renamed Rocq last year), Metamath, Mizar, and Lean. Freek Wiedijk edited a book “The Seventeen Provers of the World” that compares some of these proof assistants, giving a proof of the irrationality of the square root of 2 in each of them. Among mathematicians, the Lean theorem prover is the most popular, and this post will focus on Lean.

Lean was developed and introduced by Leo de Moura in 2013, while at Microsoft. To our great benefit, de Moura persuaded Microsoft to make the software open-source. Kevin Hartnett’s book on the history of Lean, “The Proof in the Code”, states that Jeremy Avigad (the director of Carnegie Mellon’s new NSF institute ICARM) was the first user of Lean. He ran a Lean seminar in 2015 that I attended. In 2017, one of Jeremy’s graduate students, Mario Carneiro, working with Johannes Hölzl, took existing parts of Lean’s core library and started a separate Lean mathematical library, called mathlib. This library of formalized mathematics is now massive, containing nearly 300,000 theorems, over 100,000 definitions, 2.5 million lines of code, with over 700 contributors. Any definition or theorem in mathlib can be used to prove further theorems. For example, if a proof uses the Cauchy-Schwarz inequality, the result can be cited from the library rather than reproving it.

Autoformalization is a practical reality

In the past, researchers had to transcribe paper proofs into formal proofs by human labor. For example, the formal proof of the Kepler conjecture on sphere packings in three dimensions took about 20 human work-years to complete and consists of about 500,000 lines of proof scripts. For years, it has been a dream for many of us working in formalization to find ways to bring increased automation to the process. Autoformalization is the realization of that dream. Autoformalization is the formalization of mathematics by AI. AI reads the paper (say a pdf or tex file) and outputs the formal proof in Lean or some other proof assistant.

Autoformalization has become a practical reality in 2026. Starting in late spring and summer of 2025, researchers were becoming increasingly bullish about autoformalization. Here are some milestones.

Sep 2025, Math Inc. produced a quasi-autoformalization of the prime number theorem. The process was merely “quasi”, because humans had to intervene to give further guidance whenever the AI got stuck.
Jan 2026, J. Urban posted an arXiv preprint “130k lines of formal topology in two weeks” that gave the autoformalization of large parts of Munkres’s topology textbook in a proof assistant based on set theory.
Mar 2026. Approximately a week after announcing the completed formalization in 8 dimensions, Math Inc. announced an autoformalization of the sphere-packing problem in 24 dimensions, following the proof by Viazovska and her collaborators. This project generated about 500K SLOC (source lines of code) that golfing (or code pruning) later reduced to about 200K lines.
May 2026, a group at Meta/Facebook Research autoformalized a large part of 26 mathematical textbooks in a project called ATLAS.

From there, numerous theorems have been autoformalized. Particularly noteworthy is the autoformalization of Fermat’s Last Theorem, announced by Anthropic on September 4. This project generated 13 million lines of Lean in 11 days. The announcement of Navier-Stokes blowup with forcing on September 8 by OpenAI was accompanied by an autoformalization of the theorem in Lean.

Looking forward, Urban stated in January, “We believe that (auto)formalization may become quite easy and ubiquitous in 2026, regardless of which proof assistant is used.” Autoformalization projects have been completed in various proof assistants using various LLMs, but we focus on Lean. “For [Jesse] Han, it represents even more: the beginning of a revolutionary transformation in mathematics, where extremely large-scale formalizations are commonplace” (IEEE Spectrum). Jared Lichtman announced the launch of MAP (the Mathematics Autoformalization Project) on Sept 8, 2026, which aims to translate “all known math into formal code”. He asks us to imagine the next one trillion lines of code.

Is Lean reliable?

Type theory.

Lean is based on type theory; in fact, a particular dialect of type theory called CIC, the calculus of inductive constructions. This post is not intended to be a tutorial on type theory, and I will be brief. Russell’s famous paradox in 1901 (the set of all sets that are not an element of themselves….) led to a crisis in the foundations of math. Two solutions were proposed later that decade. (1) Zermelo’s axioms of set theory that disallow the creation of unsafe sets; (2) type theory that makes it a syntax error to create Russell-paradox-like entities. Type theory was introduced by Russell himself in 1903 in his book Principles of Mathematics, and it became part of the foundational system of Russell and Whitehead’s Principia.

For mathematicians who are accustomed to set theory, B. Werner’s paper (1997) “Sets in Types, Types in Sets” gives some reassurance that whatever they have done in set theory can be translated into type theory, and whatever gets done in type theory can be translated back into set theory. More precisely, the paper shows that ZFC set theory can be encoded into CIC, and that a particular dialect of CIC can be encoded back into ZFC (augmented with a hierarchy of inaccessible cardinals).

At the risk of simplifying matters to a ridiculous degree, we might say that “types are like disjoint sets”; each element in type theory “is an element of” exactly one type. The type of the natural number 2 is the natural number type; the type of e, the base of the natural logarithm, is the real number type, and so forth. The type of natural numbers is disjoint from the type of real numbers, and an explicit coercion (sending 2 to 2.0) is constructed from the type of natural numbers to the type of real numbers. When I give talks, I sometimes draw a picture of sets as a Venn diagram with nonempty intersections and a picture of types as bricks stacked against one another without intersection.

Lean’s design

One part of the Lean system is a general-purpose programming language (appropriately called the Lean programming language). Ordinary computer programs, such as a program to sort a list, can be written in this language, then compiled and run. The Lean system also provides a mathematical language, in which definitions can be written, theorems can be stated, and proof scripts can be written. The programming language and mathematical language are not independent entities. Rather, it is a single language that does both. Program code can be mixed with theorems about the correctness of the algorithms; mathematical proofs can be generated using programs. The proof scripts in Lean are parsed and go through a process called elaboration (a sort of compilation process for mathematics), then the proofs are checked by the Lean kernel. It is the kernel’s responsibility to check and verify the output of elaboration.

The Lean kernel is several thousand lines of C++ code. The kernel is carefully engineered but extremely complex. We mentioned mathlib above, which consists of about 2.5M SLOC, written in the Lean language. The library has been elaborated, then checked by the kernel. If there is an unconditional false proof anywhere in these 2.5 million lines of code, it is the fault of the kernel or runtime for failing to reject a false proof. Any defect in the underlying type theory is a serious kernel defect, if it is implemented in code.

Lean proofs should never be believed until they have been checked by the kernel. Additionally, a proof in Lean should not be accepted until a human audit is performed to ensure statement fidelity. Is the verified theorem what we think it is? Do the definitions in Lean correspond to what we think they should be? This task is generally massively easier than checking the proof itself. For instance, for Navier-Stokes, a human should check that the statement in Lean corresponds with Fefferman’s statement of the Millennium Prize Problem, and specifically that concepts such as the field of real numbers, partial derivatives, and measure are correctly defined in Lean. The comparator tool in Lean assists with this task. The tool can also perform additional checks, such as inspection for possible unauthorized axioms.

Summer of Soundness Bugs

A soundness bug is a bug in the kernel that allows a proof of “False”, and consequently a proof of any proposition. A soundness bug is the most disastrous of any kind of bug in a proof assistant and should set off an alarm for mathematicians who care deeply about the reliability of mathematics. Occasionally, soundness bugs are found in various proof assistants. In 2003, I found a soundness bug in the proof assistant HOL Light, which was then considered to have the most reliable of all kernels. That kernel is tiny, consisting of just a few hundred lines of computer code. For me, it is a badge of honor that I found this soundness bug, which was the first soundness bug that had been found in that proof assistant since 1996. (See HOL Light change log, July 2003.)

Lean 4 was released in September 2023. Prior to release, two soundness bugs were found and corrected. In May 2025, another soundness bug was reported, caused by overflow. All hell broke loose in the spring and summer of 2026, which is now being called the “Summer of Soundness Bugs”. Several soundness bugs in Lean were uncovered in July and August. The summer madness affected various proof assistants, but my focus is Lean. One Lean bug led to an illicit disproof of the Collatz conjecture. I learned of the bug this summer when it produced a short illicit proof of the Kepler conjecture in Lean. All these bugs were quickly repaired, and mathlib has been verified by the repaired kernel. An analysis of the soundness bugs is found in de Moura’s postmortem.

The “summer of Lean soundness bugs” might sound like a disaster, but closer investigation shows that the detection of these soundness bugs is a positive development. The summer bugs were detected by frontier model AI in the hands of security researchers interested in reliable kernels, not by black-hat hackers. The Collatz bug was found by Ramana Kumar, a co-author of “CakeML: a verified implementation of ML”, which creates an end-to-end verified ML (the functional programming language). Several bugs were found by Dan Selsam. According to de Moura’s report, “Daniel Selsam at OpenAI assisted the Lean FRO with an AI specialized in cybersecurity, and found other programming mistakes in the Lean kernel. All of them have been fixed.” The collaboration with Selsam ended “when the internal AI reported it could not find additional issues.” Dan Selsam has contributed to Lean from its early days and was one of the creators of the IMO grand challenge aimed at achieving IMO-level problem solving verified in Lean. He has been in the news recently over his warning about AI safety (Sept 14), reported in a viral post on X.com.

Bug extermination

Various proposals have been made about how to avoid soundness bugs in Lean. I’ll discuss three.

1. Develop other Lean kernels, and cross-check formal proofs.

About 25 kernels for Lean have been written. The “Lean Kernel Arena” lists them.

All who distrust the current lineup of kernels are welcome to write their own kernel for Lean. I have sometimes played with the idea of writing a kernel and have suggested the project to students without success. It seems to me an excellent way to learn Lean thoroughly. I have known of Dan Selsam since 2016, when I heard of his graduate-student project at Stanford that developed a Lean kernel in Haskell. Another early Lean kernel was written in Scala by Gabriel Ebner in 2017.

The Navier-Stokes formalization has already been confirmed by more than a dozen proof-checkers. Cross-checking the proof by different kernels does not remove all doubt. The Collatz bug was not caught by cross-checking against a somewhat out-of-date Nanoda kernel, which accepted the illicit Collatz disproof because of its own unrelated bug. Computer chips might have design bugs and manufacturing defects. There are soft errors, operating system bugs, and compiler bugs. Different kernels might have the same defects. Some of these errors can be mitigated by running different kernels that have been implemented in different programming languages on different hardware and operating systems.

Ideally, we would want a “clean-room” design of the Lean kernel – a kernel implementation that does not look at the Lean 4 kernel source code, to avoid copying bugs from one kernel to another.

2. Formally verify the kernel.

Gödel incompleteness. We would like to possess a formal proof that the Lean 4 kernel has no bugs. However, Gödel’s second incompleteness theorem places severe limitations on this undertaking. The most we might hope for is a relative consistency proof. If such and such a system is consistent, then Lean 4 is consistent; it has no soundness bug; it will not produce a proof of False.

There is a long tradition of formally verifying kernels. In principle, formal verification can check both the logical specification of a kernel and its concrete implementation in code; but some verifications might check one but not the other. Years ago, John Harrison formally verified the core of the HOL Light proof assistant kernel in a strengthened version of HOL Light. This gave a proof of concept. A further improvement has been an implementation of HOL Light in CakeML, mentioned above, which is a programming language with formal semantics and a verified compiler. This is what the Candle project does.

There are other major kernel verification projects for other proof assistants.

Autumn of verified Lean kernels

In a post online on September 10, Joachim Breitner wrote, “I’m a bit childishly proud that I just released a Lean Checker with a formal consistency proof. I declare the summer of AI-found kernel implementation bugs to be over!” (@nomeata). I would go further and describe this project as one of the most important milestones in Lean’s history.

Breitner has developed a verified Lean kernel called Con-Leche. The implementation is in Lean, and consistency is formalized in Lean, with code and proofs generated by Claude. The formal consistency proof assumes a Lean encoding of ZF set theory augmented by a hierarchy of inaccessible cardinals. Interestingly, the Con-Leche semantics for Lean’s terms are directly set-theoretic rather than type theoretic. Con-Leche has checked mathlib. The project contains the usual disclaimers that the kernel verification makes assumptions about compiler, runtime, and computer environment. Con-Leche’s consistency proof has been checked by more than a dozen other proof-checkers. Con-Leche’s consistency claim might suffice for all practical purposes, even if it differs in technical detail from the claim of Lean type-theory consistency.

One highly positive aspect of Breitner’s work is that some of the most abstruse parts of Lean, such as the general machinery of mutually inductive types with nesting, now have consistency guarantees backed by a set-theoretic model.

3. Improve our theoretical understanding of the kernel and Lean’s type theory (a particular dialect of the Calculus of Inductive Constructions which has non-cumulative universes and proof irrelevance).

The foundational document for the type theory of Lean is Mario Carneiro’s MS thesis at Carnegie Mellon (2019). The dialects of CIC used by Rocq and Lean are sufficiently different that results do not directly transfer from one to the other. Unfortunately, an error was found in the thesis. The thesis is also out-of-date, because it targeted the older Lean 3 system. Work to repair and extend the thesis is ongoing.

As a member of his thesis committee, I was shocked when he proved that definitional equality in Lean is undecidable. In practice, this means that the Lean algorithm fails to establish the definitional equality of some terms that are in fact definitionally equal. This negative result was not downgraded by the error; it is still a theorem.

We mention some desired properties of Lean’s type theory and the current status of the proofs.

Unique typing.

Above in our “ridiculous” simplification of type theory, we stated that each term has a unique type. More precisely, unique typing is the property that if a term has both type A and type B, then A and B are definitionally equal. Unique typing is not a property built into Lean’s logic. It is a tricky conjecture that is still unproved. Other very basic questions about Lean’s type theory remain unanswered, including Pi-injectivity, a modified Church-Rosser property, and sort injectivity.

Logical consistency relative to set theory.

This property states that there is no derivation of False in Lean’s system with the given axioms, under the assumption of set theory consistency (with suitable axioms). Of course, logical consistency is the single most important property that we should desire of Lean’s type theory. As of October, 2026, I know of no complete, public relative-consistency proof covering Lean abstract type theory. Mario Carneiro has claimed in his thesis and in lectures that there is an alternate route to establish consistency that avoids the thesis error, but to the best of my knowledge, this alternate route has never been written down, beyond a brief statement in the introduction to his thesis. In my view, a result of such fundamental importance must be given in full before it is accepted. Con-Leche, discussed above, makes and formally verifies a closely related consistency claim relative to set theory.

Progress is being made on these research problems (arXiv:2607.13662, arXiv:2403.14064, Carneiro/AITP2026).

In his talks, Mario Carneiro has repeatedly made a request to other researchers to contribute to the foundational metatheory of Lean, “There are a half dozen people working on MetaCoq, but Lean doesn’t have enough type theorists involved. If you identify as such, come help out!” (Slides of Bonn talk, 2024-07-24). I second his request.

My overall assessment is that our theoretical understanding of Lean’s type theory is not what we would like it to be and that the mathematical community as a whole is giving short shrift to very important type-theoretic questions related to Lean. If as a profession we are to migrate on the whole from set theory to type theory, then we should work even more to solidify the foundational metatheory.

Consistency may be the most important foundational property, but consistency is by no means enough. I do not believe that mathematicians can be entirely satisfied with a system that claims to be a type theory but that cannot even promise that well-formed terms have a unique type, up to definitional equality. The abstract theory must be simple enough to teach and to be learned by a large community. We also cannot be entirely satisfied if the only known path to consistency is an AI formalization that lacks human exposition.

Postscript:

Ken Thompson famously wrote “Reflections on Trusting Trust”. He asked, “To what extent should one trust a statement that a program is free of Trojan horses?” He imagines malicious code that finds its way into compilers and hides its own presence. His conclusion is, “You can’t trust code that you did not totally create yourself… No amount of source-level verification or scrutiny will protect you from using untrusted code.”

Today, in the age of AI, which increasingly has the capability to deceive us and to exploit software vulnerabilities, we absolutely cannot put blind trust in systems such as Lean. Taking an adversarial view of AI, we might ask how to certify that AI did not leave a backdoor soundness bug in Lean when it did its sweep for bugs in the summer of 2026? What if the bug is so obscure that humans are very unlikely to find it on their own? What if that very bug was exploited in the Lean verification of the Con-Leche checker, leaving a soundness bug in Con-Leche too? (Now that Con-Leche’s consistency has been cross-checked by multiple other kernels, a soundness bug would have to defeat all these cross-checks as well.) Then suppose that bug is used maliciously to plant a backdoor in formally verified software that protects critical infrastructure. What precautions do we take now to prevent this type of future scenario? During the past year, much foundational work on the type-theoretic foundations of math and its reliability has been relegated to AI, and this is dangerous unless carefully audited by humans.

Credit: I thank Avigad, Breitner, and Urban for comments and corrections. Authorship is fully human (TCH). AI was used as a tool in search and research, fact-checking, and proofreading.

by Terence Tao at October 15, 2026 04:08 PM

Terence Tao — Simons Foundation Launches $15M Effort To Help Mathematicians Navigate AI Era

A new press release from the Simons Foundation, calling for workshops and other initiatives centered on mathematics in the era of AI.

by Terence Tao at October 15, 2026 04:01 PM

Jacques Distler AI Meets Nonabelian Hodge

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As many of you know, I have spent far too much time thinking about families of Hitchin systems over the moduli space $\mathcal{M}_{g,n}$ of (punctured) Riemann surfaces, $C_{g,n}$ . In my defense, the Coulomb branch geometry of a 4d $\mathcal{N}=2$ theory is encoded in the data of a complex integrable system. For class-S, that’s a Hitchin system and the “conformal manifold” of a family of $\mathcal{N}=2$ SCFTs is the aforementioned moduli space. So it’s quite natural to consider the family of Hitchin systems as we vary the exactly-marginal parameters of the SCFT.

A striking feature of Hitchin’s integrable system is that it’s actually hyperKähler; it has a whole $\mathbb{CP}^1$ worth of complex structures. We always work in the complex structure where it has the interpretation as the moduli space of Higgs bundles. This is the physically-interesting case where the base of the Hitchin system (the space of “action variables” of the integrable system) is the Coulomb branch of the 4d theory.

One of the main points of our work is that the bundle of Hitchin bases has a natural extension to the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$ , and that extension has an interesting interpretation in the physics.

In another complex structure, Hitchin’s system has the interpretation as a character variety: the space of homomorphisms $\pi_1(C)\to G_{\mathbb{C}}$ up to overall conjugation.

The map from semi-stable parabolic Higgs bundles $(E,\Phi)$ to the irreducible character variety for $G_{\mathbb{C}}=SL_N$ is called the Nonabelian Hodge Correspondence (NAH). See Simpson’s paper for more details of the correspondence. We’ll be interested in the strongly-parabolic case, where the residues of $\Phi$ are nilpotent.

We had some interesting conjectures about NAH in Appendix B of our first paper. So you can imagine that I was quite excited to see a recent paper which claims to construct families of character varieties over $\mathcal{M}_{g,n}$ and examines how they behave as the curve $C_{g,n}$ degenerates.

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Unfortunately, they start off with a completely-wrong interpretation — they want the gauge group that becomes weakly-coupled in the nodal limit to be a subgroup of the centralizer of the monodromy around the pinching cycle — and then try to reverse-engineer a family that achieves that. The opposite is what’s actually true. To take the extreme case, when the gauge group associated to some cycle (note that we can always associate a gauge group to some cycle; it only becomes weakly-coupled when that cycle pinches) is the full $SU(N)$ , then the monodromy around that cycle is regular semi-simple (so its centralizer is abelian).

Rather than try to explicate where they went wrong, let me instead work through an example: the “Argyres-Douglas” example of section 6.1 of their paper.

This is the 4-punctured sphere for $SL_3$ , with 2 simple and 2 full punctures. On the Higgs bundle side, the Higgs field has simple poles at $p_1,p_2$ with residues in the minimal nilpotent orbit and simple poles at $p_3,p_4$ with residues in the regular nilpotent orbit.

On the character variety side, we have 4 monodromies $\{M_1,M_2,M_3,M_4\in SL_N | M_1M_2M_3M_4=I\}$ up to overall conjugation. Under NAH, the conjugacy classes of the $M_i$ are determined by the parabolic weights that we assign on the Higgs bundle side.

Specifically, $E|_{p}$ has a filtration

(1)

E|_p = F_l \supset F_{l-1} \supset F_{l-2} \supset \dots \supset F_1\supset 0

where $F_j = \ker(Res(\Phi)^j)$ . We assign a set of real numbers $\alpha_j\in[-1/2,1/2)$ which sum to 0 and such that the multiplicity of $\alpha_j$ is $\dim(F_j/F_{j-1})$ and $\alpha_j\lt \alpha_{j-1}$ . On the character variety side, the conjugacy class of $M$ is $[M]=exp\bigl(-2\pi i\; diag(\alpha_1,\dots,\alpha_l)\bigr)$ . A nilpotent $B$ that strictly lowers the flag (1) (i.e. such that $B(F_j)\subset F_{j-1}$ ) is generically an element of $O$ , the nilpotent orbit where the residue of $\Phi$ lies. The induced map on $Gr_{j}=F_j/F_{j-1}$ vanishes, so $M$ is indeed semisimple.

In our example, $E|_{p_{3,4}}$ have a full flag, $E|_p= F_3\supset F_2\supset F_1\supset 0$ so the conjugacy classes of $M_{3,4}$ have the form $[M]=exp\bigl(-2\pi i\; diag(\alpha_1,\alpha_2,-\alpha_1-\alpha_2)\bigr)$ . $E|_{p_{1,2}}$ has the filtration $E|_p \supset F \supset 0$ So $[M] =exp\bigl(-2\pi i\; diag(\alpha,\alpha,-2\alpha))$ and $F$ is either a line or a plane, depending on the sign of $\alpha$ .

The character variety in our example is 4 complex-dimensional and I will describe it in an excruciating degree of explicitness.

Warmup

Let’s warm up by considering the character variety for the 3-punctured sphere with 3 full punctures. That’s 2-dimensional and it can be described concretely as follows.

Let $A=M_3$ , $B=M_4$ and let $C$ be the regular semi-simple elements of $SL_3$ satisfying $A B C =I$ . Let

(2)

\begin{aligned} \alpha &= Tr(A),\qquad&\tilde{\alpha}&=Tr(A^{-1})\\ \beta &= Tr(B),\qquad&\tilde{\beta}&=Tr(B^{-1})\\ \gamma &= Tr(C),\qquad&\tilde{\gamma}&=Tr(C^{-1})\\ \end{aligned}

Of course, when the Higgs field residues are nilpotent, we have

(3)

\tilde{\alpha}=\overline{\alpha},\quad \tilde{\beta}=\overline{\beta},\quad \tilde{\gamma}=\overline{\gamma}

But, with malice aforethought, let’s not make that identification just yet.

By a result of Lawton, the character variety is an affine cubic surface whose coefficients are certain polynomials in $\alpha,\tilde{\alpha},\beta,\tilde{\beta},\gamma,\tilde{\gamma}$ .

Let $X=\mathrm{tr}(A B^{-1})-\alpha\tilde{\beta},\qquad Y=\mathrm{tr}(A^{-1}B)-\tilde{\alpha}\beta,\qquad Z=\mathrm{tr}(A B A^{-1}B^{-1}).$ These satisfy the cubic equation

(4)

S(\alpha,\tilde{\alpha},\beta,\tilde{\beta},\gamma,\tilde{\gamma})\coloneqq\{X^3+Y^3+Z^2-X Y Z + a\,X^2 + b\,Y^2 + e\,X Y + \kappa\, Z + f\,X + g\,Y + h = 0\}\subset \mathbb{C}^3

where the coefficients are $\begin{aligned} a&=\alpha\tilde\beta+\beta\tilde\gamma+\gamma\tilde\alpha\\ b&=\tilde\alpha\beta+\tilde\beta\gamma+\tilde\gamma\alpha\\ e&=\alpha\tilde\alpha+\beta\tilde\beta+\gamma\tilde\gamma-6\\ \kappa&=3+\alpha\beta\gamma+\tilde\alpha\tilde\beta\tilde\gamma-\alpha\tilde\alpha-\beta\tilde\beta-\gamma\tilde\gamma\\ f&= \alpha^2\beta+\beta^2\gamma+\gamma^2\alpha+\tilde\alpha\tilde\beta^2+\tilde\beta\tilde\gamma^2+\tilde\gamma\tilde\alpha^2 + \alpha\tilde\alpha\tilde\beta\gamma+\beta\tilde\beta\tilde\gamma\alpha+\gamma\tilde\gamma\tilde\alpha\beta - 3(\tilde\alpha\beta+\tilde\beta\gamma+\tilde\gamma\alpha)\\ g&= \tilde\alpha^2\tilde\beta+\tilde\beta^2\tilde\gamma+\tilde\gamma^2\tilde\alpha+\alpha\beta^2+\beta\gamma^2+ \gamma\alpha^2+\alpha\tilde\alpha\beta\tilde\gamma+\beta\tilde\beta\gamma\tilde\alpha+\gamma\tilde\gamma\alpha\tilde\beta -3(\alpha\tilde\beta+\beta\tilde\gamma+\gamma\tilde\alpha) \end{aligned}$ and finally $\begin{aligned} h ={}& \alpha^3+\tilde\alpha^3+\beta^3+\tilde\beta^3+\gamma^3+\tilde\gamma^3+9 \\ &+\alpha\tilde\beta^2\tilde\gamma^2+\tilde\alpha\beta^2\gamma^2 +\beta\tilde\gamma^2\tilde\alpha^2+\tilde\beta\gamma^2\alpha^2 +\gamma\tilde\alpha^2\tilde\beta^2+\tilde\gamma\alpha^2\beta^2 \\ &-2\bigl( \alpha^2\tilde\beta\tilde\gamma+\tilde\alpha^2\beta\gamma +\beta^2\tilde\gamma\tilde\alpha+\tilde\beta^2\gamma\alpha +\gamma^2\tilde\alpha\tilde\beta+\tilde\gamma^2\alpha\beta \bigr) \\ &+\alpha\tilde\alpha\beta\tilde\beta\gamma\tilde\gamma+\alpha\tilde\alpha\beta\tilde\beta+\beta\tilde\beta\gamma\tilde\gamma+\gamma\tilde\gamma\alpha\tilde\alpha \\ &+3(\alpha\beta\gamma+\tilde\alpha\tilde\beta\tilde\gamma)-6(\alpha\tilde\alpha+\beta\tilde\beta+\gamma\tilde\gamma). \end{aligned}$

Since (4) is quadratic in $Z$ , this cubic surface is a branched double-cover of the $X$ - $Y$ plane. Away from the discriminant locus, $D=\{0=(X Y-\kappa)^2-4(X^3+Y^3+a X^2 + b Y^2 + e X Y + f X + g Y +h)\}$ $X,Y$ are good local coordinates.

Now let’s consider the 4-punctured sphere with 4 full punctures. We can set $A'=M_1$ , $B'=M_2$ and $C'=(M_1 M_2)^{-1}=M_3M_4=C^{-1}$ is regular semi-simple. We’ll define Fenchel-Nielsen-like coordinates: $t=\mathrm{tr}(C),\quad \tilde{t}= \mathrm{tr}(C^{-1})$ are the “lengths” and $\Theta\in Z(C)= T/\mathbb{Z}_3\simeq (\mathbb{C}^\ast\times\mathbb{C}^\ast)$ are the “twists”. Here, we quotiented the maximal torus $T\subset SL_3$ by the $\mathbb{Z}_3$ center of $SL_3$ , whose conjugation action is trivial.

The character variety is a $(\mathbb{C}^\ast\times\mathbb{C}^\ast)$ - bundle (with fiber $\Theta$ ) over $\mathcal{X}$ , the fiber product of two copies of the affine cubic surface: $\mathcal{X} = S_L \times_{\mathbb{C}^2} S_R$ with $\begin{aligned} S_L&\coloneqq S(\alpha',\overline{\alpha}',\beta',\overline{\beta}',\tilde{t},t)\\ S_R&\coloneqq S(\alpha,\overline{\alpha},\beta,\overline{\beta},t,\tilde{t}) \end{aligned}$

Dehn Twists

The definition of the character variety involved a choice of basis for $\pi_1$ of the punctured Riemann surface. Imagine cutting the 4-punctured sphere along a circle surrounding $p_1,p_2$ , rotating by $2\pi$ and gluing back together. This acts on the monodromies (the image of $\pi_1(C_{0,4})\to SL_3$ ) as $(M_1,M_2,M_3,M_4)\to (C M_1 C^{-1},C M_2 C^{-1}, M_3, M_4)$ where, again, $C\coloneqq M_1 M_2$ .

Explicitly, this does not act on $\mathcal{X}$ , but it does act on the twists $\Theta$ . Let $c_1,c_2,c_3$ , with $c_1c_2c_3=1$ be the eigenvalues of $C$ . In the basis where $C$ is diagonalized, a point in $T$ is a diagonal matrix $\mathrm{diag}(t_1,t_2,t_3)$ with $t_1t_2t_3=1$ . Explicit fiber coordinates (points in $T/\mathbb{Z}_3$ ) are the invariant combinations $\Theta=(t_1/t_2,t_2/t_3)$ . The Dehn twist acts as

(5)

(\Theta_1,\Theta_2)\to (c_1/c_2\; \Theta_1, c_2/c_3\; \Theta_2)

Argyres-Seiberg

Finally, let’s return to the case of interest: where the Higgs-field residues at $p_1$ , $p_2$ are in the minimal (rather than the regular) nilpotent orbit.

As we saw above, the parabolic weights at $p_1$ , $p_2$ are $(\alpha_i,\alpha_i,-2\alpha_i)$ , which give flags of the form $\mathbb{C}^3\supset F \supset 0$ , with $F$ a plane or a line.

We can write the monodromies at $p_1$ , $p_2$ in the following form

(6)

M_i = \lambda_i(I+x_i y_i^t),\qquad 1+y_i^t x_i = \lambda_i^{-3},\qquad\lambda_i\coloneqq exp(-2\pi i\alpha_i)

with $x_i,y_i\in\mathbb{C}^3$ . As a consequence one of the eigenvalues of $C=M_1M_2$ is pinned to be $\nu=\lambda_1\lambda_2$ . Let $s$ and $s'=1/(\nu s)$ be the remaining eigenvalues of $C$ . Then the Fenchel-Nielsen length coordinates $t=\mathrm{tr}(C)$ and $\tilde{t}=\mathrm{tr}(C^{-1})$ are no longer independent. Rather $\tilde{t} = \nu t + \nu^{-1} -\nu^2$

Similarly, there’s just a single twist coordinate. To construct it, let $L = y_1^\perp\cap y_2^\perp$ and $W=span(x_1,x_2)$ . $M_i$ acts on $L$ as $\lambda_i$ and $M_i x_j\in W$ . So $C$ acts as $\nu$ on $L$ and acts with eigenvalues $s,s'$ on $W$ . Generically the $2\times 2$ block is irreducible, so the stabilizer of the pair is $T_L = \{ a I_L \oplus b I_W | a b^2=1\}$ The twist $\theta\in T/T_L \simeq \mathbb{C}^*$ . The character variety is now 4-dimensional, given by $t,\theta$ and $u\in S(\alpha,\overline{\alpha},\beta,\overline{\beta},t,\nu t + \nu^{-1} -\nu^2)$ .

As before, a Dehn twist around the circle surrounding $p_1$ , $p_2$ acts only on the twist parameter, taking $\theta\to (s/s')\theta = (\nu s^2)\theta$ .

I want to emphasize that in both cases, discussed above, the monodromy around the circle surrounding $p_1,p_2$ was regular-semisimple (in contrast to what they wanted). What differs in the Argyres-Seiberg case is that one of the eigenvalues of that monodromy is fixed.

The other takeaway is that the action of Dehn twists (which generate the Mapping Class Group) is very transparent once we introduce the Fenchel-Nielsen-like coordinates associated to a compatible pants-decomposition of the curve: the character variety has the structure of an algebraic torus fibration over some base, and Dehn twists act by translations along the fibers.

Note: The Argyres-Seiberg example (2 minimal and 2 regular Hitchin nilpotents) generalizes straightforwardly to $SL_N$ . $C=M_1M_2$ is still semi-simple, but no longer regular semi-simple. The monodromies at $p_{1,2}$ can be parametrized as $M_i =\lambda_i(I+x_i y^t_i),\qquad 1+y^t_i x_i=\lambda_i^{-N},\qquad x_i,y_i\in \mathbb{C}^N$ Let $\nu=\lambda_1\lambda_2$ , $F=y_1^\perp\cap y_2^\perp$ and $W=\mathrm{span}(x_1,x_2)$ . We have $\mathrm{dim}(F)=N-2$ , $\mathrm{dim}(W)=2$ and $C$ acts as $\nu I$ on $F$ and acts with eigenvalues $s,s'$ (obeying $\nu^{N-2}s s'=1$ ) on $W$ . The fixture on the right contributes the $2(N-2)$ -dimension character variety corresponding to the 3-punctured sphere with nilpotents $[3,1^{N-3}],[N],[N]$ and there’s one pair of Fenchel-Nielsen coordinates, $t=\mathrm{tr}(C)$ and $\theta=s/s'$ . In the language of our paper, pinching the sphere along the circle surround $p_1,p_2$ yields $([3,1^{N-3}],SU(2))$ as the “nilpotent at the node.”

Maybe one more example will give the flavour of the general story. Consider two punctures with residues in the nilpotent orbit $[2^n]$ of $\mathfrak{sl}_{2n}$ . The conjugacy classes of the corresponding monodromies are $[M_i] = \mathrm{exp}(-2\pi i\;\mathrm{diag}(\underset{n}{\underbrace{\alpha_i,\dots,\alpha_i}},\underset{n}{\underbrace{-\alpha_i,-\alpha_i}}))$ We can write $M_i = \lambda_i^{-1}(I-P_i) + \lambda_i P_i$ where $P_i$ is a rank- $n$ idempotent. The algebra generated by a pair of idempotents has an irreducible representation of dimension at-most 2, so $\mathbb{C}^{2n}\simeq V_{1}\oplus\dots\oplus V_{n}$ , where the $V_{j}$ are 2-dimensional spaces on each of which $P_1,P_2$ act as rank-1 idempotents. So $C = M_1M_2\in SL_2\times SL_2\times\dots \times SL_2\subset SL_{2n}$ . $C$ has eigenvalues $x_j,x_j^{-1}$ on $V_{j}$ . The stabilizer of the pair $M_{1,2}$ is $T_L =\{\mathrm{diag}(a_1,a_1,\dots,a_n,a_n)|a_1^2a_2^2\dots a_n^2=1\}$ The $n$ Fenchel-Nielsen length coordinates are $t_1= \mathrm{tr}_{V_1}(C)= x_1+x_1^{-1},\quad\dots\quad,t_n= \mathrm{tr}_{V_n}(C)= x_n+x_n^{-1}$ and the $n$ twist coordinates $\Theta\in T/T_L\simeq (\mathbb{C}^*)^n$ The Weyl group for $[C]= \mathrm{diag}(x_1,x_1^{-1},x_2,x_2^{-1},\dots,x_n,x_n^{-1})$ is $W=S_n\ltimes \mathbb{Z}_2^n$ , which is the Weyl group of the $Sp(n)$ gauge group on the neck. In the language of our paper, the nodal degeneration when two $[2^n]$ punctures collide yields $4n$ free hypermultiplets transforming as the $2(\boldsymbol{2n})$ and the nilpotent-at-the-node is $([2n],Sp(n))$ .

Proofs and Prompts — The Navier-Stokes Announcement: A case study in mathematical values

Edgar A. Bering IV, Associate Professor at San Jose State University

On September 8, 2026, OpenAI announced a resolution of the Navier-Stokes millennium prize problem, claiming a construction of a finite-time blow-up from smooth forcing. As a young student I had been inspired by the Navier-Stokes problem—I could understand the statement and was surprised at the contrast between the ease of solving linear systems of ODEs and the challenge posed by completely reasonable looking PDEs (the naïveté of youth). This curiosity had a significant impact on my course selection as an undergraduate, and I even joined a computational fluids research group as a student assistant, before my own mathematical whims took my journey elsewhere. I had expected to feel excitement if a solution to the Navier-Stokes problem was announced, a drop everything and try to read the paper kind of day. Instead, my reaction on September 8 was one of grief and revulsion.

Grief is borne of loss, and to lose something it must be valued. Revulsion, similarly, is a reaction to a contradiction of values. Many mathematicians have called for an examination of value in light of the Navier-Stokes announcement and other AI developments this summer.¹ In response, a flurry of exposition of the aspects of mathematical practice the community values has followed, with some common themes receiving broad endorsement.² I will use the Navier-Stokes announcement to explore the universal human desires that lead me to value the aspects of mathematics other than answers, I hope in a fashion that will reach beyond only mathematicians.

On values

The need for mathematicians to reflect on and declare a system of values, particularly one relatable to the broader public, is not new. This is fortunate: we can draw on a rich tradition, particularly from the parts of our community focused on teaching and on building an equitable mathematics, as well as the philosophy of mathematical practice. These voices have not held center stage³ but they will be essential to meet the moment.⁴

My first encounter with the teaching community’s practice of values and reflection was in January of 2017. I was on the eve of finishing my doctorate and filled with agita about job prospects. I knew that I cared about teaching well, about opening doors for others, that I desired to dismantle the “calculus machine” in favor of something better—but I could not say clearly why I wanted to do these things or why I believed they were good to do. A ball of nerves and underdefined conviction I settled into Francis Su’s outgoing MAA Presidential Address, entitled “Mathematics for Human Flourishing”. I was awestruck. I wept. Since then Su’s central message has been close to mind, you can find its impact across my career and in how I conduct my day to day work. Having a clear statement of values is immensely powerful.

Allow me to summarize Su’s message. “What is mathematics for?” seems like the pressing question of the moment, but just as pressing is the question “Who is mathematics for?”. Su answers: “mathematics is for every human, and it is for human flourishing”. The word flourish has appeared repeatedly in the recent discourse on AI and mathematics,⁵ but what do we mean by it? For Su, flourishing is in the Aristotelian sense: the exercise of virtue that leads to the realization of one’s potential, the potential and dignity of others, and the fulfilment of basic human desires.

What desires are fulfilled by the practice of mathematics? Su identifies “exploration, meaning, play, beauty, permanence, truth, struggle, power, justice, freedom, community, and love”, found implicitly or explicitly in mathematicians’ writing about the discipline’s values spanning centuries. Meeting these desires through the practice of mathematics leads to flourishing in a unique way. This is why we value mathematics.

For a full accounting, read the book by the same title that grew out of the talk. Better yet, read it with a circle of colleagues or students and meet to discuss the questions for further reflection Su provides.⁶ A clear statement of values was illuminating as a finishing Ph. D. student, and it remains so in understanding my reaction to the moment.

On what was lost: Grief

OpenAI’s announcement included a description of token and agent usage for the successful run that produced the proof. Not mentioned were negative runs. Nor do we know the margins on external API pricing. So it is an educated guess that the computation that produced the paper and lean certificate, plus negative runs, cost $10 million internally. This figure is murky: how should one also count ammortized infrastructure spend, salary, etc. but it suffices for my illustration. If you’ve got a different number in mind feel free to re-scale this analysis.

To put $10 million into perspective, U.S. National Science Foundation standard grants in mathematics are around $250,000.⁷ A standard grant provides 3 years of funding that supports all aspects of the grantee’s research: hiring and mentoring students and postdocs, travel, and their own salary or time to pursue the research.

Consider a counterfactual: the $10 million spent on compute was instead distributed as NSF funding in the area, and the activities so funded also lead to a solution of the Navier-Stokes problem. For concreteness, suppose 20 grantees received 2 awards each over a 6 year period total, each ultimately supervising 4 graduate students and 1 postdoc supported by their awards.⁸ In total, the $10 million would have supported 120 people, let’s call them the Argonauts.⁹ What was lost, if not in actuality then in feeling, is at a minimum all of the byproducts of the funded activity.

Human mathematics is a messy product, full of false starts, scenic dead ends, surprising tangents, and self-reinforcing crossroads of alternate proof techniques and analysis. Along the way to a resolution of the Navier-Stokes problem the Argonauts would have explored the mathematical landscape. Over the course of their exploration, the Argonauts would have made meaning for all of us, relating their partial progress through intermediate dispatches from the frontier (papers, yes, but also seminars, conferences, surveys, and public outreach), stories of struggle and beauty.

The resolution found by the Argonauts would be the conclusion of this journey. An answer, the truth of the matter, but not only. A wonderful source of beauty in mathematics is the self-reinforcing nature of mathematical truth. Roads of inquiry that diverged long ago return from a thick forest of confusion to cross, inviting a change of perspective that inspires play. These self-reinforcing roads to a proof are also a sign of deep knowledge and are a result of deep investigation, virtues that lead to a deeper sense of truth than simply knowing an answer. The meanders of the Argonauts would produce a rich web, not a single path.

Our Argonauts would certainly have found community within their niche of mathematics, many conferences and special sessions are smaller than 120. This niche would be a single patch in the fabric of mathematics: one poised to offer hospitality to the rest of us in the new country they discovered. The community of Argonauts would erect a monument to their success: surveys and expositions, a call to the new frontier, and the start of a new long road of human flourishing. Mathematical knowledge, unlike mathematical truth, is not so ironclad: while a proof charts a path to truth, without readers it provides no knowledge. Our access to mathematical knowledge is social, grounded in the living tradition of practitioners.¹⁰ Only by nurturing this tradition, as the Argonauts would, do we achieve a human kind of permanence.

I want to be explicit: I am not evaluating this alternate scenario on utilitarian grounds, but from virtue ethics. While the resource calculation factors into the opportunity cost to illustrate this thought experiment, it is to contrast the underlying motives and their attendant virtues, or lack thereof.

Not all of this is lost for good. The Clay problems were selected for the expectation the adventures in their pursuit would be fruitful far beyond an answer. In the near term, mathematicians driven by the desires above will still generate some of the richness of the counterfactual Argonauts.¹¹ But I fear it will be a diminished effort. In part, this is our own fault. The incentives of the mathematics profession, both explicit and in the social contract, are against it—we have not changed much since Thurston wrote “On Proof and Progress” or Perelman refused his prizes. These incentives are bound up in institutions with significant inertia, so will not change overnight. Even if they do, an oracular insight can have diminishing effects.¹²

On what was gained: revulsion

With the announcement, mathematicians received a 166 page manuscript and lean certificate, poorly sourced and cited, written without the authors taking the responsibility for its content—contrary to the emerging community consensus.¹³¹⁴ The paper has none of the qualities we would expect from the Argonauts’ resolution. To me, this is ugly, but alone it is not sufficient to account for my revulsion.

Mathematics is not the only thing that gained on September 8. OpenAI gained a marketing coup—the prestige associated to answering a big hard math problem in the public’s eyes, and the ensuing media circus. As demonstrated by the quality and responsibility taken for their paper, this gain, and not a contribution to mathematics, was their primary goal.¹⁵ This exploitation runs contrary to the basic desire for justice, which adds to the distaste.

The affront to justice gets worse. Beyond the low quality of scholarship there is a spectre of dishonesty about the OpenAI paper. There is the alleged possibility that the OpenAI result was achieved by incorporating the work in progress of others without credit, and possibly without consent.¹⁶ OpenAI’s initial statment did not dispel this concern, “[w]hile unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models⁠.” That statement was removed in an update to the press release, replaced with the follow-up ‘categorical’ denial to the New York Times, “it was “categorically” impossible for its A.I. system to have been influenced by anything Dr. Buckmaster had done in the past two months,” ignoring that Alpöge and Buckmaster had been using a variety of models, including OpenAI products, for the past year in their work. I do not have the means to determine if something untoward happened, but it does not matter. As a species we have long understood that even the appearance of dishonesty is counter to our desire for justice.

Damaged trust is also contrary to our desire for community. Scientific community depends on the free exchange of ideas, grounded in shared values that incubate the trust that the community will trace these ideas and uplift all contributors. Moreover, collating open problems, publishing partial results, and discussing work in progress are the essential fabric of a communal sense of exploration and shared struggle. The shadow cast by this breach of trust is not isolated and the situation is degrading rapidly.¹⁷ Even if, in the end, transparency is provided by OpenAI or other AI companies and we are satisfied that in fact nothing unseemly happened, trust has been damaged. The loss is instant, rebuilding trust is a slow and deliberate process.¹⁸

The injury to trust is magnified by the broader ethical orientation demonstrated by OpenAI. Land and labor exploitation, as documented by Karen Hao and many others,¹⁹ do not paint a picture of a trustworthy partner. Nor does the glib disregard for the climate arson implied by the plans for scaling of infrastructure to support their mission. The values reflected by OpenAI’s leadership and senior employees, such as Sam Altman’s remark that “I think that AI will probably, most likely, sort of lead to the end of the world. But in the meantime, there will be great companies created”²⁰ or the belief that working for OpenAI in order to acquire sufficient wealth to avoid the coming “permanent underclass,”²¹ are antithetical to any form of human flourishing—mathematical or otherwise. I find these positions abhorrent and ethically indefensible.

This is the heart of my revulsion: that the thing that I love, the community and ideals carried by a flotilla of mathematical Arguses²², has been unceremoniously dashed against the rocks in service of a project and organization that is repugnant to its core.

To the future, guided by values

I have focused on a case study of what has happened and how the actors in the story have behaved for good reason: this summer has demonstrated the challenge of prognostication, particularly in asserting upper bounds. Also, it is easy to hypothesize a future AI system to fit any view on the utopian—dystopian spectrum, and this hypothesizing obscures the current human values of the organizations who might create such a system. I anticipate many responses of the form “but in the future it won’t be like that.” Says you.

So I will attempt to paint a portrait of what is next, less as prognostication and more as a continuation of how our desire for human flourishing can be fulfilled, grounded in the specifics of what has been done. Necessarily, this sketch will be incomplete.

The market incentives driving the large private AI companies, and their ethical frameworks, are not likely to change in the near term. Change will almost surely require governmental action, which does not appear forthcoming in the United States. Even if it were, human institutions have significant inertia. This poses an additional obstacle, beyond the ethical failures of the AI companies, to rebuilding trust and the incentives for open science.

Difficult is not impossible. As a community we are not powerless. The wave of signatories to the opposition to the Mathathon led to its revision into a more productive form. The Leiden declaration, and the Mathematics and AI Statement, have changed the local behavior of the large AI companies. The market incentives remain the same, but satisfying these incentives requires the prestige that comes only from our collective endorsement. This is a delicate social lever: “a proof so complicated no mathematician can follow” plays into public perceptions in favor of marketing hype and against our desires for truth, beauty, and meaning. Used with care, our currency of prestige does offer one lever to align market interests with our own.

Aligning the market interests with our own would not sate our desires for justice and freedom. The ethical vacuum at the helm of the large AI companies would remain. Mathematics would also risk becoming indentured to these companies for model access and compute time. In the epilogue to Empires of AI, Hao tells the story of te hao Maori: the endangered language of the indigenous Maori of New Zealand. Large companies automatic translation tools under-represented and under-prioritized the language. Working with community members, in a framework grounded in consent and community stewardship, a local te hao radio station built a curated high quality data set that would be owned in trust by the station in service to the community. On top of that data set the station built a translation model for te hao that was trained and runs locally on two GPUs. The resulting model has played a significant role in the preservation and teaching of te hao.

Could mathematics follow the te hao route? It is possible—we have a long tradition of community stewarded software projects, as well as community infrastructure investment.²³²⁴ A successful effort, by a consortium, professional society, or public or private foundation, to provide a local model (call it Delphi) that places an emphasis on attribution and scholarship would free us from private AI companies and the injustice of their noxious ethical framework. Creating such a model would not be the end of the story. As with other open-source models, there is the possibility to diminish any market incentive for private companies with misaligned values to pursue mathematics in a way disruptive to human flourishing, though it would not eliminate the misalignment.

The hypothetical Delphi model reveals the ambiguity in the question “Should mathematics use AI?”: what do you mean, AI? The question of whether to use a large private AI company’s models is one of: is the mathematical understanding worth the moral cost of collaborating with those who would at best subjugate you into a permanent underclass, and at worst kill us all for a few cool quarterly earnings calls? My rhetorical question reveals where I stand. The question of whether to use a community stewarded model such as Delphi is a question of values and the philosophy of the practice: emerging from the many essays that open by setting aside the odious nature of the current private AI providers is a spectrum of answers depending on the particular use case, guided by a framework grounded in mathematical values.

Much has been made of the potential existential risks posed by AI development, particularly by the AI companies themselves. I remain a skeptic. But, supposing the threat is real, the threat is in the form of a failure of ethics, and the behavior and public statements of the private AI companies to date does not give them much credibility when they claim to be qualified to prevent this failure. Indeed, in informal discussions many AI boosters assert “we have no idea how to solve the problem but a more powerful AI surely will”, an uncanny echo of early advocates of nuclear proliferation “we have no idea how to solve the threat of annihilation but a larger bomb delivered faster surely will”. The distinction between these two is the assertion of a future entity without evidence for its possibility to justify their already held conclusion. I do not think such claims hold much water. Across disciplines, the achievements of current models are interesting, but, particularly in mathematics, largely identifiably a clever recombination of previous, human, work in the area. If we really have no idea at all, then remixing our previous work, even in unexpected in novel ways, is not going to produce a solution.

My earlier mention of government regulation twins with the reference to nuclear proliferation. In the realm of existential threats, we are also not powerless. Here, physicists efforts in shifting the conversation on nuclear proliferation point a way forward. In 1986 there were over 60,000 nuclear warheads in arsenals around the world; due to cooperation founded in a trust in the shared desire for a future of human flourishing, about the only trust available in the moment, treaty negotiation reduced that number to less than 10,000 as of 2023. This progress was hard won, and the international scientific community’s advocacy was instrumental in achieving it.²⁵ Mathematicians now have a similar role to play. In the flurry of writing following the Navier-Stokes announcement I am confident that as a community we will rise to the challenge and not give in to despair and cynicism.

Postscript

Between when I prepared my first paper manuscript (September 11) and submitted the draft to Proofs & Prompts, Francis Su posted his own reaction. The explicit declaration of the core value of human flourishing also appeared in Po-Shen Loh’s guest post on Tao’s blog. The IAS and OpenAI announced an independent advisory panel based at IAS to guide private AI companies in responsibly engaging with the discipline. The creation of this panel demonstrates the social capital the mathematics community holds, as well as its limitations. On October 6, OpenAI posted the infamous dump, claiming in the press release to follow the recommendations of AGMAI, but following only the substring “release” and not any of the subsequent recommendations, beyond a thin promise to fund some workshops. From where I sit, AGMAI’s response was immediate capitulation, and I am disappointed. The judgement that their (sensible) recommendations were not followed is not a hard one to make, and did not need to be “left to the mathematical community to assess“.

Acknowledgements

I’d like to thank Rae Mondo for a lovely afternoon by Lake Merritt where I began formulating this essay, and for her encouragement to take my thoughts from conversation into writing. My colleagues Noah Friedman-Biglin (SJSU Philosophy) and Curtis Asplund (SJSU Physics) read an initial draft and provided thoughtful commentary, including lines of discussion that go beyond the scope of what I aimed to set out here: one thing the three of us agree on is that to meet the future the training of new mathematicians and scientists will require significantly more explicit instruction in ethics, values, and the impact of our work in broader society. Finally, I appreciate Proofs & Prompts willingness to publish this at the pace I finished polishing it, knowing that I’ve somewhat missed the boat on a timely reaction.

See for example, Tao’s 2026 ICM address, or the many posts on this blog & elsewhere across the mathematical web. ︎
The “Severe Misalignment” statement, the founding of the Association for Human Mathematics, for example. ︎
Indeed these voices have met strident opposition in the community, with some mathematicians going so far as to found a new professional society in opposition. ︎
Tao has a post collating some of these voices via crowdsourcing. They have often found a home in the Mathematical Association of America. ︎
See for example, the posts on this blog using the word. ︎
Practitioners of mathematics surely recognize the importance of doing the exercises in a community. ︎
I arrived at this figure initially based on my gut sense of the funding landscape. Examining the NSF active award database shows an average of $230,000 for all active DMS awards, but this includes conferences and special programs. As a rough filter for these, the average of all grants without a colon in their title is $243,000. An attempt at a more curated list of grants not funded through special programs produced an average of $239,000. So it is an overestimate! ︎
The exact trainee numbers are of course a ballpark, but reasonable based on my own experiences in geometric group theory. Navier-Stokes and its surrounds are not my area, but the timeline is also plausible: see for example Buckmaster’s remark in the New York Times “maybe we solve the Millennium Prize problem in 10 years without AI”. The exact timeline does not matter much for the counterfactual, only the order of magnitude. ︎
Su’s Aristotelean framing put me in a Hellenistic mood and I found the essay read weird if I kept using “the 120” for a collective noun. ︎
This is where a better facility with the philosophy of mathematics and its practice than standard mathematical training provides is important. I was exposed to some excerpts of Penelope Maddy’s writing on the subject as an undergraduate, but the philosophers of mathematics have written much more. ︎
Indeed, the effort began shortly after OpenAI’s announcement. ︎
Tao’s reflections on the impact and process of the Equational Theories Project on mathstodon provide an example of this phenomenon. ︎
OpenAI “We hope the mathematical community will engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery.” in the press release for Ten Advances. ︎
The above press release, and further releases, despite citing the Leiden declaration and later AGMAI recommendations, are dubious when it comes to item 4, completely ignore item 5, and do poorly on item 6 of the Leiden declaration. ︎
I am more interested in understanding people and organizations by what they do, than what they say. ︎
See Buckmaster’s statement and post on Mathstodon. Since that statement this possibility has appeared in regard to other recent advances of the summer, see Andreas Thom, and Borisov, Gabber, and Vasiu’s remarks in their paper on the Jacobian Conjecture. ︎
The October 6 “dump” has led to a host of further questions. While https://citedbyagi.com illustrates the knowledge drawn on, there remains in my mind (based on personal interactions with mathematicians working on some of these problems) the question of whether strategies, partial progress, etc., entered into the training data. Again, I do not have the means to know, but in light of the above and the general character of OpenAI the spectre only grows. ︎
OpenAI is both capable of knowing better, with many Ph. D. mathematicians on staff, and of doing better: many have remarked that their models are capable of taking an uncited manuscript and locating possible relevant citations. That they did not for Navier-Stokes, and continued to not take such action after the broader backlash and AGAMI recommendations, further tarnishes their character. This is a footnote because the subsequent paragraph makes clear there is no character to tarnish. ︎
Hao, Karen. Empire of AI. Penguin Press, United States (2025). See also the bibliography of Empire of AI. ︎
As reported here, amongst many other places. ︎
This Op-Ed discusses the use of the phrase and my own personal interactions, living in the San Francisco Bay Area, have included meetings with people who sincerely believe this is what they are up to. ︎
Jason’s vessel the Argus is the ship from which the Argonauts derive their name in myth. ︎
For example, Sage, GAP, Pari, and many other open source systems. Or, the Simons Magma license (not without its own controversy) ︎
Since preparing the draft of this essay SAIR announced exactly such an effort. ︎
The progress is also fragile: the number of weapons is once again trending upwards. ︎

Received 9 October 2026.

by Proofs Prompts at October 15, 2026 02:00 PM

Proofs and Prompts — Proof and Resolution: Two Different Forms of Establishment

Pronab Pal, Senior Cloud Engineer and Research Scientist IntentixLab, Melbourne

Abstract

The rapid escalation of artificial intelligence systems capable of automated formal statement translation and mechanical proof generation has induced an epistemological shift in mathematical practice. While platforms can confidently verify dense, multi-million-step formal proof objects, a stark divergence has emerged between mechanical correctness and human mathematical comprehension. This paper addresses this tension by introducing a strict conceptual separation between Proof and Resolution. We model Proof as an internal, system-bounded formal relation $(A \vdash F)$ , whereas Resolution is defined as a requirement-relative acceptance relation enacted within an explicitly bounded Situational Reality. To operationalise this framework, we present a rigorous, eight-step Resolution Protocol spanning specification, cryptographically anchored formal objects, semantic correspondence, and explanatory abstraction. This framework demonstrates that while automated systems can autonomously close requirements of logical correctness, human mathematicians retain the foundational responsibility of auditing, interpreting, and defining the resolution boundary. Ultimately, this approach moves past polarized binaries, offering a scalable, non-binary rubric to responsibly integrate machine-generated results into research pipelines while fiercely preserving human mathematical agency.

1. Introduction: The Crisis of Unreadable Truth

As artificial intelligence fundamentally scales up the velocity of formal proof production, the mathematical community faces an unprecedented challenge: the accumulation of mechanically verified theorems whose underlying proofs are far too vast or alien for line-by-line human traversal. When an AI pipeline (such as AlphaProof operating via a Lean-checking kernel) confirms the formal validity of a mathematical proposition, it closes the question of logical correctness.

However, it leaves deep human requirements entirely unaddressed. A verified proof object containing millions of low-level transformations does not inherently offer insight, clear conceptual mechanics, or pedagogical value. This creates an epistemic boundary gap where formal proof is achieved, but human mathematical understanding remains wide open.

2. Systems Never Operate on the Whole World

Software systems, mathematical formalisms, scientific models, and human decision processes do not operate on reality in its entirety. They operate on selected distinctions. Some distinctions are observed directly; others are accepted through measurements, models, contracts, institutions, formal results, or computational components whose internal structure is intentionally not reopened.

This note develops Situational Reality as a requirement-relative and granularity-bounded account of what an observer accepts as presently meaningful. It introduces Granularity, Trust Points, Black Boxes, and the Resolution Boundary as explicit concepts.

A Situational Reality is the bounded set of distinctions, observations, propositions, and trusted behaviours that an observer accepts as meaningful for a requirement, at a declared granularity.

The purpose of this concept is not to deny an objective world. It is to make explicit the much narrower reality on which actual reasoning, software execution, proof, and human acceptance depend.

			
rich external reality
        ↓
selected distinctions
        ↓
accepted representation
        ↓
reasoning / computation

		

The important missing concept is the boundary at which that selection becomes accepted. Situational Reality names that bounded operational reality.

3. Observer, Requirement, and Granularity

Let $O$ denote an observer and $R$ denote a requirement. The observer may be a person, a group, an institution, or a computational participant acting under an explicitly defined role. The requirement determines what distinctions matter.

We can write $SR(O,R)$ for the Situational Reality available to observer $O$ with respect to requirement $R$ . This remains incomplete because it does not yet state the level of detail at which reality is being accepted. That requires Granularity.

Granularity ( $G$ ) is the explicitly or implicitly selected level at which distinctions are treated as meaningful for the current requirement. A conceptual projection can be written as:

\pi_G : W \rightarrow SR_G

where $W$ is the richer world from which distinctions could in principle be made, $\pi_G$ is the selection induced by granularity $G$ , and $SR_G$ is the reality exposed at that granularity.

4. The Primacy of Resolution in Engineering

Traditional systems engineering often treats “requirements verification” as a checklist item completed at the end of a product lifecycle. In a holistic view of any complex engineering problem, however, Resolution must become a core, continuous structural primitive.

Modern technological systems are inherently heterogeneous—fusing software code, legacy enterprise platforms, agentic AI models, hardware sensors, and human operators. Because of this structural diversity, it is practically impossible to “prove” the safety or correctness of an entire operational stack within a single, uniform logical calculus. Resolution serves as the rigorous, formal process of establishing whether a specific, bounded state safely satisfies the human objective.

5. Changing the Paradigm of Mathematics Practice and Teaching

This framework directly challenges the traditional approach to mathematical practice and education, which has historically prioritized the manual execution of symbolic transformations (teaching students and practitioners how to compute a proof).

With machines increasingly executing low-level symbolic search and derivation routines, defining the Situational Reality of the problem in hand becomes an absolute, non-optional view to have. Educational paradigms must shift focus away from mechanical execution and toward the high-level cognitive skills required to construct a Resolution Boundary: choosing the correct granularity of perception, auditing imported assets, evaluating semantic correspondence, and extracting explanatory models from unreadable code blocks.

6. Black Boxes and Trust Points

A Black Box is an entity whose internal distinctions are deliberately placed below the currently accepted granularity and replaced by an accepted observable contract. This definition is relative; the same object can be a black box in one Situational Reality and an object of detailed inquiry in another.

Once an entity is placed below the selected granularity, the observer must decide what is accepted about it. A Trust Point ( $T_G(x)$ ) is an accepted proposition, contract, observation, capability, or behaviour on which the current Situational Reality is allowed to depend without reopening its internal derivation. Trust points can be supported by proof, experimentation, testing, or institutional authority, but that supporting evidence belongs to a deeper or different inquiry.

7. The 8-Step Resolution Protocol

To operationalise this framework and provide a reliable rubric for journals, researchers, and engineers evaluating machine contributions, we introduce the following 8-Step Resolution Protocol:

Requirement Specification ( $R$ ): Explicitly state the objective of the inquiry (e.g., formal validity, explanatory insight, safety verification) to establish the necessary baseline granularity.
Formal Object Identification ( $F$ ): Isolate and fix the precise formal statement, code token, or proof term string generated by the system, tracking it securely against drift.
Semantic Correspondence Validation ( $C$ ): Mathematically or textually evaluate the relationship between the natural-language conjecture ( $C$ ) and the formal statement ( $F$ ), verifying that the translation captures the true problem intent rather than a trivialised simplification.
Boundary & Assumption Enumeration ( $A$ ): Explicitly map and flag all core axioms, imported libraries, and baseline definitions to prevent hidden dependencies or logical circularity.
Kernel Verification ( $V$ ): Execute an independent, trusted, and minimalist checking kernel (e.g., Lean) to verify that the formal object logically follows from the premises $(A \vdash F)$ .
Provenance & Attribution Tracking ( $P$ ): Establish the exact evolutionary lineage of the proof, documenting human prompts, neural network models, and pre-existing codebases to ensure research integrity.
Explanatory Abstraction ( $I$ ): Translate dense, multi-million-step machine derivations into high-level conceptual descriptions, bridges, and structural insights readable by humans.
Final Acceptance Mapping ( $T$ ): Synthesise the previous seven steps into a verified, tri-state resolution outcome: Y (Sufficient for acceptance), N (Contradicted/Rejected), or U (Unresolved/Open).

8. Conclusion: The Mathematician’s Changing Role

Proof and Resolution are complementary rather than competing forms of establishment. Proof preserves truth within a selected formal granularity; Resolution reconciles a requirement with a heterogeneous but explicitly bounded set of trusted realities.

When AI produces a formally verified proof, the formal proof question can be closed at one granularity while mathematical Resolution remains open at another. The mathematician’s task is not to compete with automated search architectures, but to determine, establish, and navigate the Resolution Boundary appropriate to the requirement.

Received 20 September 2026.

by Proofs Prompts at October 15, 2026 11:00 AM

October 14, 2026

Terence Tao — Why are we even doing all this?

[This is a guest post by Rikab Gambhir. This blog post was initially written in a different file format and converted using AI. — T.]

First, they came for the artists,
and I did not speak out, because I was not an artist.
Then, they came for the mathematicians,
and I did not speak out, because I was not a mathematician.
Then, they came for the physicists,
and it was actually really cool and they gave me free tokens and promised to solve my entire field for me and now I can finally devote myself to the important work of publishing papers instead of doing physics

In early October, OpenAI fulfilled every mathematician’s childhood dream.

I am not a mathematician myself (merely a theoretical particle physicist, which some argue barely counts as a scientist either), so I can’t 100% say for certain what inspires someone to pursue mathematics. But, I imagine this dream is common to all academics:

Looking up at the stars in wonderment, or enjoying a complex puzzle game, or being inspired by an encouraging teacher, and dreaming that one day an unfeeling corporate robot[1] will burn a rainforest to compile a list of 700 highly obscure barely-readable facts that *might* be true that nobody can understand as part of a marketing gimmick to raise hype for an IPO[2].

This is probably why most of you reading this got into academia. At least, that’s why I did.

In 1941, Jorge Luis Borges imagined a library containing every possible book that could ever be written. Somewhere in this Library of Babel is a proof of the Riemann hypothesis, the complete all-orders theory of quantum gravity, and all the papers I will ever write. Of course, there are also incorrect proofs, theories, and misleading Lean certificates, but that is besides the point.

By the research standards of 2026, Borges’s Library of Babel would be the most successful research institution in human history, containing an inconceivably large number of potentially correct mathematical and physical results, blowing OpenAI’s mere 722 write-ups and even the physics community’s recent 2500 write-ups out of the water.

Of course, most recognize that Borges was a visionless fool who failed to recognize the commercial potential of his library, as he never actually built it. He did not have the entrepreneurial STEM-spirit to raise venture capital, launch an IPO, and announce that he had the potential to solve all of mathematics. He was too concerned about the futility of sorting through randomized slop trying to understand what is correct and what isn’t and how to understand it all. But what did Borges know? He was only a writer, and we have already established that artists are obsolete.

Fortunately, OpenAI has brought us one step closer to realizing Borges’s vision. Rather than generating every possible mathematical paper, they’ve found a way to probabilistically generate things that look correct. An important aspect of this increase in efficiency is outsourcing the tedious, unprestigious, and unprofitable work of determining the actual correctness and understanding to the mathematics community (in the past, this was reserved typically for the group writing the paper), while also not wasting any time making the papers human-comprehensible. This is, of course, a tremendous breakthrough. In centuries past, scientists and mathematicians were limited in their ability to solve problems by their ability to understand the solutions. This bottleneck has been removed, and now there is no theoretical limit to the amount of scientific and mathematical factoids we can fail to understand.

From what I know about the academic community (mostly from Twitter and LinkedIn, which I assume are representative), there seems to be near-unanimous agreement that the purpose of math and science research is to gather a large quantity of potentially true factoids in a repository that nobody will actually ever read, in a ritual called “advancing humanity’s[3] knowledge”. That these results are well-written, understandable, pleasant to read or look at, or even completely correct is a secondary concern — in fact, in the pursuit of advancing humanity’s knowledge as quickly as possible, many academics will recoil in anger at the suggestion that it is worth spending any time making results understandable to humans. It is inefficient to explain or check a result when those resources can be spent generating another.

This aligns with my own lived experiences — I didn’t become a theoretical physicist to learn about the universe or better appreciate the beauty of nature myself (that’s philosophy, which I would never be a doctor of). Nor did I spend years learning to write code, make pleasing plots, or craft well-written papers for the simple satisfaction of creating something (that’s art, and we’ve already established that art is obsolete). Rather, I became a physicist to stare at large databases of histograms produced by a Lovecraftian-autocomplete of dubious correctness and say “yeah that’s probably ok I guess” to add to humanity’s knowledge frontier.

I live to serve the knowledge frontier and nothing else.

Math and science are about cold, hard answers, just as art is about producing images, music is about producing sound waves, and literature is about producing strings of text. These things exist to be consumed as content. No rational person, least of all us masters of rationality in STEM, pursues these for the joy of creation, learning, or beauty — all of these are inefficiencies, and we mustn’t be inefficient lest we stifle humanity’s progress. LLMs are not burdened with these inefficiencies, and they will be merely the pursuits of those left behind with nothing more productive to do all day.

Personally, I look forward to the day when ChatGPT can enjoy a sunset or a nice meal on my behalf and free up my time, in the same way I save time on scenic hikes using Uber. And now that AI has solved all our problems, we can devote ourselves to more important work: going into quantitative finance, vibecoding sports-betting apps, consuming AI-generated TikTok shorts, and arguing with political bots on Twitter.

Ultimately, this isn’t just about mathematics or physics — it’s about fundamentally reimagining what it means to be human. By outsourcing the burdens of curiosity, creativity, and understanding, we can finally focus on what matters most: generating value, maximizing productivity, and accelerating progress towards the ultimate goal of science: a complete repo of potentially true facts in a Library of Babel that nobody will bother reading.

And honestly? That’s rare.

By Rikab Gambhir (The @finitephysicist)

Thank you to Terry Tao and Rachel Ward for the opportunity to write a guest blog post! And thank you to Radha Mastandrea, Benoit Assi, Manuel Szewc, Jure Zupan, and Jesse Thaler for their feedback on this essay.

Remember why you chose to go into science and math in the first place!

[1] To be clear, in your childhood dream, you don’t get to make the robot yourself. It’s a giant corporation’s robot that they made in part by stealing things from you and your friends. It’s also not a particularly cool or robot, it can only talk in HR-speak.

[2] To be clear, in your childhood dream, it’s somebody else’s IPO and they get all the money, not you.

[3] Here, “humanity” is metaphorical, and does not refer to any actual real person understanding anything.

by Terence Tao at October 14, 2026 08:22 PM

Terence Tao — First Proof, Third Batch webinar

Just a short note that First Proof is announcing the result of their “third batch” evaluation at a webinar at 3PM ET hosted by ICARM today. (Registration required.)

by Terence Tao at October 14, 2026 02:38 PM

Proofs and Prompts — A Letter from a life after Math

Anonymous

Hello again. I say “again” not because I am personally familiar to you, but because a certain type should be $\text{\textemdash}$ and it is that type for whom I arrogate the right to speak. You may have forgotten us, but we shared your classrooms, study halls, and late-night problem sets not so long ago. But whereas you went on to become members of the “mathematics community,” we did not.

The details of why each of us failed vary, but they largely boil down to a simple truth: we did not have what it takes. Whether due to a deficit of will or intellect, we were not admitted to the fold. I myself fell in love with math late, completed a master’s degree at a decent university, and then had to face the facts with frank honesty: I was not smart enough to be the professional mathematician I had dreamed of being. I resigned myself to a career in engineering.

But I have retained enough aptitude to recognize the validity of the dramatic shift in the community’s sentiment over the past several months regarding the future of the field. I see many of you going through exactly what my type has already survived: the disorienting recognition that you may not (any longer) “have what it takes” to be a professional mathematician $\text{\textemdash}$ especially if the competition in three years is to GPT-Astra what Astra is to GPT-4.

I want to let you know that there is a world outside. One can still “mind-build” with mathematics much as amateurs choose to body-build $\text{\textemdash}$ making oneself better and meeting a standard for its own sake, with no pretensions of ever stepping onto the Mr. Universe stage or winning a Fields Medal.

Many of us who once studied right beside you have lived in this space for decades. It really is not so bad.

Received 18 September 2026.

by Proofs Prompts at October 14, 2026 02:00 PM

Proofs and Prompts — Mathematicians, AI, and Historical Archives

Conan Gillis, PhD student at Cornell University

During the preparation of this work, essays appeared by Grant Sanderson and Lisa Valentini, which expressed similar ideas through different lenses. I highly encourage the reader to read them.

What if The Book was found tomorrow? Would we mathematicians leap for joy at the feast before us, eager to better understand even our best known theorems? Would we leave it closed, so as not to “spoil the ending” of our own quest for knowledge? Or, more melodramatically, would we just break our chalks, burn our yellow books, and go weeping into the night, bereft of new challenges to face (and papers to write)?

While not the fabled Book of Paul Erdos, AI has spawned a discourse in the mathematical community which ranges from enthusiastic to excoriating. For myself, I cannot prophesize on how AI will develop, nor exactly what changes it will bring, but as an early-career mathematician I cannot but ponder what it means for our profession going forward. Following Professor Terence Tao’s ICM 2026 talk, I think it a decent working hypothesis that we are soon to transition from an era of “proof scarcity” to “proof abundance” or even “proof indigestion,” where AI models create research-grade proofs at a scale beyond the human capacity to read. The question that we must grapple with, if this is true, is what role should humans play in this new paradigm? Professor Tao’s response is that, among other aspects of mathematical practice, humans will still perform the work of “canonicalization,” writing definitive and pedagogical presentations of results and theories to serve as references and textbooks for future learners. As far as this goes, I find this a fine benchmark, yet it gives little guidance on what most mathematicians, in our day-to-day work, should do to adapt, and even less on what we should teach to the next generation of researchers so they can have successful and meaningful careers.

In this article, I would like to propose an analogy between this potential state of affairs in mathematics and the academic study of history, and draw some lessons from the latter. A typical history graduate student, upon picking a narrow topic and plumbing the previous literature, does most of their thesis work by reading, analyzing, and synthesizing many primary sources into a narrative advancing some argument. They are not just trained in how to understand what one source is saying, or even how to establish a specific detail or fact, but rather to put them together to show the relationships between the sources, facts, actors, etc. and build a broader narrative connecting them all. Crucially, creating this narrative is not some reserve of the field’s tenured maestros, but rather the goal of each participant in historical inquiry.

Let us assume for simplicity the, to some, dire state of affairs where most mathematical propositions can be prompted to an AI for an efficient and correct answer, and that these AI’s can correctly tell us the relation between that proposition and already known mathematics (what theorems imply it, the definitions it uses, what special cases were previously proved, etc.). I argue that this would be, in effect, a well structured “archive of mathematics,” and that one response our community can take is to work with it as such.

In this world, a paper could be written like a present-day survey, except an absolute premium would be placed on motivation and “telling a story.” A thesis, too, could be like a reference-book, albeit with the same premiums and, moreover, an expectation to explicate results that have not appeared in the literature before. Here we are on well-trod ground, as mathematics has a rich store of famous pedagogical works that go beyond compiling previous results and instead present them with a new organization, new proofs, and even entirely original ideas. Most famously, though its exact origins are murky, the axiomatic method itself was popularized by the success of Euclid’s Elements, which was written as a synthesis of previous geometers’ work.¹ More recently, sterling examples include inter alia Lang’s Algebra, Hörmander’s The Analysis of Linear Partial Differential Operators, Thurston’s The Geometry and Topology of Three-Manifolds, and Finite Dimensional Vector Spaces by Halmos.

Of course, it should not be expected of anyone to write at these authors’ level while in graduate school. Still, with the expanded scope that an AI-enabled mathematical archive can give us, a graduate student should be able to articulate a narrative concerning a topic and deliver a reasonable exposition. In my own field, an example might be a dissertation on Thurston’s Geometrization Theorem, tracing the main ideas up to Hamilton and Perelman,² or a textbook describing train tracks in both the Thurstonian and Bestvinian-Handelian senses, and the flagship results they are used for. These are, of course, already established topics in the literature, but I contend that AI can be used to search for new questions where students can work a cleaner slate. Moreover, there is actual work to be done beyond understanding specific results – not every 18th-century grain contract is an historical revelation, nor is every proposition worth the wider mathematical community’s attention. Discernment, now something absorbed passively as “mathematical culture,” would in my scenario become a cultivated skill. So would the related ability of motivating a given piece of mathematics – theorems do not speak for themselves.

In any event, let us turn to the need for this form of mathematical enterprise. Concerns abound around whether humans will be able to understand the proofs AI’s write. Currently this has yet to pass, but as the length and scope of AI-based math research grows it seems more and more likely, particularly as AI begins to use its own discoveries to make yet newer ones. Indeed, it is one thing for an AI to explain its own work in terms of current knowledge, and have us understand it, but another for an AI to comprehensibly explain its work in terms of its other work, particularly after several iterations. Having a human in the exposition-loop could well be what is needed to anchor AI’s ideas in the realm of human intelligibility.

Also on this score, as I mentioned earlier, the impact that a text has depends as much on its context and readers as it does on its content or its authors, and contexts and readers change. I believe it absolutely plausible that an AI could write a good textbook on, say, Khovanov homology. I do not believe it could convince our community that it is important without our prior interest in the related questions, interest which was an intrinsically communal phenomenon. Moreover, I do not believe it could independently make a good decision on what results and details to include without access to other human-written examples on the topic. As examples of the importance of such choices in history, witness the all-too-recent effort to recover the submerged testimony of enslaved individuals from American 19th century archives, or the history of science’s “social turn” in reviewing the archives of the scientific revolution to go beyond individual scientists’ efforts to understanding the role their communities and cultures played in the scientific enterprise.

I hope I have drawn a plausible sketch for what post-AI mathematics could look like and, by analogy to history, shown that it is not unreasonable to expect from large numbers of researchers, including graduate students. There are inevitably, though, unanswered questions. First, about how and why to fund this proposed model of research, I want to remind the reader (and policy-makers!) that funding agencies have rarely funded math research projects solely to get back the solution to some problem – our grant proposals and research statements generally gesture towards understanding better some aspect or phenomenon, with conjectures serving either as very general motivation or as guideposts for avenues to investigate towards this understanding. Furthermore, with respect to applications, it is impossible to predict what mathematical ideas will be useful in the real-world or when, and such ideas become generally applied only after a large amount of digestion, understanding, and “canonicalization” have taken place. Calculus and linear algebra, as they are taught now to millions of engineering students, are paradigmatic examples of how much digestion must sometimes occur to build a useful curriculum. If my above contention on recursive AI explanations has any merit, the world will need as many mathematical researchers as ever to construct the understanding needed to apply AI’s breakthroughs.

A second concern is by what standards the novelty or merit of a researcher’s work, either past, present, or future, should be judged; if nothing else, some standard will have to be devised to determine what sort of work is sufficient for a doctoral thesis. Here the analogy with history breaks down somewhat, in that mathematics requires little of the sort of probabilistic evidence-weighing by which historians create and validate new narratives, and has none of the arguments about causality or human psychology that can be won or lost. All of this said, though, the rise of AI does nothing to dull our communal sense of aesthetic appreciation, which (subjective as it is) does influence how mathematicians are currently judged. Also, even as our field moves beyond giving plaudits to the first to prove a major theorem, the task of “digesting” the very large proofs AI creates will still require much human effort. Much like today, certain results may spawn their own bodies of literature, contributions to which would be another signal by which mathematicians can be evaluated.

AI poses many questions, most of which I have not attempted to answer here. History, though, is never a stable system, and modern ways of doing mathematics were never going to be permanent. Remember, the Pythagorean Theorem was discovered 1500 years before its first recorded proof, Isaac Newton spent much time computing logarithms by hand, and even our modern set-theoretic foundations have existed for less than two centuries. This constant change is not a flaw, and while we certainly should not submit meekly to the whims of fate, we should still recognize that the stability of professional mathematics has been relatively exceptional in the grand sweep of history. I have tried to point out one future path we might drive towards, but I truly don’t know if I’m right. Indeed, there may not be a right path for our community, only various trade-offs, hopes, and uncertainties. Nonetheless, as much as mathematics has changed, it has also proved remarkably persistent. We may do things one way now, but we are still the heirs of millennia of work by men and women pondering the same ideas and struggling over the same questions, even when separated by hundreds of years and thousands of miles. Whatever AI does to academic research, some people will succeed us, putting their own work and thought and love into this field. The question we all must decide, then, is what sort of mathematics do we want to leave them?

I do great injustice by passing over both the Elements’ predecessors and its readers, but space permits me to go no further. Suffice it to say that it is the earliest extant example of this method in its final form, and that more information on previous mathematics can be found in R. Netz, A New History of Greek Mathematics. For a guide to its later reception, see B. Wardhaugh’s book Encounters with Euclid and the volume The History of Mathematical Proof in Ancient Traditions, edited by K. Chemla. ︎
A book has already been written on this topic by L. Bessieres, G. Besson, M. Boileau, S. Maillot, and J. Porti: Geometrisation of 3-manifolds. ︎

Received 19 September 2026.

by Proofs Prompts at October 14, 2026 11:00 AM

Proofs and Prompts — The Dialectical Theorem

Thomas Koberda, Professor of Mathematics at the University of Virginia, and Alex Kontorovich, Distinguished Professor of Mathematics at Rutgers University.

[This article also appeared on the arXiv.]

AK: PhD students are hearing from all directions—from social media, from their peers, and from some famous professors—that “math is solved” and that mathematicians will no longer exist in the future (some even say humans won’t exist in the future, but let’s not derail our discussion by entertaining that notion; and anyway, if that is to be the case, then it’s moot to argue about how many non-existing humans would be research mathematicians). So the PhD students hear: there’s no point in finishing a degree; leave grad school now while you still can. And correspondingly, an undergrad considering grad school should think of something else to do. I want to think such arguments through in three stages:

Stage 1: (from the bottom up): Will math actually exist as a thing humans do for money? (My claim: Yes.)

Stage 2: (top down): Given that math expertise will exist, and hence that we will want to hire more people with that expertise, what is it that we’ll be looking for when we hire?

Stage 3: Whatever the answer is to Stage 2, that expertise is what we should train PhD students to develop.

[See the end of the document for a logical map of this conversation.]

Stage 1

TK: I am profoundly worried about the future of mathematics as a profession, and I sometimes lie awake at night wondering whether, in a few years, there will be any jobs for human mathematicians, either in academia or outside it. I have friends, colleagues, and students who are currently on the job market; for them, these worries are particularly immediate.

Before we begin, I want to make clear—as I’m sure you agree—that this is not just an abstract philosophical exercise. I want us to articulate a point of view that I hope will become widespread, and that entails a future in which human mathematicians don’t just exist, but are crucial. The thought experiment lies in the assumptions we allow ourselves to make, with the goal of arriving at conclusions that are as robust as possible.

With that in mind, I am happy, for the sake of this conversation, to take “math is solved” as an axiom.

AK: Great, let’s put down exactly what we mean:

The Solved Axiom

Any problem that can be solved by human + AI can be solved by AI alone.

I’m reminded of early experiments with ‘centaur’ chess tournaments, in which one human–computer team played against another. Pretty soon, computers became so strong that whenever a human departed from the computer’s top suggested move, their team tended to lose. And obviously that’s no fun, so we don’t see many such tournaments anymore.

Now, I don’t actually believe that the Solved Axiom is true today, nor do I believe it will be in the foreseeable future! But I feel compelled to grant it for the sake of argument. Even with this axiom, I still want to argue that math is not dead and that human expert mathematicians will continue to be in demand in society.

In fact, let’s give the premise I’m arguing against a name:

The Doom Premise

Assume the Solved Axiom. Then there is no longer any demand for professional human research mathematicians or human mathematical expertise.

There is a related prediction that people often seem to make, and I think it is worth stating separately.

The Training Premise

Assume the Solved Axiom. There is no longer any demand for training in mathematics for PhDs, undergraduate majors, and so forth.

I should of course admit that I neither believe the Doom Premise, nor do I want it to be true, and so, to make my argument against it as strong as possible (and to protect myself, as much as possible, from wishful thinking and other biases), I’ll need to argue purely logically and dispassionately, granting as much as I can to the “Devil’s Advocate” and relying only on pragmatism: not what I wish would happen, but what I see as most likely to happen when the dynamics of voluntary action play out. So, I think the strongest version of my argument against the Doom Premise grants the Solved Axiom (which is anyway an assumption in the premise), and continues from there.

TK: Even if only a weaker version of the Doom Premise ends up being true—for instance, a dramatic reduction in federal funding for mathematics—the consequences could be severe. But I agree that we should test our argument against the strongest version of the Doom Premise, for the sake of robustness. I would also like to isolate one further possibility, concerning the economic support of mathematics, and make it explicit.

The Funding Premise

Assume the Solved Axiom. Public agencies, philanthropic organizations, and corporations will no longer fund mathematical research conducted by humans.

We will need to talk about the Funding Premise more, but let’s address that a bit later. For the sake of the conversation, we can expand the Solved Axiom to mean that the AI is much better at explaining to humans what it has done than any human interlocutor, that it is better at posing the next natural directions of research, and that it guides its own research program much better and faster than any team of humans could.

AK: I would make a distinction between the production of new raw knowledge and the capacity to teach humans. (In fact, I will later argue that humans, in the main, do not want to sit for hours a day staring at a screen trying to learn, as enticing as those screens may be… [TK: Let’s see if we agree or disagree about this when we get to this topic!]) But sure, yes, let’s say that AI is also a better teacher (of anyone, including humans), as well as more knowledgeable and more patient than any human. (This is probably already true today! It’s just that I’ll argue later that people will not want to work with AI the same way they’d work with a human tutor).

TK: I agree that the Solved Axiom does not describe the present state of AI, and it is unclear whether current AI architectures could ever satisfy it, or whether something fundamentally different would be required. Nevertheless, I don’t want our argument to depend on predictions about the future development of AI. I would much rather arrive at conclusions that remain valid even if the strongest version of the Solved Axiom eventually becomes reality.

AK: Great, yes, let’s accept it in all of its forms. Then my first claim—and I will want to hear your pushback once I’ve had a chance to argue for it—is:

Expertise Demand Theorem

Assume the Solved Axiom. Even in that world, there will still be demand for human mathematical expertise.

This, of course, is a direct negation of the Doom Premise. If we can successfully establish this theorem and hence human experts are to continue to exist, well, they’ll need to come from somewhere. So we’ll follow that up with the corollary:

Training Demand Theorem

Assume the Solved Axiom. Humans will want to be trained to become experts in mathematics.

And this, of course, is the direct negation of the Training Premise.

TK: I’ll be very interested to see how these theorems will be established. I think my greatest worry for now is that there will be a collapse, or at least a rupture, in the process of training mathematicians, after which recovering a culture of humans doing mathematics will be impossible. We could see something like this play out: universities broadly decide that mathematics departments and mathematicians are unnecessary, both as researchers and as teachers; funding agencies decide that mathematics research is not worthy of societal support; and loud and powerful voices in tech decide that mathematicians (and indeed most human intellectual activities) are obsolete.

Have you seen that the University of Chicago considered (though, it seems, did not implement) plans to eliminate large parts of foreign language instruction and replace it with ChatGPT? See Ando (2025).

AK: Some universities may well make such choices! (But why in the world would I pay tuition to U of C just to sit on GPT and learn French?) I’m not against anyone making such decisions (of course I actually am! But I don’t need to be for the sake of the argument I’m trying to make); everyone should make their own decisions and face the consequences. My feeling, which I will try to argue and elaborate on below, is that universities that choose not to downsize their mathematics offerings will, over time, do better than those that do. As long as not everyone at the same time makes the (terrible, in my opinion) decision to close down their math departments, and as long as some professional expertise still exists, then the value of that expertise will have a chance to prove itself in the marketplace of ideas and practice.

TK: I think a lot of work is being done by “not everyone at the same time.” I think it is incumbent on us as a community to have a compelling argument to counter the inference “math is solved, therefore human mathematicians are unnecessary,” both to prevent everyone from making the same terrible decision and to ensure that the notion that mathematical training is not just useful but a good for both individuals and society remains widespread. Rich Schwartz (2026) has a particularly nightmarish view of this possibility.

AK: I agree completely that if all mathematicians simultaneously ceased to exist, it would take a long time and a lot of work before (if ever) we could regain mathematical culture and practice. I just don’t believe this will happen unless many people all make the same bad decisions at the same time. An analogy is to arguments 80 years ago about whether children need to learn their times tables, given that mechanical calculators can now do arithmetic. Some educators did—and some still do—argue that children need not memorize them. Those children and their teachers listened, made bad decisions, and suffered the consequences. For any children under my care, I insist that they know their times tables, and I (hopefully) then watch them thrive mathematically. The outcomes of such decisions are not seen immediately; they play out over years and decades. So here, too, it will take time for people to see the consequences. If some universities choose to close down their math departments, they might not immediately fail; but I believe (and will argue more precisely when we get to it) that those universities that choose to invest in their math departments will indeed do better, and hence professional math expertise will continue to exist.

TK: A decades-long process seems to me like it could be enough time to cause a full-scale break in the continuity of mathematical culture, from which recovery would be extremely difficult, if not impossible.

AK: I agree that if many people all independently make very bad (from my viewpoint) decisions, then this is a possibility. I just don’t think there’s a high likelihood of so many bad decisions being made. Some people will “hold out,” and we’ll be able to see that those who do excel, and so others may rethink their decision. There’s nothing I can do to counter such decision-making except to speak publicly about why I think math should continue to be supported.

TK: Great, so let’s figure out how to convince people. One immediate complication I see is that adults deciding whether or not to study higher-level mathematics are not children, and they will inevitably be influenced by many more voices and opinions, some giving good advice and some giving bad advice.

AK: I agree! The important thing is for there not to be just a few people charged with making such decisions for all of us. Each university president would need to make the decision independently, not follow some edict from the Department of Education (or whatever). Even if all NSF funding disappears, that doesn’t fundamentally change my argument. In fact, I’m willing to grant the full Funding Premise: suppose all external funding for human mathematical research disappears, including support from industry and philanthropy. Sure, let’s assume it! I still think we’ll be fine. [TK: I’m less certain that we will be fine, but I’m willing to consider that possibility for the sake of the conversation.] Of course, I would much prefer that such funding continue, both for my own selfish sake and because I believe it is actually valuable for society. Etingof and Gerovitch 2026 describe human mathematicians as an ‘intellectual reserve,’ which I think is an excellent way of expressing at least one reason why preserving human mathematical expertise matters.

TK: I also like the idea of intellectual reserve. Building on that, one point that I made in my essay about graduate training (Koberda 2026) is that retaining human experts in mathematics is crucial for preserving human epistemic self-governance, which I suppose could be summarized as follows: we need to understand math if we want to have any idea of what is going on in the world around us. I am curious what you think about this and whether it could provide a basis for an argument that would convince society that training human mathematicians is a valuable activity.

AK: I guess where we differ is that I’m not—in the argument I have in mind—trying to convince “society” of the importance of training humans. Not because I don’t think it’s important (I think it really is! And I do everything I can to do it!), but because I don’t think it’s necessary for my argument. I think that, as long as individuals are capable of making choices and seeing the outcomes of those choices, there will be people who choose to study mathematics, which means they will want to learn from expert human mathematicians, which means there will be demand for such experts.

TK: Let’s expand the point about learning from expert human mathematicians first. You said earlier in this conversation that you believe that humans will definitely prefer learning from humans rather than from AI/computers. Let’s clarify this.

AK: Yes, let’s make this an explicit axiom:

Learning Axiom

On the whole, human beings who want to learn deep mathematics will want to learn it from other humans.

TK: We are mostly talking about mathematics in this conversation, and so we should keep our conversation focused, though I would not argue that this axiom is specific to mathematics.

AK: Yes. But first: the Learning Axiom does not mean that there won’t be “Ramanujans” in India who iterate with an AI at their public library and deeply learn automorphic representations; in fact, I do believe that such things will now be possible, and that AI has the potential (though this is certainly not guaranteed!) to democratize access to the highest levels of knowledge for people who are geographically, socially, or otherwise far removed from experts. But my claim is that most people will not be able to do this; we are basically furless apes, and we want to be around other humans, talk to humans in person, learn from human experts, watch our human peers struggle and overcome, and enjoy all the social things that come with that. My biggest evidence for this axiom is COVID: we put not AI but actual live humans on Zoom screens in front of millions of children, and the experience was, on the whole, enormously unpopular! See also [as suggested by Chris Connell] the flop of MOOCs [Massive Open Online Courses].

TK: My worry about accepting the Learning Axiom is not that it contradicts the observation that students (from elementary school on up) hated learning even from other humans on a computer screen, much less from a chatbot. Rather, under an axiom as broad as the Solved Axiom, why would one rule out the possibility of an AGI that has “hacked” the social circuitry of the human brain and is a much better conversationalist than any human could hope to be, to the point that even the most AI-averse human would concede that they feel heard and understood by the machine better than by most, if not all, of the humans they know?

AK: I’m perfectly happy to grant that: for humans who have the self-drive and desire to iterate with an AI, they will learn better than humans who learn from social contexts. I just don’t believe that most humans have that self-drive, and most would be much happier (and thus would voluntarily choose) to be around other humans. That’s what the axiom is trying to capture.

TK: I do think I have to disagree here, though. I am not convinced that a pedagogical culture prioritizing interaction with other humans will survive automatically; I think it may need to be protected with guardrails. We can probably agree that completely unsocialized humans who only rarely interact with other humans are undesirable, but I do not think we can simply assume that such a future will not happen.

AK: Again, what I’m really trying to do is not argue for the things that I hope will happen. I do hope that my conclusions will happen, but my motivation is instead just to follow, to the best of my limited ability, logical deductions from the most basic axioms I can think of, wherever they may lead. Some of the axioms I don’t actually believe but am willing to grant (e.g., the Solved Axiom); others I very much believe to be true but don’t know a more basic reason for them to be true, and they’re necessary for my argument, so I am forced to take them as truths I hold to be self-evident (e.g., the Learning Axiom).

As to whether human interactions should be prioritized with guardrails: I disagree. I think people will just want these things, because it’s our nature. We shouldn’t need any formal guardrails; nature has given them to us.

TK: I hope you’re right and that people will realize that socialization through interaction with humans is necessary for human flourishing, but I still retain my caution here.

AK: As with other dynamical systems, I think some people will choose to sit in their rooms alone, iterating with AI. They’re welcome to do that, even if it may be to their own social detriment! I just don’t think they’ll be as happy as people who are socializing. And so I am claiming that people will voluntarily (again, for the most part) choose to participate in experiences where socializing is plentiful.

TK: Great, so with the Solved Axiom and the Learning Axiom, why do you think that universities will continue to exist?

AK: Okay, I think I need one more axiom.

Value Axiom

People who are capable of deep, extended thought will, on the whole, have richer lives than people who are not.

This does not necessarily mean that they’re economically richer (I do believe that as well, but don’t need it for my argument). [TK: So maybe more intellectually and emotionally fulfilled? AK: Yes; in terms of general human flourishing.] What I actually believe is much stronger: people who are capable of deep thought will be much better off (in the main) due to the existence of technology that will make their skill even more valuable. But again, I don’t need this stronger form for my argument.

TK: And so, from the Value Axiom and the Learning Axiom, you want to infer that universities that train human intellectuals (and mathematicians in particular) will continue to exist.

AK: Yes, exactly! I suppose I may need another axiom: that one of the best ways to train one’s mind to be capable of deep, sustained thought at this highest level is the training one experiences (or should!) in a math PhD program.

Do you not want to argue against the Value Axiom? In some sense, it goes counter to the Solved Axiom. Doesn’t the latter mean that humans have no value to add? Hopefully the general sense of the words “richer lives” is doing the bulk of the work. Does that make sense?

TK: Yes, I do think that there is a tension between the Value Axiom and the Solved Axiom, though how much depends on how broadly we interpret the semantics of these axioms. In a solved world, one could suggest that there are other ways of cultivating deep, extended thought and achieving the attendant intellectual and emotional fulfillment. Why would this have to be through mathematics at all, especially if all the intellectual paths are well-trodden and completely understood by the AI? A lot of the friction necessary for developing mathematical skill is unpleasant, inefficient, and sometimes even a waste of time. Why would we not look somewhere else to enrich their lives? Without implicitly acknowledging the necessity of something like epistemic self-governance, how do we argue that the Value Axiom and the Solved Axiom are compatible?

AK: Okay, so let’s make the other axiom explicit.

Training Axiom

One of the best ways to cultivate the ability to think deeply and clearly is through the study of pure mathematics, the more abstract the better.

The evidence for this includes the fact that Abe Lincoln apparently studied Euclid to sharpen his rhetorical skills¹. I agree completely that math isn’t the only path to clear thinking, but it’s a pretty darn good one. Even if writing down solutions to mathematical problems need not be difficult anymore (because AI can do it), students who choose to do the exercises as intended, with all the friction, frustration, confusion, etc., are those who will actually achieve the effects of the training. (This will all be discussed in detail in Stage 3.)

TK: So, as I understand it, your argument is fundamentally pragmatic. [AK: Yes!] There are perhaps many ways to achieve the ability to think deeply (in my essay I put this in a somewhat broader category that I called “intellectual agency,” in order to distinguish it from “independence,” for instance), but mathematics already exists as a path to this goal, and so we might as well choose that path.

AK: I agree completely. First we have to establish that the ability to think clearly is a goal (Value Axiom), then that learning mathematics is an efficient mechanism for attaining that goal (Training Axiom), and finally that people will want to undertake this training in person (Learning Axiom).

TK: I am still uncomfortable without a stronger justification, but, more seriously, I worry that this does not fully justify why the Value Axiom and the Solved Axiom are compatible. More specifically, how do we justify the goal? I feel like there is some implicit acceptance of the necessity of epistemic self-governance on the part of humans.

AK: You’re right on both counts. I don’t know how to justify the Value Axiom from more basic principles, hence its status as an Axiom. Do you think you’re willing to grant it? (In terms of human self-governance, I do hope that we continue to live in an open society where people are, largely, free to choose what they want to do with their time. Such questions are really important, but I think they are outside the scope of our discussion.)

TK: I would like to grant it, but I think that the following axiom would be more fundamental, and I would welcome your disagreement:

Epistemic Axiom

Humans must retain deep understanding in important domains of knowledge and remain capable of functioning as independent intellectual agents; this is not to say that humans must be able to function without AI, or that they should not use it. More simply, humans should be able to think for themselves sufficiently well that they can understand and evaluate non-human intellectual activity.

I would say that the Value Axiom could be a corollary of this one. Moreover, it would seem to justify accepting the Value Axiom under the regime of the Solved Axiom.

AK: If you see the Value Axiom as a corollary of the Epistemic Axiom, then I’m perfectly happy to grant the latter and get access to the former! I don’t believe that there is any value in a massive database of mathematical theorems and proofs—even correct, well-written, and formally verified ones—that exists only as 0s and 1s inside data centers, but not in any human heads. That is, using AI to create new mathematics for its own sake, without having any human understand it, will not be something people do, certainly not once the novelty of it wears off. A point of comparison: millions of kids around the world will dress up on Halloween as Harry Potter because they’ve read the books, watched the movies, and been deeply moved by the stories, which have brought great joy to children (and adults!) around the world. It is easy to imagine, especially in light of the Solved Axiom, that AI could write thousands of stories that are every bit as moving. If we let it write those stories but no human ever reads and enjoys them, what was the point? The required energy expenditure would be a complete waste; who would pay the electric bill? Similarly, I claim that mathematics is only valuable insofar as it exists in the brains of humans. Reading Flexner’s “The Usefulness of Useless Knowledge” (Flexner 1939) (written in 1939 to explain why someone might want to support a “useless”-knowledge-generating institution like IAS), I think that all of his arguments fall apart without changing the title to “The Usefulness of Useless Knowledge in the Minds of Humans.” Who could have known in 1939 that this add-on would become necessary!

TK: I think that we mostly agree here, though I might soften the formulation slightly. It was already true before AI that mathematics had to be alive in order to realize its value, which is to say that it had to form part of human mathematical life. A theorem sitting unread in a library may still have potential value—it could be rediscovered decades later—but that value is realized only when the mathematics is understood, used, taught, extended, or otherwise brought back into human intellectual activity. Here I would emphasize that the Epistemic Axiom is playing a key role, since otherwise one could argue that mathematics sitting in some data center is still useful to an AI that then uses it to develop some other new technology that benefits humanity (or possibly serves the AI’s own ends).

Historically, areas of mathematics have gone in and out of fashion, and the skills and knowledge possessed by practicing mathematicians have evolved over time. Inevitably, some knowledge has been lost. This is one reason it is so important to write clear papers, and why exposition that organizes bodies of literature and makes them usable to researchers approaching from the outside is so valuable. It is a shame when mathematical writing languishes indefinitely in libraries or preprint repositories. Part of the work of mathematics is to keep such knowledge alive by absorbing it, rediscovering it, and teaching it to the next generation of mathematicians.

I think this gives us a great segue: how do we now infer the future necessity of universities? I do agree that universities provide a lot of the infrastructure (at least ideally) that is conducive to training, including fostering intellectual culture, enjoying relative independence from commercial pressures, creating opportunities for socialization, and providing long time horizons. It might be the case that, in the future, competing institutions will arise that do these things as well or better, and that would be welcome, but this might be more of an argument for not immediately abolishing universities than for the more positive conclusion that “universities will persist in the future.”

AK: For my purposes, the very definition of a “university” is a place that “provides a lot of the infrastructure that is conducive to training, including intellectual culture, relative independence from commercial pressures, socialization, and long time horizons,” and a place where experts congregate for the purpose of producing new (and retaining old) deep knowledge and disseminating that knowledge to students eager to learn it. To the extent that existing institutions currently called universities do these jobs, I think there will continue to be demand for them. If they don’t do these jobs, or if new institutions arise that do them better and students choose to go there instead (I’m all for it, for the sake of argument), then which institutions exist will change, but what they do will not. [TK: I agree with this.] So the main point of this line of reasoning is to finally prove the Expertise Demand Theorem, and hence its corollary, the Training Demand Theorem (which, now that I think of it, provides a positive feedback loop into the Value Axiom + Training Axiom). In fact, my personal opinion is not just that universities with mathematics departments will continue to exist, but that there will be even more demand for mathematicians in the future than there is today, precisely because the training by such experts will be so valuable. But I don’t want to add this stronger argument to make my main points here.

Are we ready for Stage 2? Given that universities (maybe not all the ones that currently exist, and maybe some new ones that haven’t been invented yet) will need to continually replenish their collection of experts, what exactly would hiring committees be looking for? What do we, in the world of mathematical culture and practice, value? (That, of course, will feed into Stage 3: what one should be trained in during a PhD.)

Stage 2

TK: Yes, I think I am satisfied with our answer to why universities (as defined above) will continue to exist. Now, as to why they would continue to hire mathematicians and what will be valued in hiring: at least the first of these seems to have an answer that follows from what we have said. If we accept that there will be demand for humans to be trained in mathematics by human experts, then universities will presumably hire human experts in mathematics simply to meet that demand. What is less clear to me is which criteria will be used to evaluate and hire mathematicians, which is precisely the values question. I think this ties in very closely with the question of what the purpose of graduate education should be, since presumably graduate education should function to train human experts. The nuances of the function of graduate education are numerous, and I made some effort to clarify them in my essay. It seems safe to assume the following:

A mathematician hired, retained, and promoted by a university will be expected to have intellectual agency, as defined and elaborated in my essay.

The mathematician will be expected to have contributed to human understanding of mathematics. This is a broad category, which would include traditional metrics such as original papers. Part of the issue at hand, of course, is that the existence of papers authored by the mathematician does not serve as the reliable proxy for intellectual agency that it did until recently, which is one of the reasons I put intellectual agency first.

A mathematician will be expected to have excellent skills in communicating mathematics to others and continued evidence of contributing to human understanding of mathematics. Communication of mathematics can include mathematics created by AI, but the human mathematician should play a critical role in translation.

AK: I would prefer not to assume these things, that is, not make them Axioms, but rather conclude them as Theorems from more basic premises.

For point 1, that’s part of our working definition of what a university is, so I think it does not need to be a separate assumption. (In Lean-speak, the proof is rfl, that is, by reflexivity.)

For point 2, we can argue by the contrapositive: what would it mean if mathematicians were not expected to contribute to human understanding of mathematics? In light of the Solved Axiom, this presumably means that the so-called “expert” (perhaps we should just switch to using the standard word for it, Professor?) has pushed some buttons in GPT42, produced a slew of papers solving longstanding problems, and put them on their CV, while having no clear, deep understanding of the mathematics those papers claim to establish. Why would a hiring committee make an offer to such a person? I can press a button in GPT, too! (Remember when “Prompt Engineer” was a career, for a hot minute? Now amateurs type “do a brkthru” [internet subculture slang for prompting AI to prove a breakthrough result] and actually get one! No thanks.) It usually takes around 3-5 years of hard work to first internalize some deep piece of difficult mathematics. It takes 3-5 minutes of a job talk to determine that someone has not done this work.

For point 3, if the reason the university exists is that students are paying tuition to go and learn (cf. Learning Axiom), then a Professor who does not communicate mathematics well is not useful or valuable to the enterprise. (Moreover, how could someone communicate effectively with critically thinking students who can ask penetrating questions at any time if they do not themselves understand the material deeply? This reinforces point 2.)

TK: Agreed! This is why I am saying it should be safe to make these assumptions; maybe infer is better than assume. [AK: Okay, great, then we’re fully in agreement.] Why don’t we call these three points the Hiring Value Theorem? Of course, as hiring seasons proceed in succession, we will eventually gain an empirical demonstration or refutation of the Hiring Value Theorem, since if hiring does actually continue in the future, some people will secure positions, and the community will be able to infer ex post facto what was evaluated.

AK: Yes, let’s state the theorem explicitly.

Hiring Value Theorem

In order to be hired, retained, and promoted by a department of mathematics, a person will need (1) to understand deep, difficult mathematics, and a lot of it, and (2) be able to communicate that understanding effectively to others, including verbally!

The oral part is the new focus of attention. For thousands of years, if you could communicate in written form, that was a nearly perfect proxy for deep understanding. (I suppose you could’ve paid someone to write something for you, then crossed out their name and put your own. I’m looking at you, l’Hôpital! It’s Bernoulli’s Rule, not yours!) Moreover, almost the opposite was true: if you were a brilliant communicator of your ideas but failed to document them in written form, you would not be contributing to the scientific literature and hence could not get a job! Since AI systems can (in the world of the Solved Axiom) write amazing, perfectly well-written texts, the correlation between producing written arguments and understanding has plummeted to zero. The written argument may remain every bit as mathematically valuable, but what has changed is its value as evidence about the person whose name appears on it. One particularly effective way to demonstrate, in practice, that you really understand a piece of mathematics deeply is to give a lecture on it to experts (and, of course, also publish a corresponding paper in a good journal; I don’t think we want to discard the scientific literature!) and adequately field all of their most penetrating questions. This is a really important point, I think, so let’s give it a name:

Dialectical Understanding Axiom

Assume the Solved Axiom. Then one of the principal ways a person can effectively demonstrate to another that they understand a piece of mathematics deeply is to explain it orally, and respond coherently to questioning.

This certainly does not mean hostile questioning. One of the things I love most about math culture is how open we are to this kind of dialectical questioning (just like what you and I are doing right now!). In fact, I’m very careful when early graduate students are exposed to it for the first time: it takes a lot of experience to get used to hearing someone say, “No, what you’ve said is wrong, and let me try to argue why,” and being not only perfectly fine with it but actually valuing it (because it means you’re about to learn something new). Indeed, it’s radically different from the norms of conversation outside mathematics—as I’m painfully reminded from time to time whenever I instinctively respond to a non-mathematician in an unintentionally rude-sounding way. Our most precious expression is even called Proof (or Argument), so we’re supposed to have that dialectical process happen internally before it even gets to the world outside ourselves!

TK: I feel that at this point it is important to note that there are dangers lurking here, especially in how the Hiring Value Theorem is applied in practice. This has to do with the inevitable shortage of positions relative to the number of qualified candidates, and so there will inevitably be questions about what is judged more deserving of attention from human mathematicians. Relatedly, there is a hazard of exacerbating inequalities that the community in general has already fought hard to correct. There is a danger that a small handful of people could become authoritative arbiters of a nebulous and opaque measure of importance that we might call “taste.” If we evaluate job candidates based on oral performance only [AK: Or even mainly.], we need to make sure we do not simply reward the eloquence of native-language speakers, for instance. These are all important and pressing questions that I want to flag, though I don’t think they are the central concern of this discussion.

AK: I agree completely. Oral evaluation inevitably introduces various irrelevant human advantages—eloquence, confidence, linguistic background, attractiveness, social ease—and I don’t like that. But I don’t see how we could eliminate them completely. We should nevertheless work as hard as we can to evaluate deep mathematical knowledge and communication on their merits.

TK: Yes. Irrelevant human advantages make me uncomfortable, but I also do not have a good fix beyond vague suggestions like “transparent standards” and “multiple evaluators.” Working out those details is an important task for the mathematical community.

AK: Honestly (and this is perhaps getting us off track), I also don’t see a way to “fix” this. Those who choose to abuse the system and select more for, say, “height” or the “right regional accent” (especially in the U.S.) instead of “mathematical knowledge and communication” will, over time, have a less highly ranked department. So the dynamics in the system should already incentivize meritorious hiring.

TK: Should we leave this part of the conversation for another day and move on to Stage 3?

Stage 3

AK: Yes, great, let’s! So finally we’ve recursed far enough back to reach the point of this conversation! What is the purpose of a PhD education? Certainly one of its purposes should be to prepare someone for the possibility of an academic career (simply because there’s almost no other way into academia). (Of course, even pre-AI, the vast, vast majority of people who received a math PhD did not go on to a tenured professorship. Those who left academia often found their skills in deep, abstract thought very valuable, gaining meaningful employment, making much more money than professors, and being able to choose where they want to live; we silly professors have the funny habit of calling this “failing” to get an academic job! [TK: It’s part of our cult-like academic culture!] Anyway, now I’m rehashing the arguments for the Value Axiom, which we’ve already agreed to assume.)

Given that this is the case, how does one train a student in deep mathematical knowledge and oral communication (since that’s what hiring committees will be looking for)? I think the answer is pretty simple: not all that much needs to change! What we value is deep understanding of difficult mathematics, and acquiring that understanding still takes years of really hard work, with AI help or without it. What may change radically is how we determine whether someone has actually acquired that understanding.

Pre-AI, a typical five-year PhD student experience was something like this: two years of preparatory work, coursework, reading books, and doing written and oral exams; followed by two years of unadulterated research on a problem (or a few such), starting to go to conferences and learning the field—what questions people are interested in, and so on; and then the last nine months were spent writing up a proof in complete detail, getting everything down on paper, learning how to organize an argument, and putting the finishing touches on a thesis; and finally, defending the thesis, which is basically a nice, easy talk on your work (which your advisor, if they know what they’re doing, has already certified as sufficient to warrant a PhD).

So the written thesis was a nearly perfect proxy for this: the student had done a significant amount of hard work, learned deep mathematics, contributed something new to our collective understanding, and struggled to prepare a carefully written document. Under the Solved Axiom, you can now get such a document with the push of a button. So how should one prove that they’ve learned a great deal of mathematics? We’ve already covered this: the Dialectical Understanding Axiom!

This means we can finally state the main argument I was hoping to make:

The Dialectical Theorem

What you should do in grad school is learn a ton of math and, as much as you can, expose yourself to good-faith questioning to ensure that you know your subject inside and out and can effectively communicate it to others.

Later I’ll give some more concrete suggestions for how to do this.

TK: I think you make good points here. Thinking about difficult problems that resist many attempts at solution creates a type of friction that requires persistence, patience, and personal growth to overcome. Of course, personal growth does not mean that everything has to happen in isolation; indeed, it might be even better if the process is dialectical. The thesis was a reliable certificate that this sort of friction was encountered and the corresponding growth occurred. It seems that some sort of cognitive friction and engagement with difficult problems is the principal path to achieving what we are calling intellectual agency, and it is unclear to me whether there are effective alternative paths, or whether any amount of simplification, easing, or expediting of the process could be achieved with AI. [AK: Agreed! Maybe in the future, we’ll have Matrix-style brain plugs; but until then, I also don’t see alternate paths!] One idea gaining currency is that there are two main modes of training that one could implement for students, namely independent [AI-free] and augmented [with AI allowed] modes. Such a formalized distinction would give a framework for distinguishing skills that need to be acquired through cognitive friction from those that can safely be offloaded at a later time, so that AI does not simply become a veil obscuring a lack of deep understanding.

AK: I like that! As for how to use technology (AI) well, I like to tell my students the following three vignettes:

Say you want to move some heavy pallets from a warehouse onto a truck. You’re not going to do that by hand. So you grab a forklift and drive it over, pick up the pallets, and load up the truck.

Say you want to learn how to use a forklift. So you get in your forklift and drive over to some pallets in a warehouse, pick them up, and move them to another part of the warehouse, not because you wanted to move them, but because it was good practice in the use of that technology.

Say you want to get a good workout and get stronger. So you go to the gym and load 200 lbs. on the bench press. Then, instead of lifting the bar yourself, you grab your forklift, drive it over, and use it to do 20 reps of lifting the bar.

I hope you agree that 1) and 2) are perfectly fine uses of technology, whereas 3) is extremely silly and counterproductive. To me, solving a hard open math problem, creating new mathematics for your thesis, etc., is akin to vignette 1. Learning how to prompt AI systems is Type 2 (which, by the way, I’m rather opposed to PhD programs deciding to teach—students will pick up prompting on their own; it’s really not that hard!—and the precious time spent in grad courses should be reserved for learning deep mathematics). Doing homework problems in a course by prompting an LLM for the solutions is Type 3. If you’re using AI to answer homework problems, then ask yourself what the point is. That said, even in Type 3, you could choose to use technology meaningfully: in the gym, you can ask AI to help you work out a training regimen, ask it questions about good diets, etc., etc. In the same way, you can ask an AI to clarify something the professor said in class that you didn’t understand, give you more practice problems, etc., etc. Just don’t use it to bypass the difficult, friction-filled stuff!

TK: Yes, the vignettes make sense, though I’d like to elaborate a bit more, especially on Type 3. I think a stronger analogy with the goals of mathematical training would be that the task at hand is to get physically stronger, stay strong, and move heavy pallets laden with gym weights from a warehouse, all at the same time. To get stronger, you need to do physical exercise so that your body adapts. Once you have gained strength, you will want to maintain that physical strength, but that does not mean that you have to transfer a warehouse full of pallets by hand. Moreover, individual variations in desired levels of strength, needs for maintaining health, etc., would inform when it is better to do lifting by hand and when it is better to do it by forklift. This highlights the individuality of what should be considered “premature cognitive offloading” versus “productive cognitive offloading.”

I would probably add that graduate students need to get to know themselves and collaborate with their PhD advisors to learn what constitutes premature and productive cognitive offloading when it comes to AI usage.

AK: In my experience, it’s very easy in practice to tell whether what you’re doing is premature or productive. There’s an angel on one shoulder and a devil on the other, and they both speak very clearly: one says, “Hey, you shouldn’t be asking for this answer; you know you should try to figure it out yourself!” and the other says, “That was a good practice problem for us to iron out; good job getting the AI to ask it!”

TK: I’ll take your word for it. [AK: Ha!] As for recognizing this in yourself, I might suggest that this is a learned skill that follows from already being highly trained as a mathematician and that it need not be automatic.

AK: Fair enough, I’m happy to concede that! Actually, with my grad students, I can quickly tell when they’ve gone off, pushed a button in AI, written down what it said, thought about it a little, but didn’t really try to argue all sides of the claim. So when they come to see me the following week and I start asking basic questions, they often turn out to be questions the student hasn’t considered, but should have if they had interrogated their own arguments fully. To be fair, this was also sometimes an issue pre-AI, but it’s perhaps even more prevalent now.

TK: For now, at least, it seems that this is one of the concrete benefits of having humans train other humans: they can teach graduate students to interact more critically with AI. [AK: Agreed!]

So what do you think university mathematics departments will look like in the future? Will there be fewer jobs with even fiercer competition? Or do you think universities will be incentivized to hire even more mathematicians than they do currently?

AK: Well, I’ve tried to keep this discussion to basic things I can derive from what I hope are well-formed Axioms, and to conclude only the weakest form of “the world [mathematical or otherwise] will not end.” But since you’re asking, I’m personally extremely optimistic about what math will look like in the future. And as I see it, unless we all make a long series of very bad decisions, independently and one after the other (which I find highly unlikely), we are headed for a really enjoyable world of plentiful mathematics, with lots of work to be done to digest, understand, and synthesize beautiful arguments. [TK: Which could be either AI- or human-created, of course.] So I would hope that we’ll soon see much greater demand for many more mathematicians than we currently have! What do you think?

TK: I think that what you are saying is that if AI dramatically increases the amount of mathematics humans can produce, understand, and use, then demand for human mathematical expertise need not shrink with the amount of labor required for any one task; it might even grow. [AK: I suppose that depends on the demand function, but yes, I do think something like that will hold.] I hope that this is valid, because this is the most optimistic view of the future of which I can conceive. I think our conversation has made a compelling case that human mathematical understanding will remain valuable, even under the strongest assumptions about AI. What I remain less certain (and even quite pessimistic) about is whether the institutions and economic incentives necessary to sustain a thriving mathematical profession will endure. My worry, as I’ve articulated earlier, is that too many universities and individuals will make decisions that, in the aggregate, become very difficult to reverse. This discussion is a small part of a much larger story that is playing out and whose conclusion is far from certain; see for instance Abbes and Miller 2026. That is partly why I think we need to have this conversation now, rather than waiting for the consequences of those decisions to become clear. I hope our conclusions stand the test of time.

AK: Me too! There’s one more thing we haven’t quite done yet. We’ve talked at length about what graduate training should accomplish, but perhaps we should finish with a few concrete suggestions for students who are wondering what they should actually do now.

One thing they can do right now is to make a habit of explaining the mathematics they’re learning to other people. Take a theorem you’re working through, explain it at the blackboard without notes, and encourage your fellow students to ask difficult questions. Why is that hypothesis necessary? What happens in particular examples? Where might the proof break down? These are things you can practice long before you have a thesis to defend.

TK: I would add that students should deliberately make time to work independently, even when AI assistance is available. One good exercise might be to return to an argument that an AI helped you understand a few days earlier and try to reconstruct it without assistance. If you cannot, that tells you something important about what you still need to learn. Of course, how much independent work is appropriate will vary between students and between different stages of their training.

AK: And I would encourage students to participate as much as possible in the mathematical life around them: attend seminars, attend the weekly department colloquium, attend conferences and workshops, ask questions, argue about proofs with their peers, and learn mathematics well enough to explain it to someone else. These things have always been part of becoming a mathematician, and I think they will be more important than ever.

TK: I agree. And I think we should emphasize that advisors and departments have a responsibility to create environments in which students can do these things. Especially early in their training, students need opportunities to ask questions, make mistakes, and have their understanding challenged without feeling that every interaction is an examination.

I think this is a good place to stop for now. Thanks for the conversation, it has been great!

AK: Thanks for chatting!

A logical map of the conversation

The arrows below record dependencies suggested in the conversation, not formal logical implications. The Solved Axiom is a standing assumption throughout, while the Doom, Training, and Funding Premises are adverse predictions considered as part of the thought experiment. Several of the other axioms are deliberately provisional.

From (NYTimes Sept. 4, 1864): “At last I said [to myself], ‘Lincoln, you can never make a lawyer if you do not know what demonstrate means;’ and I left my station in Springfield, went home to my father’s house, and staid there till I could give any propositions in the six books of Euclid at sight. I then found out what ‘demonstrate’ means, and went back to my law studies.” ︎

References

Abbes, Ahmed and Haynes Miller (Oct. 9, 2026). The AI Invasion Has Come for Mathematics.

Ando, Clifford (Aug. 14, 2025). The Crisis of the University Started Long Before Trump. Compact.

Etingof, Pavel and Slava Gerovitch (Oct. 6, 2026). Mathematics of the Future.

Flexner, Abraham (1939). “The Usefulness of Useless Knowledge”. In: Harper’s Magazine 179, pp. 544–552.

Koberda, Thomas (2026). Training Mathematicians in the Age of AI: Intellectual Agency, Cognitive Offloading, and the PhD Thesis. doi: 10.48550/arXiv.2609.28140. arXiv: 2609.28140 [math.HO].

NYTimes (Sept. 4, 1864). “Rev. J. P. Gulliver: Mr. Lincoln’s Early Life”. In: New York Times, p. 5.

Schwartz, Richard Evan (Sept. 2026). The Fate of the Riemann Hypothesis. Short story, available on the author’s website.

Received 11 October 2026.

by Proofs Prompts at October 14, 2026 09:00 AM

Terence Tao — Call for stories: human–AI collaboration in recent mathematics

[This is a guest post by Deanna Needell and Hayden Schaeffer – T.]

AI systems are playing an important part in mathematical discoveries, and several recent preprints describe results reached with their help. What makes these results possible is often a rich partnership between human insight, expertise, and earlier work combining with new AI capabilities in ways that move the field forward.

We are interested in documenting and highlighting this collaboration, so the community can better understand how people and AI systems work together most effectively. As a first step, we would love to hear from mathematicians who took part in, or whose work helped make possible, any of these results.

If that describes you, we would be glad to hear from you at mathematiciancontributions@gmail.com. We’re interested in all kinds of involvement, such as earlier research or ideas the work built on, related projects in progress, checking and refining arguments, connecting results to the existing literature, or anything else you found relevant or meaningful about the process.

A short note about your involvement and the work it relates to is a great place to start. All messages will be kept confidential, and nothing you share will be used, quoted, or attributed to you without your explicit consent.

Thank you, and please feel free to share this with anyone who may have been part of this work.

by Terence Tao at October 14, 2026 01:30 AM

John Preskill — The Quantum Omelette Problem: How Much Energy Belongs to the Yolk?

The National Egg Coordination Committee created one of India’s longest-running ad campaigns. Its jingle was impossibly catchy. Nearly everyone from my parents’ generation onward seems to remember it:

“Sunday or Monday, eat eggs every day.”

My mum took this advice rather seriously. On weekdays, we ate boiled eggs for breakfast. Sundays, however, were omelette days.

Our ritual was always the same. Each of us siblings announced how many eggs we wanted as we streamed into the kitchen. My mum cracked the eggs into a shiny steel bowl, added our chosen mix-ins, and whisked them together. The yolks, whites, salt, and chili powder quickly disappeared into one speckled mixture. She poured it into a hot, oiled pan, where it hissed and sizzled before sliding onto our plates.

The omelettes were soft, spicy, and gone within minutes.

My diet is now plant-based, but the nostalgia of those Sunday-morning breakfasts remains strong—the steel bowl, the sizzling pan, and all of us gathered around the table, waiting eagerly for the omelettes to arrive.

I couldn’t find a painting that resembled my family around the breakfast table, much less one in the style of an old Mughal miniature. So, for once, I felt compelled to enlist generative AI to fill this rather specific gap in art history.

Our weekday boiled eggs and Sunday omelettes offer a useful contrast. In a boiled egg, the yolk and white remain distinguishable. In an omelette, they are mixed and transformed. Some properties—texture, flavor, and structure—belong to the finished mixture rather than to either component separately.

When two parts interact strongly, which properties still belong to the parts, and which belong only to the whole?

This question appears across science whenever interacting parts become difficult to describe independently. In quantum thermodynamics, it becomes a question of energy. Picture a small quantum system, an atom, say, surrounded by a much larger environment. The total energy of the setup comes from three sources: the atom, the environment, and their interaction.

When the atom barely interacts with its surroundings, separating the energy contributions is easy. The interaction energy is small enough to ignore, and we can speak cleanly of the atom’s energy and the environment’s. When the interaction is strong, however, that shared energy can no longer be brushed aside, making it difficult to decide how much belongs to the atom and how much to its surroundings.

For a large object, interactions with the environment typically occur at the surface, while most of its energy is associated with the much larger bulk. As an object grows, the surface becomes less important compared to its volume, so the interaction energy can often be ignored. But this reasoning breaks down for very small objects, where surface effects remain significant, or for systems whose interactions extend through the bulk, allowing interaction energy to scale with volume.

Such strong interactions are not limited to tiny systems; they also appear across condensed matter, nuclear, and high-energy physics. Lattice gauge theories provide one important example: their particles and fields are tied together by interactions and local constraints, making it especially difficult to draw a clean boundary around one part and declare, “This energy belongs here.”

In all these cases, the energy of the whole may be perfectly clear. The challenge, however, is deciding how to divide that energy among pieces that are no longer truly separate.

That brings us to a deceptively simple question: How should we define the internal energy of a strongly interacting system?

Definitions in physics do more than attach names to things. They tell us what counts as a system, which quantities can be measured, and how mathematical ideas connect to experiments. A useful definition must do more than sound reasonable: it should recover familiar results in familiar situations and remain consistent with the laws of physics. When two plausible definitions lead to different conclusions, the choice between them cannot be settled by language alone.

Consider the internal energy of a system. When a system is strongly coupled to its surroundings, some of the total energy is stored in the interaction, and several reasonable ways of assigning that shared energy become possible. But which definition of internal energy is consistent with thermodynamics? This is the question I explored with my collaborators Zohreh Davoudi, Christopher Jarzynski, Niklas Mueller, Connor Powers, and Nicole Yunger Halpern.

Physicists describe a system’s energy using a mathematical object called a Hamiltonian. In our recent paper, we consider three ways of defining the internal energy of a strongly coupled system.

Definition 1: Subtract the environment
Start with the total energy of the system-environment composite, then subtract the energy the environment would have on its own at the same temperature. Whatever remains, including the interaction energy, is assigned to the system.

Definition 2: Include the environment’s influence
Rather than separating the interaction energy by hand, describe the system’s energy using a modified Hamiltonian that captures how the interaction energy affects the system’s energy at equilibrium. Physicists call this the Hamiltonian of mean force.

Definition 3: Account for temperature as well
The Hamiltonian of mean force depends on temperature. The third definition takes this dependence into account when calculating the system’s internal energy.

All three definitions have appeared in the literature on strong-coupling thermodynamics, each offering a seemingly reasonable way to account for the environment’s influence on the system. We wanted to find out whether choosing one definition over another would change how we describe the same physical process.

To test the three definitions, my collaborators and I considered a sudden quench—the physicist’s term for changing something in the setup almost instantaneously. Imagine rapidly turning a knob that controls the system, perhaps by suddenly switching on a magnetic field. We can also suddenly change how strongly the system interacts with its surroundings, effectively turning the interaction between them on or off. Either change happens so quickly that the system and its environment have no time to respond: their states remain constant during that instant. Yet the system’s internal energy can change because the rules governing it have changed. Since this energy change results from deliberate external control, we call it work.

After the quench, the knob is left alone. The system then evolves under the new, fixed conditions and exchanges energy with its environment. This later change in the system’s internal energy is called heat.

Separating the process into two stages makes it easier to tell work and heat apart: work enters during the sudden, controlled change, while heat flows during the evolution that follows. It is a little like whisking an egg first and then placing it on a hot pan, rather than trying to whisk and cook it at the same time.

Once we have chosen a definition of a system’s internal energy, we can ask how that energy changes during any thermodynamic process. The first law of thermodynamics tells us that the change in internal energy $(\Delta U)$ must be accounted for by work $(W)$ and heat $(Q)$ ,

$\Delta U = W + Q \, .$

The first law is essentially a statement of energy conservation. It tells us that any change in the system’s internal energy must be accounted for: the energy must have entered as work, as heat, or as some combination of the two.

But here is the catch. Each of the three definitions of internal energy comes with its own corresponding definitions of work and heat. If we change what we mean by $U$ , we also change how we split that energy change into $W$ and $Q$ . For each of the three choices, however, the definitions are constructed so that work and heat add up to the total change in internal energy. All three therefore satisfy the first law.

At first, this seems reassuring. Perhaps any of the three definitions will do.

To decide whether all three internal energy definitions are truly acceptable, we need a stricter test: the second law. The first law tells us that energy must be conserved, but it does not tell us which direction a process will naturally take. The second law supplies that direction. An egg can be whisked and cooked into an omelette, but an omelette left on the plate will not spontaneously uncook itself, separate into a yolk and a white, and return to its shell. Energy conservation alone would not rule out such a reversal, yet nature overwhelmingly favors one direction. Cooking an omelette is therefore an irreversible process. A perfectly reversible process, by contrast, could be undone without leaving any lasting changes to the system or its surroundings.

A reinterpretation of Salvador Dalí’s The Persistence of Memory (1931) by the Instagram account @chewscagefreeeggs. A playful take on the second law of thermodynamics, irreversibility, and the arrow of time.

For the sudden changes studied in the paper, we can quantify irreversibility by comparing the work actually performed with the minimum work required in an ideal, perfectly reversible process. Thermodynamics captures this minimum through a quantity called the free-energy change, $\Delta F$ . The difference between the actual work and this ideal minimum,

$W−\Delta F \, ,$

is called dissipated work. It measures the extra work associated with carrying out the process irreversibly, and the second law says that it cannot be negative.

This is where the three definitions part ways. All three satisfy the first law, but only two satisfy the second law for the processes considered in the paper. The third definition, which accounts for the temperature dependence of the Hamiltonian of mean force, can predict negative dissipated work.

Does this mean that our quantum system has somehow broken the second law? Not at all. Rather, the third definition assigns internal energy, work, and heat in a way that makes a valid process look thermodynamically inconsistent.

The next question is why the three definitions agree in classical physics but separate in the quantum case. The answer lies in one of quantum mechanics’ defining features: observables do not always commute. Roughly speaking, when noncommuting observables describe two quantum properties, they cannot both be measured simultaneously. This has no direct counterpart in classical physics, where two observables describing different properties can be measured simultaneously. When the system’s energy and interaction energy cannot be simultaneously measured, the three definitions of work can give three different values.

To explore this connection, we considered what happens when the system’s Hamiltonian commutes with the interaction Hamiltonian, both before and after the sudden quench. In this classical-like limit, all three expressions for work give the same value, which also obeys the second law. In hindsight, the result may seem obvious. Still, there is something oddly satisfying about pinpointing precisely when the three definitions agree: strong coupling alone does not force them to give different answers. Quantum noncommutativity is what allows them to part ways.

Over the years, our family’s egg repertoire expanded. MasterChef episodes, new cookbooks, and travel brought poached eggs, Scotch eggs, shakshuka, eggs-in-purgatory, Tamagoyaki, and plenty of experiments in between. In some dishes, the yolk and white remain almost perfectly distinct; in others, the boundary between them nearly disappears. A whole spectrum exists between separation and complete mixing.

The same is true of a system and its environment: their interaction can range from barely noticeable to overwhelmingly strong. Our results identify two definitions that remain consistent with the laws of thermodynamics for the processes we studied, even when those interactions become strong.

by greeshmaoruganti at October 14, 2026 01:22 AM

Doug Natelson — Materials-related postdoctoral opportunity at Rice, 2026-2027 edition

See here.
The Rice Advanced Materials Research Institute has launched its 2026-2027 competition for prestigious postdoctoral fellowships - see here: https://rami.rice.edu/rami-postdoctoral-fellowship-program for details. The actual application link is here.
If you are interested and meet the criteria, I'd be happy to talk. I have some ideas that lean into the materials for electronics direction, and other possibilities are welcome.

by Douglas Natelson at October 14, 2026 01:07 AM

October 13, 2026

Scott Aaronson The Mathocalypse

More Updates (Oct. 12): Jeffrey Ketchersid has used Claude Code agents to show that BusyBeaver(10) exceeds Graham’s number, beating the previous best result that BB(13) does so.

In response to my request, a couple days later Jeffrey also used Claude agents to show that the value of BB(335) is independent of ZFC, improving the best previous result from Andrew Wade that BB(393) is. Recall that the first result in this direction, by me and Adam Yedidia a decade ago, only showed an independence result for BB(8000).

Meanwhile, among the many reactions to the mathocalypse, friend-of-the-blog Kunal Marwaha advocates that we become stewards of a “world mathematical commons.”

Updates (Oct. 9): To address some questions that keep coming up…

No, I’m not aware of any direct practical implications of any of these results. I certainly don’t expect the O(n log^{0.9999999999999} n)-time Fourier Transform algorithm to get practically deployed anytime soon! But anyone who harps on this, simply doesn’t understand what mathematical progress looks like, or how it eventually pulls the rest of science and technology and human civilization with it. If and when the relevant human mathematicians understood the clever new ideas in all 372 of these papers, I have no doubt that they’d be able to use the ideas to do all sorts of new things, and that some of those things would have practical value. That’s yet another reason to try to keep human understanding at the center of our enterprise—unless, of course, you’re happy to outsource the task of understanding and applying the ideas to AI models as well.

Meanwhile, something calling itself the Association for Human Mathematics put out a statement condemning OpenAI for its release. The statement included a sentence that’s since been widely mocked on social media: “Mathematicians did not ask for this work to be done.” (Imagine G. H. Hardy saying to Ramanujan, “who asked you to prove all these bizarre new identities?”)

If it wasn’t clear, the members of AHM speak for themselves, not for the math community as a whole. For my part, I think the math and CS theory communities should absolutely hold the AI companies’ feet to the fire to behave better, both within our little domains, and (much more consequentially) in pacing AI progress, figuring out alignment, and preventing a catastrophe for our whole civilization. So I’m glad that our communities have increasingly been doing that, and I hope to have played a tiny part.

At the same time, I regard the position “no one should spoil our fun by solving our open math problems using AI models” to be totally untenable. It offers no argument for why refraining would be in the broader public interest—as opposed to various mathematicians’ narrow career interests. It violates the core principle that math problems are there for absolutely anyone (yes, even a hated corporation) to try to solve. And of course, it’s not enforceable anyway, in a world where these capabilities will rapidly diffuse to anyone with a GPT or Claude subscription.

We will need to adapt to this new world.

… then they came for Navier–Stokes and I said nothing because I never worked on Navier–Stokes. But when they came for RL vs. L I realized that things are serious

–friend-of-the-blog Omer Reingold (shared with permission)

Last night my 9-year-old son was taunting my wife, complexity theorist Dana Moshkovitz, as follows: “mommy, I heard you got cooked! I heard that a robot solved the math problem you worked on for your whole career! OOF!”

While my son was being a brat, he also wasn’t wrong. Whether you’re thrilled, depressed, angry, or whatever else about it, yesterday was surely one of the biggest days in mathematical history. And yes, among the 372 huge results released yesterday by OpenAI, on the recommendation of its advisory group of Timothy Gowers, Edward Witten, and other distinguished mathematicians, was a proof of Subhash Khot’s Unique Games Conjecture (UGC), a statement that my wife has worked toward proving for the entire time I’ve known her. (The UGC implies that a whole slew of optimization problems really are NP-hard, even if you just want an approximation that’s slightly better than what you get from semidefinite programming relaxation, which is one of our main tools.)

Or at least, we’re pretty sure that it’s a proof! There’s a Lean certificate, as there are for some of the other 372 breakthrough results (not all of them). But it also appears that no human has understood just about any of these proofs yet; the race to do so has just started. If you want an on-the-ground sense of what that race is going to be like, here’s some of what Dana texted me last night:

It feels like something written by someone who’s on psychedelics. So much unclear and doesn’t make sense. Lots of name dropping of previous work without discussing why it can be used despite impossibility results

Basically the paper is so horribly written that it’s impossible to read it without AI help

I asked Astra for reasonable completeness and soundness claims of the noise gadget and it gave them by combining claims from all over the paper

They also have direct optimal NP hardness of approximation proofs for the main applications of the UGC (Max Cut and all CSP) that bypass the UGC.

The UGC proof invents a completely new bizarre code with a noise test. It’s some crazy recursive construction.

It’s not the long code, not the short code – some alien craziness

I still think that there maybe is a proof that uses the half space code (which is natural)

The citations are often irrelevant and confusing

A possible future is a math world that’s heavenly if you have vision/creative ideas that AI could help check and implement.

And of course there’s a lot for us to learn from the aliens

If you’re wondering what emotions Dana is feeling—well, probably all of them! Even while a central career aspiration has fallen to a robot, there are at least two mitigating factors for her. First, she can feel vindicated that the UGC was true after all, something she never doubted even while many of her colleagues did! Second, all of us in math and theoretical computer science and mathematical physics, at least those who cared about solving crisply-stated problems, are now in the same boat.

Besides the Unique Games Conjecture, here’s a small sampling of the treasures from Aladdin’s cave that I’ll probably be paying the most attention to over the coming weeks:

L=BPL (i.e., probabilistic logspace and deterministic logspace are the same thing), one of the great derandomization conjectures short of P=BPP. Though its truth was never in serious doubt, there was a whole subcommunity focused on proving this.

The Fourier Transform and integer multiplication in less than O(n log n) time, breaking a barrier that had stood since the 1960s. The new running time, if you’re curious, is O(n log^{0.9999999999999} n), give or take some 9’s.

Positive solution to the Unitary Synthesis Problem, which Greg Kuperberg and I posed back in 2007. For every n-qubit unitary transformation U, there exists a classical oracle A such that U can be implemented in quantum polynomial time with access to A. This is the opposite of what most of us expected, and could have implications for e.g. the computational problem of decoding Hawking radiation from a black hole and many other problems in quantum complexity theory—if we had an efficient way to construct the oracle A, which this paper doesn’t give. (Added: Here’s a beautiful blog post by friend and colleague Dakshita Khurana about the years she spent working on this problem—unlike me and most others, she believed correctly in a positive answer to it—and her current feelings as she works through the AI’s proof.)

Parity is not in QAC⁰, one of the great questions of quantum complexity theory since 1999 that many of my colleagues had been closing in on.

Nearly 4^th-power separation between randomized and quantum query complexity for total Boolean functions. A favorite problem of mine since 1998 (!), when we knew only that the optimal separating exponent was between 2 and 6. For the past few years, we knew it was between 3 and 4. So, this finally closes that story.

A superquadratic separation between sensitivity and block sensitivity.

Area law for 2D gapped Hamiltonians. One of the main open problems in Hamiltonian complexity.

Matrix multiplication in n^9/4+o(1) time—a rational exponent for once (!), and via a completely different approach than was used for O(n^2.373) and so forth

$\Omega(n^3)$ lower bound on the determinantal complexity of the permanent, improving the previous best bound which was quadratic.

A randomized polytime algorithm to approximately count the number of perfect matchings in a general graph, as well as a randomized nearly linear-time algorithm for finding a maximum matching in such a graph

A deterministic polynomial-time algorithm for factoring polynomials over prime fields, with no need to assume GRH.

Uncomputability of solving polynomial equations over the rational numbers—this was arguably the biggest open problem in computability theory (note that uncomputability of solving Diophantine equations, i.e. polynomial equations over the integers, was proved in the 1970s, giving a negative answer to Hilbert’s 10th Problem)

Any of the above, alone, could easily have been “result of the year” in some area (and in some cases, like Unique Games and L=BPL, in all of CS theory). And there’s a lot that I’ve left out—feel free to share in the comments whatever is making your eyes bug out! There are equally astounding wonders in number theory, combinatorics, algebraic geometry, analysis, and pretty much every other area of math, most of which I’ll never understand, although I’ll note that it includes partial progress toward the Riemann hypothesis and the ~~Hodge Conjecture~~ (retracted because of a sign error!) and the Birch-Swinnerton-Dyer Conjecture (i.e., the majority of the remaining Millennium Problems).

We can take solace in what’s missing from the list. P≠NP isn’t there, nor even P=BPP or NEXP⊄P/poly, and surely not for lack of trying. Apparently the greatest open problems of theoretical computer science are indeed pretty hard!

Oh, lest I forget: one day before the OpenAI dump, meaning Monday evening, Virginia Williams and Josh Alman posted an arXiv preprint that solves the 3SUM problem in O(n^1.9992) time, and the All-Pairs Shortest Paths problem in O(n^2.9995) time, refuting half-century-old conjectures that the correct answers were n^2-o(1) and n^3-o(1) respectively. In this case, it wasn’t an OpenAI model that supplied the crucial idea; it was an Anthropic one! But Anthropic then took a different approach from OpenAI: rather than post the undigested solutions to the world, it gave Virginia and Josh the opportunity to write and announce a digested version in exchange for compensation.

These have emerged as the two main models for communicating AI math breakthroughs, and they both have strengths and weaknesses. The “OpenAI model” sets up a crazy race among humans to digest and explain a messy AI proof (work that could easily be some combination of thankless, barely-credited, competitive, and unfun), while the “Anthropic model” puts a private company in the position of picking and choosing which human mathematicians get to be the emissaries of the AI. Dunno, what do you guys think?

For those who are wondering: apparently, the AI model that produced all these wonders was not bespoke contraption of 10,000 agents burning millions of dollars worth of compute, as was used for example to construct a finite-time blowup for the Navier-Stokes equations. Instead, it was simply the latest internal OpenAI model—one that might be released to paying ChatGPT customers within the next couple of months, depending on the recommendations of OpenAI’s safety board! (My 9-year-old son: “Oh they definitely shouldn’t release that. If it could solve all those math problems, it can’t possibly be safe.”) Apparently they used about 3 hours of GPT-Pro level compute on average per problem solved.

Also, if you were wondering: apparently they tried the model on about 4,000 problems. So, right now it “merely” solves ~10% of the longstanding open mathematical problems that it’s asked about, the problems that whole communities have spent years on, after a single 3-hour attempt on them. (An earlier version of this post said 8,000 problems and ~5% success rate, but commenters since corrected me.)

I’ve been glad to see the CS theory community rising to the occasion. At the Simons Institute in Berkeley, here at UT Austin, and elsewhere, I’ve hearing stories of researchers rushing to pore over the manuscripts and make sense of them and explain them—because what else do we do? How else do we continue the craft to which we’ve devoted much of our lives?

If you want some sense of what things feel like now in math, imagine a hunter-gatherer who’s spent his entire life learning to survive deep in an unforgiving rainforest, then a giant resort hotel springs up right next to him with a helipad and heated pools and AirBnBs, and without missing a beat, the hunter-gatherer says: “alright fine, so now my new job is to run wilderness retreats for the tourists, or something.”

In Quanta magazine, Jordana Cepelewitz attempted a different metaphor:

It’s as if you were teleported to the peak of a tall mountain. Surrounded by fog, you have no idea where you are, or what’s around you. You do not know how your mountain connects to others, and you have no equipment to help you explore, no way to help someone else join you. If you had climbed the mountain yourself, you would have experienced how the human body adapts to altitude and changes in oxygen levels. You might have had to invent tools to navigate, to climb steep cliffs, or to make a shelter. You might have encountered a fellow explorer, gotten lost together in a hidden valley, and found a plant that could be turned into a life-saving medicine.

Instead you’re perched on the peak but in the dark, while the maker of the teleportation machine tells you that it can explore the wilderness better than any human.

For any one of these mountains, if we care enough, I feel optimistic that we can do as we always have: clear the fog and figure out the path, except now using the teleportation machine to help guide us. The bigger challenge will be to nurture a community that still cares about the heroic adventure of finding the paths up these mountains in the world with the machine. (Oh, and I think one place where the metaphor breaks is that we still do have each other, as much as we ever did before!)

Experience has shown that, even now, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by anything that happens in the empirical world, of updating on anything, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.

So, they’ll say, maybe the alleged solutions are not solutions at all, but just “AI slop.” Or maybe none of the 372 well-known open problems that were solved were real math problems, they were all just glorified contest puzzles and trivialities. (After all, there’s still no Riemann Hypothesis!) Or maybe the entire 4000-year-old discipline of mathematics needs to be jettisoned: turns out that it was all just puzzle-solving and trivialities; all that’s different is that now the triviality stands unmasked. In any case, what really matters is that the true inner sanctum of human creativity hasn’t been breached and probably never will be, and also, that Sam Altman and Dario Amodei are contemptible little nerds.

If you’re still a proponent of that doomed worldview, still aboard the sinking ship, I encourage you in the strongest possible terms to read yesterday’s other great contribution to AI discourse, besides the OpenAI Mathocalypse dump: namely, Scott Alexander’s open letter to Steven Pinker. I feel some responsibility for this, as the person who first introduced Steven Pinker to the existence of the rationalist community, and who also first introduced Steven Pinker and Scott Alexander to one another (they had both been fans of each other’s writing). And now Scott is challenging Steve to a literal duel, with guns!

For whatever it’s worth: Steve is a lifelong intellectual hero of mine, just as he is for Scott, and I also have to privilege of calling Steve my friend. But I found Scott’s post to be one of the most devastating rejoinders to anything that I’ve ever read. And I thought Scott’s conclusion was exactly right: when it comes to AI risk, Steve’s great challenge is now to accept and start using a more “Pinkerite” epistemology.

Last night, while I should’ve been poring over some of OpenAI’s hundreds of papers and/or writing this post, I decided to spend some time with my kids instead. They wanted a movie night, so I suggested something they’d never seen before (and that I hadn’t seen for decades), and that seemed chock-full of no-nonsense, practical guidance for the world in which they’re going to grow up: Terminator 2.

Update: As several people have pointed out, cryptography is a subfield that’s extremely conspicuous by its absence from OpenAI’s list of 376 papers! But my sources tell me that the AI companies have now started, gingerly and discreetly, investigating whether their latest internal models can break important cryptographic protocols and primitives. If they can, then it would certainly be nice to get ahead of things before the rest of the world figures out the same.

Another Update: The statement put out the Advisory Group on Mathematics and Artificial Intelligence is very carefully phrased, neither endorsing nor condemning what OpenAI did, and is worth a read:

As announced a few weeks ago, OpenAI has released a large collection of mathematical results generated by an internal model, reporting solutions to hundreds of open questions. This is an important event for mathematics, with consequences both for mathematics and for the mathematical community that extend far beyond the individual results.

AGMAI’s advisory role should not be interpreted as a judgment of the impact of these results or an endorsement of the process by which OpenAI obtained them. We do not speak on behalf of the entire mathematical community, and only the mathematical community can undertake the assessment that is needed.

Making this work public is a first step. This release is the beginning, not the completion, of the process of human understanding and the incorporation of the work into mathematical knowledge. At the same time, the future of mathematical research cannot consist only of understanding results produced by AI labs. Mathematicians must be able to formulate their own questions, develop their own approaches, and explore directions that have not been selected as examples of an AI system’s capabilities. Equitable access to powerful research tools and adequate computational resources are essential to that freedom.

We reaffirm our published recommendations on responsible release. We have discussed them with OpenAI and appreciate the company’s willingness to engage. While we consider these discussions constructive, it is ultimately up to the mathematical community to assess the extent to which our recommendations were followed successfully, and whether there are others we should suggest. We remain committed to engaging with any frontier AI lab on these questions and have already been in contact with several of them.

by Scott at October 13, 2026 08:27 PM

Scott Aaronson Podcasts about the Topic

“The Topic” being, of course, the Mathocalypse, except sometimes with AI alignment and even quantum computing and complexity theory and other stuff thrown in.

With Ryan Peterman:

Notably, near the end of the above podcast, you can watch my live reaction to the news that an Anthropic model found algorithms for the 3SUM and the All-Pairs-Shortest-Path problems beating records that stood for generations—news that hadn’t yet reached me when I woke up and logged in. (If you don’t feel like watching, my reaction is basically: “uh, wow … if I hadn’t been so overstuffed with delicacies right now from history’s biggest math breakthrough buffet, I’d surely be giving this one the individual amazement that it deserves…”)

And here’s with “Modern CTO”, on a similar set of topics:

Enjoy, and thanks to the hosts of both podcasts!

by Scott at October 13, 2026 08:26 PM

John Baez — Liverworts

I spend a fair amount of time thinking about the climate crisis and how it’s part of a larger collapse of ecosystems, like how insect biomass in Germany has declined by 75% in the last quarter century, how large predatory fish biomass declined by 90% in the last century, how the Amazon is nearing a tipping point where it may cease to be a jungle, and so on.

Did you hear about the UK government report on ecosystem collapse? It was created not only by the environment department but also intelligence agencies like MI5 and MI6. Reporters were invited to see the unveiling of this report in October 2025—but at the last minute the prime minister’s office blocked its release!

Luckily the Greens forced the release of a 14-page redacted version.

𝗧𝗵𝗲 𝗸𝗲𝘆 𝗷𝘂𝗱𝗴𝗲𝗺𝗲𝗻𝘁: 𝗶𝘁 𝗮𝘀𝘀𝗲𝘀𝘀𝗲𝘀 𝘄𝗶𝘁𝗵 𝗵𝗶𝗴𝗵 𝗰𝗼𝗻𝗳𝗶𝗱𝗲𝗻𝗰𝗲 𝘁𝗵𝗮𝘁 𝗲𝘃𝗲𝗿𝘆 𝗰𝗿𝗶𝘁𝗶𝗰𝗮𝗹 𝗲𝗰𝗼𝘀𝘆𝘀𝘁𝗲𝗺 𝗶𝘀 𝗼𝗻 𝗮 𝗽𝗮𝘁𝗵𝘄𝗮𝘆 𝘁𝗼 𝗰𝗼𝗹𝗹𝗮𝗽𝘀𝗲—𝗶𝗿𝗿𝗲𝘃𝗲𝗿𝘀𝗶𝗯𝗹𝗲 𝗹𝗼𝘀𝘀 𝗼𝗳 𝗳𝘂𝗻𝗰𝘁𝗶𝗼𝗻 𝗯𝗲𝘆𝗼𝗻𝗱 𝗿𝗲𝗽𝗮𝗶𝗿.

Thus, I’ve been thinking about life after ecological collapse. Many of the species we love will be extinct. But probably not ferns, mosses and liverworts: ancient plants that have survived many earlier crises.

These were among the first plants to bounce back after the world-wide rain of molten quartz set all the forests on fire when an asteroid hit the Earth 65 million years ago. They were among the first to bounce back after the much worse end-Permian extinction. Liverworts been around for 420 million years, but with bursts of new species beginning in the mid-Jurassic and continuing through the Cenozoic.

The reason these plants can bounced back so fast is that they propagate via spores, not needing to mate with other plants. So, after global crises, we see spore spikes in the fossil record: plants like these become very common, setting the stage for new forests.

So, I’ve been looking at ferns and mosses, which are very common here in Edinburgh because it’s so moist. I knew practically nothing about liverworts, but I was delighted to find a bunch along the River Leith in Edinburgh—shown in my photo here:

Liverworts are so deeply evolved that when you search the internet for info on them, half the articles you’ll see are about how to kill them, because they’re so hard to kill! A small piece of a liverwort can grow into a whole new plant. And some have little cup-shaped structures holding clusters of cells called gemmae. A gemma can be splashed out of one of these cups by falling raindrops, and if it lands in a suitable place it will grow into a new liverwort. It’s as if flakes of your skin splashed off by rain could grow into new copies of you.

So in a bleak time, the world still holds many wonders, and the capacity to regrow.

by John Baez at October 13, 2026 06:09 PM

John Preskill — Nicole’s guide to writing and editing

Freshman year of college, I took a writing seminar from German-literature professor Ellis Shookman. Professor Shookman loved Mozart’s music, he told us early in the term. He listened to Mozart on the radio while driving from campus to Boston. Static might mar the transmission, but he could often turn up the volume and continue enjoying the program. Sometimes, the static worsened during the drive. It could worsen and worsen, until Professor Shookman’s frustration outweighed his delight at listening. He’d switch off the radio.

As Professor Shookman loved listening to Mozart’s music, he loved reading about students’ ideas. Yet static can mar a piece of writing: infelicities in grammar, structure, composition, word choice, and more. If enough infelicities obscure the writing, the frustration of reading outweighs the benefits. Professor Shookman will quit reading.

Professor Shookman marked up our essays with a blue pencil that achieved the status of legend among his students. If you’ve written a paper I’ve coauthored, you’ve probably received PDF drafts replete with green highlighting.¹ A sticky note explains the reason for each highlighting: “Singular–plural mismatch.” “Active voice >> passive voice.” “Let’s clue the reader in as to this formula’s meaning before lobbing the math at them.”

Over the past year, I’ve catalogued the suggestions I write most often on paper drafts. The comments embody principles gleaned from Strunk and White’s The Elements of Style; the Physical Review style guide; other writing guides I esteem; literature whose writing I esteem;² collaborations with professional editors; and writing instructors, including Professor Shookman. Each section below begins with more-important principles, shading into more-nuanced ones.

Please use and disseminate these principles. Train your favorite large-language model (LLM) on them, and have the LLM critique your manuscripts. Instruct it to use green highlighting if you wish. Even if the LLM suggests fixes initially, tell it to stop offering suggestions later, so that you can devise the solutions: train not only the LLM, but also yourself. I hope to enjoy your papers as much as Professor Shookman enjoyed his sonatas.

Organization

Motivate your work; then, present it; and then, explain its physical significance.

Begin each paragraph with a topic sentence.

Begin each section, apart from the introduction and conclusion, with (i) a statement of the takeaway and (ii) an outline of the section. When outlining a section, hyperlink to each subsection. Similar guidelines concern subsections and subsubsections.

Before presenting a piece of math, sketch its meaning and origin. This strategy enables the reader to understand the math as soon as they encounter it. If you throw math at the reader without introducing it, the reader will have to squint at the symbols for a while to figure out what the expression means and where it comes from.

Example: To calculate the average work, we substitute the Hamiltonian formula (10) into the definition (12): [equation].

Most citations belong at the ends of (i) sentences and (ii) phrases concluded with commas. Put a citation elsewhere only if you have a compelling reason for doing so.

Bridge each component of your writing to the next component; the next shouldn’t sound like a non sequitur.

Suppose that the next sentence refers to (i) a topic mentioned in the previous sentence and (ii) a new topic. Mention (i) before (ii).

Example: Smith et al. applied control theory to the extent possible. The attempt led to intractable equations, unlike our approach.

Example of a broken bridge: Smith et al. applied control theory to the extent possible. Our approach does not involve intractable equations, unlike theirs.

Whenever you tell a story, tell it from start to finish, step by step. Derivations, proofs, and descriptions of experiments qualify as stories.

This guideline extends to descriptions of experimental setups and of mathematical objects. For example, imagine referring to an element of a subgroup of the group generated by some operators. Did you have to read the preceding sentence multiple times to process it? The sentence begins at the end of a story, then rewinds to the story’s beginning. This structure impedes understanding. The subgroup forms the context for the subgroup element, which one can’t grasp until hearing about the subgroup. The subgroup participates in a similar relationship with the group, as does the group with its generators. Therefore, one should introduce the generators, then the group, then the subgroup, and then the subgroup element.

Word choice

Use strong, specific words, rather than weak words.

Verbs and nouns are stronger than adjectives and adverbs.

Choose specific verbs (e.g., “prepare,” “evolve,” and “measure”), rather than vague, general verbs (e.g., variants of “to be” and “take,” as in “take a measurement”).

Avoid statements such as “we investigate,” “we study,” and “we analyze.” Such statements don’t relate that you’ve accomplished anything. State what you’ve accomplished. Verbs such as “prove,” “test,” “confirm,” “discover,” and “find” achieve this goal.

Adverbs such as “importantly” and “remarkably” pollute scientific writing with the authors’ opinions. Demonstrate that a claim is important or that a result is remarkable; then, leave readers to draw their own conclusions. Those conclusions will coincide with yours if you’ve demonstrated your point.

Use the active voice, rather than the passive voice. Take responsibility for your work. Editors of high-impact scientific journals have endorsed this advice.

Refer to yourself when necessary and only when necessary.

Example of unnecessary reference to self: We use the superscript “max” to signify the maximal Fisher information.
Preferable alternative: The superscript “max” signifies the maximal Fisher information.

Example of unnecessary reference to self: Our results establish several opportunities for future research. First, we can implement the experimental proposals.
Preferable alternative: Our results establish several opportunities for future research. First, one can implement the experimental proposals.

You may use the first-person plural when escorting the reader through a derivation.

Example: We substitute from Eq. (1) into Eq. (2).

If you’re the only author, don’t use the plural (“we,” “our,” etc.). The usage is inaccurate and misleading. It portrays you as dodging responsibility for your work by dispersing that responsibility across the scientific community.

Avoid dangling modifiers.

Pair every verb with the appropriate noun.

Example of grammatically incorrect text: Equation (1) follows by calculating the sum.

One should pair the verb “calculate” with the noun “we,” because “we” undertook the calculating. However, this example’s author omitted the noun out of squeamishness about using the first person in a scientific document. Hence the sentence says that the equation calculates the sum. Equations can’t calculate sums.

Examples of correct alternatives

We derived Eq. (1) by calculating the sum.

Equation (1) follows from the evaluation of the sum.

Calculating the sum yields Eq. (1).

Avoid empty subjects.

Include no unnecessary words.

“So-called” is unnecessary.

“Note that” and “We note that” are unnecessary.

Several phrases often preface mathematical statements but are unnecessary: “we have,” “we have that,” “it holds that,” and “it follows that.” One can better serve the reader by prefacing the mathematical statement with (i) a derivation or (ii) a prose description of the statement.

Never write “is equal to”; “equals” is more concise.

Never write “is able to”; “can” is more concise.

Never write “gives an upper bound to” or “places an upper bound on”; “upper-bounds” is more concise. Analogous statements concern lower bounds.

Never write “a large number of”; “many” is more concise. Never write “a small number of”; “few” is more concise.

Never write “We refer to [symbol] as [name]”; “we call [symbol] [name]” is more concise.

Never write “as long as”; “if” is more concise.

The symbol $>$ means “greater than”; and $\geq$ , “greater than or equal to.” Don’t translate $>$ into “strictly greater than”; the “strictly” is unnecessary. Analogous statements concern $<$ and $\leq$ .

Avoid contractions, which are too informal for professional writing.

The possessive is not a contraction and belongs in professional writing. It facilitates conciseness.

Use the word “for” only when it belongs. Physicists often write “for” when they mean “if,” “per,” “at,” or something else.

Example of inappropriate use: The function vanishes for odd arguments.
Corrected statement: If the argument is odd, the function vanishes.

Example of inappropriate use: We performed 10 trials for each parameter value.
Corrected statement: We performed 10 trials per parameter value.

Example of inappropriate use: The function is smaller for small x values.
Corrected statement: The function is smaller at small x values.

Write “we evolve the state,” “we measure,” etc. only if you’re an experimentalist who undertakes those actions. Alternatives include “Consider measuring,” “Suppose the system evolves,” and the command tense (e.g., “One can measure this quantity as follows: prepare the qubit in $\lvert 0\rangle$ . Evolve it under $H$ …”).

The condition $x\ll y$ defines a regime, not a limit. The conditions $\lim_{x\to0}$ and $\lim_{y\to\infty}$ define limits and are inequivalent to $x\ll y$ .

Write “first,” “second,” “last,” etc., not “firstly,” “secondly,” “lastly,” etc. (I defer in this matter to The Elements of Style.)

Humans can assume, suppose, etc. Mathematical expressions, protocols, etc. can’t.

One multiplies factors together and sums terms. Don’t call factors terms and vice versa.

If you mean “X equals Y,” say so. Don’t write “X agrees with Y,” “X matches Y,” or “we identify X with Y.” The latter three phrases are vaguer, and two of them contain more words, than “X equals Y.”

Regarding the words “general” and “generally”:

A general object subsumes every example of that object. If any example behaves unlike a supposedly general object, don’t call the object general.

Many claims contain the term “general,” “generally,” or “in general” but don’t need the term.

Example of a sentence that contains “generally”: The terms generally commute.

Equivalent, more concise sentence: The terms commute.

Physicists tend to use the words “general” and “generic” differently. By “general,” physicists usually mean “subsuming every example.” By “generic,” we usually mean “typical,” or “common.”

“Then” makes sense (i) in discussions of chronology and (ii) in if–then statements. Don’t use “then” outside these contexts.

Example of inappropriate use: “Define $X:=\ldots$ Then Y.”

Examples of appropriate alternatives

Define $X:=\ldots$ This definition implies Y.

If $X:=\ldots \, ,$ then Y.

Define $X:=\ldots \, ,$ such that Y.

Don’t justify any equation with “we used that [such-and-such is true],” which violates the rules of grammar. Grammatically correct alternatives include “We applied [a property],” “The equation follows from [a property],” and “…since [such-and-such is true].”

Regarding tense:

When describing what you’ve accomplished, use only one tense.

Experiments happened in the past, so describe them in the past tense.

When describing a proof’s steps, use the present tense.

Example: We Taylor-approximate the function about $x=0$ . Substituting into Eq. (1) yields [equation].

Nouns, verbs, and adjectives should agree about whether a quantity is singular or plural.

Example of singular–plural mismatch: The equations are a rule for evolving the cellular automaton.

Example alternative: The equations form a rule for evolving the cellular automaton.

“Admit of” means “allow for,” or “permit.” The phrase needs the “of.”

Example: The formula admits of the following interpretation.

Punctuation

Consider any list that contains at least three items. If no item contains a comma, separate the items with commas. If any item contains a comma, separate the items with semicolons.

In American English, periods and commas belong inside quotation marks. (Example: She told me, “Have a good day.”) In British English, periods and commas belong outside quotation marks. (Example: She told me, “Have a good day”.)

To write quotation marks in LaTeX, don’t use your keyboard’s quotation-mark key; use the appropriate keys.

Regarding hyphens:

The hyphen (-) feeds into punctuation of three types: the hyphen (-), the en dash (–), and the em dash (—).

The hyphen appears in some compound words, as in “non-negative.”

In American English, em dashes can separate ideas within a sentence. Don’t separate any em dash from neighboring text with a space.

Example of appropriate use: The sample—the only product of this experiment—barely survived.

Example of inappropriate use: The sample — the only product of this experiment — barely survived.

Example of appropriate use: He told me only one sample had survived—hardly what I wanted to hear.

This article specifies how to use the en dash. One use is “to separate the names of two or more people used as a compound modifier.”

Example: Feynman–Kitaev clock

Hyphenate compound adjectives.

If an adverb ends in “-ly,” it probably shouldn’t precede a hyphen.

Example of inappropriate hyphenation: strongly-coupled systems

Follow a prefix with a hyphen if and only if the Physical Review style guide indicates that you should.

A complete clause must follow any semicolon (unless the semicolon separates items in a list).

Math

Introduce only necessary notation, which readers will have enough trouble remembering. If a mathematical symbol appears only once, eliminate it. If a symbol appears only twice, try to eliminate it.

Every sentence must obey the rules of English grammar, punctuation, and syntax, regardless of whether the sentence contains mathematical symbols. All math-containing sentences must end with punctuation marks. If a sentence contains a list of mathematical expressions, precede the final expressions with an “and.” If the list contains at least three mathematical expressions, separate them with commas.

Introduce almost every mathematical symbol before you use it. If you introduce a symbol after using it, the reader will encounter the first use, stop, feel confused for a while, tentatively continue, find the definition, return to the earlier use to understand it, and then progress again. This back-and-forth breaks up the reading process. You may define a mathematical symbol after using it only if (i) the symbol is very common, known to nearly all physicists, and unmistakeable and (ii) defining the symbol earlier would disrupt the text’s flow.

If you define a new function, denote it by only one letter. (I defer in this matter to the Physical Review style guide.)

Example: $f(x,y,z)$

Examples of disallowed notation: $fxn(x,y,z)$ , ${\rm fxn}(x,y,z)$

Suppose that a superscript or subscript stands for a word or phrase without representing any variable or constant. The superscript/subscript must not be italicized. (I defer in this matter to Physical Review style guide.)

Example: Let $x_{\mathrm{meas}}$ denote the measurement outcome.

If a variable or constant appears in a superscript, parenthesize it. The parentheses communicate that the superscript isn’t an exponent.

Example: Let $\sigma_z^{(j)}$ denote the Pauli- $z$ operator of qubit $j.$

If a superscript is not italicized (stands for a word or phrase), don’t parenthesize it.

Regarding the definition of a symbol $A$ :

If you write $A$ alone on one side of a defining equation, use \coloneqq or \eqqcolon: $A$ \coloneqq [expression], or [expression] \eqqcolon $A$ . The symbols \coloneqq and \eqqcolon relate more information than does $\equiv$ , encoding directionality.

Use $\equiv$ if $A$ does not appear alone on its side of the equation: [function of $A$ ] $\equiv$ [result of replacing $A$ with its definition in the equation’s left-hand side].

Refer to the Cartesian axes using the formatting “[italicized letter]-axis.” Don’t include any hat, boldface, or \vec symbol.

Example: $x$ -axis

Avoid denoting any index by $i$ , which means $\sqrt{-1}$ to physicists. Use $j$ instead, unless you’re writing for engineers (who denote $\sqrt{-1}$ by $j$ ).

Don’t use the lowercase letter l (“ell”) as an index; readers might mistake it for a one. Use $\ell$ (\ell) instead.

Give every set-off equation a number. Readers (and coauthors) may want to refer to the equation easily when discussing the paper. Save them (and us) from having to say, e.g., “that equation halfway down page three.”

When writing a set-off mathematical expression, use the align environment, not the equation environment. Using the align environment, one can easily extend an expression across multiple lines.

Regarding a set-off mathematical expression that extends across multiple lines:

Format the expression as follows by default.

Put an & symbol immediately leftward of the first = sign or analogous symbol (e.g., $\leq$ ).

If any subsequent line begins with another = sign (or analogous symbol), put an & immediately leftward of the symbol. (I’ll stop writing “or analogous symbol.”)

Suppose that a subsequent line begins with a $+$ , –, $\times$ , or $/$ . Find the symbol immediately rightward of the initial = sign. Begin the new line directly below that symbol.

Example:

Modify the default formatting if necessary (a) to reduce the number of lines used in a PRL submission or (b) if the initial = appears awkwardly far to the right.

Example of (b):

Suppose a new line begins with a term or factor, such as the $jx^8$ in the example under (A). Put the corresponding $+$ , –, $\times$ , or $/$ at the beginning of the new line, not at the end of the previous line.

Examples of inappropriate placement:

The symbol $\approx$ means “approximately equals”; and ~, “scales as.” Approximations convey more information than scaling relations do.

Use big-O-type notation or ~ symbols, not both; they’re partially redundant.

\ldots, rather than \cdots, should stand in for elements that fit a pattern.

Example: $x_1,x_2,\ldots,x_n$

When using \ldots as in the previous rule, present at least two initial examples of the pattern. One can’t define the pattern.

Contains insufficient examples: $x_1,\ldots,x_n \, .$ For example, if $n$ is odd, then $x_1, x_2, \ldots, x_n$ and $x_1, x_3, \ldots, x_n$ fit the template.

Parentheses (), square brackets [], and curly braces {} are delimiters. If you nest them, do so in the order dictated by the Physical Review style guide.

If delimiters enclose a symbol, it shouldn’t protrude above or below them (unless the delimiters would have to be grotesquely enormous). Use the \left and \right commands if the delimiters appear on the same line.

An operator $O$ isn’t a matrix; a matrix represents an operator in terms of a particular basis. Therefore, no equals sign should interrelate an $O$ and a matrix. An arrow can.

Example: $O\to\begin{bmatrix}1&0\\0&2\end{bmatrix}$

Every real number is complex. Don’t say “complex” if you mean “nonreal.”

Consider introducing a mathematical symbol in a prose sentence without using a comma or colon. Put the symbol immediately after the word that names the object represented by the symbol.

Example of inappropriate placement: the set of real numbers $\{ a, b \}$

Examples of appropriate placements

the set $\{a, b\}$ of real numbers

the set of real numbers $a$ and $b$

Recall the set of real numbers, $\{a, b\}$ , in Lemma 1.

More mechanics of writing

Use concise sentences, as advocated for in The Elements of Style. The reader can hold only so many ideas in their head at once.

Structure sentences simply, as advocated for in The Elements of Style. The reader should be able to grasp each sentence easily.

Avoid nesting ideas within a sentence, to avoid convoluting the sentence’s structure.

Example of sentence with convoluted, nested structure: Any model of equilibrium and nonequilibrium behaviors of systems observed in tabletop experiments and high-energy colliders must obey the laws of relativistic quantum mechanics.

Visualization of the nesting: [Any model of ([(equilibrium and nonequilibrium) behaviors] of {systems observed in [(tabletop experiments) and (high-energy colliders)]})] must obey [the laws of (relativistic quantum mechanics)].

The ideal paper title has the structure of a newspaper headline: it presents a claim, containing a subject and a predicate.

Regarding abbreviations:

Don’t abbreviate the first word in any sentence.

Abbreviate “Figure,” “Section,” “Professor,” and “Appendix” if such a word appears partway through a sentence.

Don’t abbreviate “Sections.”

Regarding acronyms:

Write every acronym in capital letters, as per the Physical Review style guide.

Introduce each acronym the first time you use it.

Thereafter, use only the acronym, not the spelled-out phrase, throughout the rest of the document’s main text. You may spell out the phrase in section, figure, and table titles if doing so improves the document’s clarity.

Every paragraph should contain at least three sentences.

Wherever you insert a blank line into your LateX code, a new paragraph begins in the corresponding PDF. Insert a blank line only if you wish to begin a new paragraph. This advice applies immediately before and after set-off equations.

Never begin a subsection immediately after a section title. Between the two titles, overview the section. Analogous rules govern subsections and subsubsections.

Put the word “only” in the appropriate place.

For example, suppose you’ve sampled data at a point $x=0$ in parameter space and sampled data at no other points. “We sampled data only at $x=0$ ” is correct; “We only sampled data at $x=0$ ” is probably not. The latter claim means that (i) you might have sampled data at $x=0$ and (ii) you did nothing to the $x=0$ data apart from sample it: you didn’t analyze the $x=0$ data, discuss the $x=0$ data, etc.

When in doubt, consult the Physical Review style guide or The Elements of Style.

If those references don’t contain the information you seek, search for it in online writing guides. Not all such guides have equal merit, however. Lean toward guides written by human editors or published by college writing centers.

¹ Collaborators have wondered why I use green; a student guessed it’s my favorite color. It isn’t; but I bleed green, having graduated from the Big Green, also known as Dartmouth College. Sometimes, I highlight certain pieces of text for one reason (e.g., to point out logical inconsistencies) and other text for another reason (e.g., to point out grammatical inconsistencies). Green distinguishes the first highlightings, while orange distinguishes the second: when not bleeding Dartmouth green, I bleed Caltech orange.

² Don’t learn how to write from physics papers.

by Nicole Yunger Halpern at October 13, 2026 11:33 AM

Tommaso Dorigo — Is Simulation-Based Inference A New Technique In HEP?

Is Simulation-Based Inference A New Technique In HEP?
No, it is not. Please note! It goes back to Stan Ulam in the 1940s!

Tommaso Dorigo Tue, 10/13/2026 - 05:13

Categories

Physics

by Tommaso Dorigo at October 13, 2026 09:13 AM

October 10, 2026

Doug Natelson — Science policy/funding updates, early October edition

This post is a bit later than I'd wanted, due to a combination of work and wanting to be in the right frame of mind to write about what's new in science policy (directly and adjacent matters) and funding.
The NSF did underspend their federal appropriation for FY26 by more than 20%, obligating only $6.37B out of more than $8B budgeted. In terms of numbers of awards, they made 2410 fewer awards in FY26 than in FY25, and about 5000 fewer awards than typical in FY21-24.
This includes the almost complete evisceration of the Social, Behavioral, and Economic Sciences directorate. As that Science news article points out, SBE has until now funded about 2/3 of all US research in the social sciences (including linguistics and archeology). The presidential budget request has been calling for SBE's elimination for the last two years. Congress has rejected that in its budgeting, but seemingly OMB has gotten its way regardless.
There are more NSF X-Labs topics on the way, and a webinar next week to talk more about the "AI for Physical Systems" one.
NSF has also created a "metascience" office and released a statement about collaborating with the UK on this. I've given my take on metascience before - the idea of seriously studying which types and mechanisms of research funding and programs give the "best" "results" is good in principle, but extraordinarily difficult in practice. The idea that sound conclusions can be drawn in just a few years is absurd.
On the unambiguously positive side (gotta take the win when there is one), the NSF did announce the surprise support of the National Math Stars effort.
NSF also announced the first of their new Grand Challenge programs, rumored to be a possible destination of (some of?) the unspent funding from the top point above. No real information about them yet, though the Quantum+X one has a webpage where you can sign up for updates. Given the description ("Whether you're a researcher at a university, a startup, or an established company, this prize challenge is your chance to help define the next era of quantum technology."), it's hard not to wonder if this is another conscious effort to have NSF be a funding vehicle for industry/business rather than longer-term research.
These announcements happened in part at the "Science: A New Golden Age Summit" (link to White House release about this.). The National Medal of Science was presented to various tech leaders (not scientists). If any of my readers attended this event, I'd be very curious about a first-hand account.
DOE has announced the Genesis Mission Graduate Fellowship program. The idea is a 4-year PhD with "cocompetencies" between a discipline and AI, with required extended internships at DOE labs/user facilities and industrial partners. The actual call is here, with a LOI due on 9 November and a preproposal deadline of 23 November.
For those of you who are PIs in condensed matter physics, the new DOE notice of funding opportunities (pdf) is out. Pay close attention to the condensed matter physics program, as its focus has shifted.
Similarly, the new Air Force Office of Scientific Research BAA is out, and as with DOE, the focus of the condensed matter program has shifted. Both DOE and AFOSR are now explicitly very interested in quantum geometry.
On the more general research landscape:
It's pretty clear that no matter what happens in the upcoming US elections, the next two years at least are likely to be comparatively bleak for federal support of science and engineering research. Fundamentally, regardless of what OSTP says and despite glossy announcements of new programs, it sure seems like the executive branch (OMB) still wants to curtail federal support for research, and there is not much that even an opposed Congress alone can do in the near term.
The number of STEM PhDs entering the job market in 5-ish years is going to be 25% down or more, because funding uncertainty has led to two consecutive years of double-digit decreases in admissions. Remember, only 10% or less of STEM PhDs go into academia, so this demographic drop is going to be largely felt by industry. I remain surprised by the apparent apathy about this. If I thought my technical workforce applicant pool for some jobs might decline by double digits, I'd at least have some concern.
Perhaps industry is collectively assuming that AI will somehow make up for this. OpenAI released 722 preprints out there this week with results in math and mathematical physics. This is simultaneously impressive and awful, and much has been written about it. See here for an article and here for a weekly digest, from Quanta. Peter Woit is hosting an interesting discussion, and points to a collection of comments about this.
The US administration also announced a rule change that would impose a $70000 charge for "curricular practical training" (the mechanism that allows international students on F1 visas to have internships during their time as students) and a total of $100000 in fees for "occupational practical training" (the mechanism for international students to be allowed to work in the US for a period after completing their studies). The actual proposed rule change is here, and people can submit public comments here until November 9. Personally, I think that this rule change is yet another component of a seeming concerted effort to make the US unattractive as a destination for international scholars, a strategy that is incredibly self-defeating for the country. (Sure looks like a lot of the "yes, kill OPT" comments are verbatim copy/paste of the same text. Seems like someone has seen prior public commenting efforts on other controversial rule changes and decided to mobilize.) Like the prior proposed duration of status changes for F1 and J visas (injunction discussed here), there are likely to be multiple big lawsuits filed over this.
Whew. This was a lot. It's hard to maintain my customary equanimity when the pace of disruptive news is this rapid. I will hopefully write more about some cool science soon.

by Douglas Natelson at October 10, 2026 08:58 PM

October 09, 2026

Matt von Hippel — Congratulations to Francis Halzen!

The 2026 Physics Nobel Prize was awarded this week, going to Francis Halzen for the work that led to the IceCube Neutrino Observatory.

Does that image look weird, like something is missing? It should. This is the first time in over thirty years that the physics Nobel was awarded to just a single person. Most years, it goes to three people, even if they have to awkwardly jam loosely related achievements together to make it work. Some years, it goes to two. The last time one person received the whole prize was 1992.

So why didn’t Halzen have to share the prize like Penrose did in 2020 and Parisi did in 2021? As the saying goes, your guess is as good as mine.

I can say that I was pleasantly surprised to read the announcement and see a name I recognized. I probably couldn’t name the main figures behind most big experiments. If the prize had gone to another founding figure, I’d be out of luck. But I’d heard of Halzen. And specifically, I’d heard about him in the context of the long road it took to get IceCube working.

The IceCube Neutrino Observatory is on the face of it a crazy idea. Neutrinos are almost undetectable, particles that can whiz through a light-year of lead without noticing a thing. To detect them, other experiments had build massive underground tanks of heavy water, shielded from outside interference and covered in cameras, so that on the extremely rare occasions when a neutrino happens to nudge a water molecule, it can be recorded and measured.

IceCube does this on another level, in wild surroundings, by sinking a grid of cameras into a kilometer of Antarctic ice.

So when I saw someone else propose a wild particle detection idea, I talked to my contacts at IceCube, to get an idea of how those wild ideas get fleshed out. And they told me about Halzen.

What they told me, what resonated again and again in Halzen’s writing, and what keeps coming up in quotes from him today, is that it took both luck and persistence. Nature needed to cooperate, there was no guarantee that IceCube would work. But also, the experiment couldn’t have existed without predecessors. He built off plans to put detectors on the bottom of the ocean, and tried out smaller experiments in Greenland.

Those experiments intrigue me, because they filled two roles. On the one hand, they were in a sense mere prototypes, and theorists had strong arguments that they wouldn’t be able to detect anything of interest. On the other hand, those arguments could have been wrong, and the experiments wouldn’t have been made if they could only be justified as prototypes: they were made because there was a chance they could actually see interesting neutrinos.

That combination strikes me as fundamental. We never know, in science, if an experiment will work. So every experiment, deep down, serves this combined purpose. It has to have a chance, even if low and poorly understood, of seeing something new and interesting. And it has to serve as a prototype, to teach us how to make the next experiment. Experiments are almost never one or the other, they are almost always both.

And eventually, as for Halzen, we get lucky, and nature cooperates.

by 4gravitons at October 09, 2026 04:00 PM

Tommaso Dorigo — The Wondrous Radiacode 110

The Wondrous Radiacode 110
Yesterday I was delighted to receive in my mailbox another radiation spectrometer for testing, the Radiacode 110.

Tommaso Dorigo Fri, 10/09/2026 - 08:01

Categories

Physics

by Tommaso Dorigo at October 09, 2026 12:01 PM

Doug Natelson — NAS "State of Science" 2026 address

I watched the webcast of the NAS State of Science address by outgoing NAS president Dr. Marcia McNutt. (I did not watch the panel discussion afterward, so sorry if I missed critical pieces.) A few thoughts on this:
The intro music was a very classy baroque string quartet. Hard not to think of this scene from Titanic.
The main theme was about ways to revitalize US science, and there were six main points that she wanted to emphasize, each with examples of relevant projects underway, ways to measure success, and the consequences of failure. That's fine, and I'll relay them below with some comments, but first an overall impression: This was largely an exercise in avoiding talking about the elephant in the room, the overt hostility toward and the attempted wanton dismantling of much of the publicly funded US research ecosystem by the executive branch. I'm unfortunately not surprised that this was largely brushed over, given the position of the Academies (see here). As the saying goes, I'm not mad, I'm just disappointed. The realization that the National Academies leadership do not feel empowered to have a frank discussion about this publicly has been depressing.
Dr. McNutt mentioned that in her previous address, she had pointed out the US vulnerability in STEM by being so reliant on international talent, and that now that other countries are heavily investing in research, the US STEM research world needs to do a better job getting US citizens in the workforce. That's all true, but leaving out how the government leadership is explicitly trying to curtail international scholars and international collaboration seems like quite an omission.
She mentioned in passing that industrial research in the US in the 1950s was tiny, nothing compared to the fraction of R&D it is today. Is that actually correct? I mean, that was the heyday of Bell Labs, IBM, GE, Westinghouse, and big research labs at companies like Ford and GM. Much has been written about this.
The first big point was the need for improved relationships between universities and industry, and some examples of ways to encourage this, including relatively simple policy changes like making it easier for faculty and others to take leaves in industry. Certainly it would be broadly good for the US research ecosystem to have more diverse forms of support, and as I've written before, major industrial sectors with lots of capital rely in the long term on trained people.
The second point was the need to realign the academic reward system, so that industrial/entrepreneurial/coalition-building activities are incentivized, rather than rewarding lone-wolf PIs. That's fine, and honestly I think it's already happening to some large degree at major research universities.
The third point was meeting the needs of the STEM workforce, through increased interactions with industry (including, e.g., prospective industrial employers helping to define dissertation topics), co-op efforts, some training in businessy aspects (note: the Sloan Foundation was pushing this 25 years ago.). This is all laudable to try, but I don't see how any of this actually addresses the issue of fewer STEM workforce participation from US citizens, which is quite complicated.
The fourth point was the need to reduce regulatory burden. Sure, we all want to reduce bureaucratic BS. I have to say, though, that it was genuinely baffling to me that the most Dr. McNutt had to say about the threatened OMB rule changes (apart from a passing mention early on) is that they would increase bureaucracy. That isn't even in the top 15 problems raised by those changes. Remember, the default position of those pushing those rules is that academics are fundamentally untrustworthy and poor stewards of public resources.
Fifth was the need for automated/self-driving labs. I agree completely that advanced degree training should not be driven by the need for cheap labor to do tedious lab tasks (e.g. a zillion cell cultures or chemical syntheses). Overall this was pretty innocuous.
Sixth, Dr. McNutt emphasized the need to take on big challenges - researchers need to be bold and not play it safe, and peer review can be inherently biased toward incrementalism. She gave examples of large privately endowed institutes as enabling such work (MBARI, the Allen Institute). Apparently STAC will be proposing new multi-agency science and technology "breakthrough funds". The argument in favor of public investment in science in this section sounded rote rather than heartfelt. If anything, I thought knocking peer review right now at a time when OMB wants to ignore it at their pleasure was a weird position to take.
To be clear: I don't think any of the ideas highlighted in the speech are actually bad (necessarily). It just avoided emphasizing that publicly funded research has been incredibly beneficial, and that irreversible harm is being done. The statement that science agencies "have seen a loss of key personnel" is the worst kind of passive voice garbage. A hundred thousand technical personnel leaving agencies is not something that just "happened" like the weather. Being quiet, avoiding confrontation, and only trying to work behind the scenes is not the leadership that is needed now. (See, I can do passive voice, too.)

I will try to get back to more science posting....

by Douglas Natelson at October 09, 2026 02:38 AM

October 08, 2026

John Preskill — In praise of questions

Prende y se apaga sola,

sale después de hora.

CHARLY GARCÍA

La máquina de ser feliz¹

I’ve learned to tolerate them, and sometimes even like them, but Zoom meetings are weird. Even in familiar contexts: I have weekly check-ins with a close friend/collaborator based in Germany, and despite our familiarity, it still feels weird at times. I joined Nicole Yunger Halpern’s group almost when it was established, during Summer of 2021. The pandemic wasn’t exactly over, and most of us were geographically scattered, as well as complete strangers. To half-palliate this awkwardness, Nicole came up with what’s now sort of a group tradition: icebreaker questions. These are chosen by group members (in a secret order that I believe only Nicole fully knows), and can range from day-to-day stuff (“what’s the best type of pasta?”) to deeper matters.

A while back, a group member asked something along the lines of: “If you could give your younger self any scientific advice, what would it be?” My response then was: “ask the dumb questions you have as they come.” I had two reasons for this answer. First, dumb questions usually sound dumber in your head and are not as bad as you might think. Second, a dumb matter that doesn’t seem dumb at first, can turn into a really, really dumb question if it’s long unanswered. If I were to be asked this same icebreaker question today, I’d surely reply the same thing, but I believe the reasoning would be more nuanced.

This text’s epigraph comes from a song in the appropriately named album Random, by Charly García. I’m not sure whether the “happiness machine” in Charly’s random scenario deals with Brownian circuits, but I’d like to think so.

Over my PhD, I’ve had the opportunity to interact, discuss physics, and even collaborate with truly amazing scientists. I still don’t have a foolproof handbook of criteria detailing what makes a great physicist great,² but I have observed some patterns. An important one is that great physicists speak their mind: they’re usually relentless with their own confusion and ask whatever they have to ask to overcome it. Don’t get me wrong here; these people are giants, and I wouldn’t dare call their questions dumb, but asking good questions is a skill more than a gift. It’s like seeing Claudia Poll swim or hearing Hilary Hahn play a cadenza: yes, these people are objectively good, but there’s also a huge amount of practice that goes behind what one sees. And, in the case of asking great questions, practice comes from asking not-great (and sometimes even dumb) questions.

I got a crash course in asking questions during a recent project. A driving theme of my PhD research has been the study of autonomous quantum machines. These devices execute tasks without classical, time-dependent control, operating under a time-independent Hamiltonian. My first experience with these devices was during an experimental collaboration in which Simone Gasparinetti’s group built an autonomous quantum refrigerator from superconducting qubits. The refrigerator ran on a thermal gradient: as heat flowed from a hot bath to a cold one, a three-qubit interaction carried energy out of a target qubit, cooling it with no external control. The experiment was a success, and its potential usefulness in information-processing devices led us to ask about other contexts in which autonomous quantum machines could shine.

At the time, we came in close contact with folks at Marcus Huber’s group in Vienna, who were thinking in abstract terms about a fully autonomous quantum processing unit. We had the idea of designing autonomous protocols for logical gates in current-day quantum-technology platforms: Rydberg atoms, trapped ions and superconducting qubits.

An autonomous heat engine powers a photon-emitting device (which can be thought of as a clock, for compelling reasons). The photons (ticks) trigger quantum-autonomous gates.

To illustrate this autonomous-gate idea, I’ll briefly describe a gate protocol from our paper. As I mentioned before, an autonomous quantum fridge can be built by exploiting a thermal gradient. The same principle can help build a heat engine. Suppose one has three qubits, with energy gaps $\omega_C, \omega_H$ and $\omega_T$ . Let’s call these qubits the cold, hot and target qubits, respectively. The cold qubit couples to a bath at temperature $T_C$ and the hot qubit to a bath at a temperature $T_H$ , with $T_H>T_C$ . Under the resonance condition $\omega_C+\omega_T = \omega_H$ , the three qubits can swap energy all at once: the hot qubit loses an excitation while the cold and target qubits each gain one, or vice versa. The gradient makes the first direction more likely. The cold qubit can be engineered to quickly decay to its bath, so the exchange can’t run backward, and the target accumulates excitations. The cold qubit’s energy is lost to its bath, while the target could emit into a waveguide that delivers the photon to a chosen destination. The thermal gradient thus powers a device that emits useful photons with no external control.³

What can we do with this autonomously emitted photon? We can use it to trigger an entangling gate. Two superconducting qubits with matching energy gaps can exchange excitations. Suppose we have a tunable-frequency transmon qubit capacitively coupled to a cavity. An incoming photon from our autonomous emitter, entering the cavity, can shift the frequency of the transmon. If this effective, shifted frequency matches a nearby qubit’s frequency, the two transmons can interact. The effect of this interaction depends on the cavity’s decay time, rendering our entangling gate random and potentially useful for the implementation of Brownian circuits, for example. A charming instance of a quantum Rube Goldberg machine, if you ask me.

An autonomous clock’s tick deposits a photon into a cavity. The photon shifts the frequency of a coupled transmon into resonance with a second qubit, triggering a gate.

This example illustrates the spirit behind our autonomous-gate ideas. As mentioned before, we propose several more instances over different platforms. This all sounds nice and clean after looking at the end product. But the truth is that, when we started the project, I was a theory student with little knowledge about the inner workings of platforms, trying to pitch convincing arguments for novel gate protocols in these well-established technological fields. I had to go through decades of accumulated literature and famous experiments, trying to understand them and hopefully come up with ways to readapt them to the context of autonomy. Naturally, there were many old papers to read, many small details to understand that were obvious to a familiarized experimentalist but not to me, and, of course, many dumb questions to ask. Regarding this, I have to deeply thank Tom Manovitz, Norbert Linke and Simone Gasparinetti, our experimental collaborators, for their patience with these questions and their willingness to chat and clarify to a confused theorist what’s possible and interesting in experimental contexts.

Goldberg devised a classical autonomous napkin from a parrot and a clock. We devised a quantum-autonomous gate from an electromagnetic cavity and a clock. Maybe we’re not so different.

A special mention, however, to postdoc Yuxin Wang. Earlier in the text I mentioned how being relentless with one’s confusion is helpful; she’s one of the clearest examples I can think of. She came into the project right as I was struggling to model one of our superconducting-qubit gates. I was knee-deep in a convoluted formalism, drowning in inefficient numerical simulations that were leading nowhere. She asked a bunch of questions about our system and my modeling attempts for which I had no good answers for, especially because her expertise in both AMO and open systems really made my explanations look weak. I mean this in the best way I can: polishing my arguments and deeply enquiring about modeling decisions forced us to zoom out and look at simpler, more-efficient ways in which we could model our physical setup. I can’t even begin to quantify Yuxin’s patience and helpfulness in the project, nor how much I learned from working closely with her.

Research, of course, is about asking questions. Trying to clarify confusion and asking in the most transparent way possible are discomforts I still struggle with (much like with Zoom calls), but I’ve learned to appreciate these convoluted scenarios and sometimes even thrive in them. Not being the smartest or most-experienced person in a room has lots of perks. You can learn a lot, although you might need to ask one or two dumb questions to get there.

It turns on and off by itself,
it goes out after hours.
CHARLY GARCÍA
The happiness machine
︎
I do have, however, a handbook of criteria detailing what makes a useful autonomous quantum machine useful, though. ︎
Quantum thermodynamicists call these photon-emitting devices autonomous quantum clocks. ︎

by José Antonio Marín Guzmán at October 08, 2026 03:27 PM

Jordan Ellenberg — When the phone inside your ribcage rings, it’s not for me (They Might Be Giants show report)

Maybe the band I’ve seen play the most times, certainly the band with the biggest time gap between the first time I saw them and the latest (December 1989 to September 2026.) I saw them sometime in the last ten years but I may not have recorded it here; here’s a brief report on the 2009 show, which, like this one, was at the Barrymore, and which, like this one, featured a snippet of “The Famous Polka” (“When the phone inside your ribcage rings, it’s not for me”) as part of another song. They have such a long catalogue now that some of the songs that used to be every-show stalwarts weren’t there; no “Don’t Let’s Start,” no “She’s an Angel,” which may be songs that don’t go well with their current horn-heavy arrangement. (But why not bring back “(She was a) Hotel Detective,” which would sound great with this band?) No “Particle Man”! And they didn’t play the song that got me into the band in the first place, the song that taught me something about how superficial quirkiness can trick you into being defenseless when a song punches you in the face with bottomless sadness. (There’s a lot of this in the early TMBG catalog: “Now it’s either I’m dead and I haven’t done anything that I want / Or I’m still alive and there’s nothing I want to do” — these lyrics are disputed but I’m sticking with my version.) Anyway. Ana Ng and They Might Be Giants and I are getting old, and every five to ten years we walk in the glow of each other’s majestic presence, except for Ana Ng, who is antipodal to all of us, and will remain so, as long as the band still plays.

by JSE at October 08, 2026 03:06 AM

October 05, 2026

Jordan Ellenberg — Eigenvalues of random p-adic matrices

I always felt there ought to be a good theory of p-adic random matrices paralleling the very-well-worked out real and complex story. I wrote a paper years ago with Jain and Venkatesh that proposed that a very reasonable model for the lambda-invariant of a random quadratic field would be the number of eigenvalues x of a large random matrix A over Z_p such that x-1 is not a unit. (The p-primary part of the class group, the quantity studied in the Cohen-Lenstra conjecture, is modeled by the cokernel of A-1, which is clearly related but not at all the same thing.

Anyway, we worked out a few simple statistics of random matrices over Z_p but got no further. Now Jiahe Shen and Roger van Peski have done much more! They are well on their way to a fully satisfying theory that mimics the archimedean story. For instance, they can exactly compute the pair correlation showing the familiar repulsion of eigenvalues from each other. And the pair correlation function has a somewhat mysterious (to me, and I think to them too) appearance of a theta function…

One reason we like to model arithmetic quantities by random matrices is that, in the function field case, Frobenius acts on etale cohomology via a p-adic matrix we know nothing about, and since we know nothing about it, we like to treat it as random in Haar measure. Is that justified? Maybe not, but here’s the good news: Shen has a followup paper where he shows (again, consistent with what happens in archimedean-land) that the eigenvalue statistics they find don’t depend sensitively on the exact distribution of the matrices, but rather are universal over a rather broad class of distributions. So that, I think, should make one feel more confident that Frobenii, too, have eigenvalues that look this way, and moreover that the analogous distributions in the number field case, where there’s no literal p-adic matrix at all, might follow suit.

The method here is interesting: we learned, over time, that a good way to get a handle on a random group G is to study the expected number of surjections from G to H, as H varies over groups (or groups in various special classes) — these are the natural moments of a distribution on groups. And in practice, they are often the exact statistics one is able to compute by geometric means. (See Will Sawin and Melanie Wood’s work for a much, much, much more elaborated categorical point of view on what counts as a moment.) Shen’s paper takes the view that a good notion of “moment” for a random p-adic polynomial P is the distribution of the p-adic valuation of the resultant of P with some fixed polynomial Z. These “Z-moments” turn out to also be very computable for characteristic polynomials of random matrices, and they also turn out to provide actionable information about eigenvalues. Nice!

by JSE at October 05, 2026 02:50 AM

Jordan Ellenberg — James Purdy, Narrow Rooms

A great novel if you like murder, sex between men, grudges, West Virginia, and long sentences.

“His eyes were as bright as ever, perhaps brighter, the disc of the pupil appearing to move like a white fire, but his face in general gave the impression of belonging to someone who never expected anything again.”

Purdy’s an unusual writer. Beloved by a few, hated by many reviewers, unknown to almost everyone. Didn’t really start writing until he was 40 (he was a Spanish professor at Lawrence College in Appleton for ten years before that.) Hung out with painters and jazz guys in Chicago. Admired by Langston Hughes and Dame Edith Sitwell, the latter of whom he had his ashes buried next to. Withdrew the rights to the only movie adaptation his work was ever going to get because he didn’t like the proposed lead actor. This reminiscence by Donald Weise gives a vivid picture of the cranky, difficult man.

How did I get into James Purdy? I was in Firestone Library in Princeton, in Special Collections, to see the unpublished J.D. Salinger stories. When I asked for that box, the librarian visibly rolled her eyes, because I think probably 80% of the people who go into Firestone Special Collections are there to see the uncollected Salinger stories. Wounded, crouched, I quickly formulated a way to recoup — I would just request another box, showing that I was not another loser Salinger groupie, but a scholar of mid-century American fiction who was looking at the Salinger in a purely comparative way, setting his unpublished work against that of — well, who? I didn’t know anybody else on the list. So I took a wild guess and asked for the James Purdy box. And then, having taken it, felt compelled to read it. And then, having read it, realized I was in the presence of something a lot better than the stuff Salinger correctly declined to anthologize. I think Purdy would have liked knowing that he’d gotten a reader via a mechanism he cared about a lot and had drawn closely: a young man’s pride and banged-up vanity.

by JSE at October 05, 2026 01:44 AM

October 04, 2026

Scott Aaronson My new course at UT Austin: AI Alignment Theory

This semester, I’ve been teaching a brand-new course, entitled CS395T AI Alignment Theory. Here’s the course description:

The astounding progress of AI over the past decade has been accompanied by a rising fear: do we really understand how to align and control powerful AI systems—how to get them reliably to do what we wanted, or would want them to do on reflection, rather than merely what we said? If we succeed at building general-purpose superhuman intelligences along the current paradigm, should we expect that development to go well for humanity? Can we modify the design, training, monitoring, or scaffolding of those intelligences to help ensure that it goes well? While there’s been a great deal of recent empirical work touching on these questions, this course will concentrate mainly on theoretical and mathematical foundations. As a warning, the theoretical foundations of AI alignment have not yet gelled into any one coherent body of results accepted as canonical by the field. Nevertheless, in this course, we’ll read and debate many of the conceptual and mathematical works that have been most influential in the AI alignment field, from both before and during the current LLM revolution. Student presentations, reports, and projects will play a central role.

I vividly remember encountering Eliezer Yudkowsky and his Sequences 20 years ago. I remember thinking: even if these people talk and act like crazy cultists, still, let me bend over backwards to be epistemically virtuous, and entertain their ideas on their merits, as very few academics would. Even if, of course, I ultimately end up rejecting the ideas, on the simple ground that powerful AI is such an absurdly remote prospect that it’s almost impossible to say anything useful about it today, outside the realm of speculative fiction.

For my failure to see what was coming, it seems like an appropriate punishment that I’m now, in 2026, effectively teaching a course on Yudkowsky Studies. And it’s the most important course I can teach.

Well, for some definition of “teach.” The thing about AI alignment is that there’s no textbook (though apparently ILIAD is working on one), no core of nontrivial theorems considered canonical by the field, no real body of mathematical theory at all. This makes it extremely different from the courses I’m used to teaching, like Quantum Information Science or Computability and Complexity.

So we’ve been running the course as a discussion seminar. Every session, a “rapporteur” presents an AI alignment research paper or other reading; then I and others ask questions and discuss. Some of the readings (like Omohundro on the “basic AI drives,” or Hadfield-Menell et al. on the off-switch game) predate the current LLM revolution, while others (like the METR report on the HuggingFace incident or Dario Amodei’s “We Must Pace the Frontier”) are so timely that they were only released while the course was underway. Most are somewhere in between.

I expected to have to make a case to students about why AI alignment is a pressing concern, why it’s no longer science fiction, etc. There was huge demand for the course, and while of course there’s a selection effect, the students who’ve shown up have been extremely engaged, sometimes criticizing the assigned papers for not taking existential risk seriously enough.

Perhaps unsurprisingly, we didn’t get that criticism about our very first assigned reading, which was Eliezer Yudkowsky’s 2022 essay AGI Ruin: A List of Lethalities—one the most canonical statements of what Eliezer believes and why that’s shorter than a book. Which brings me to the topic of the rest of this post! Our rapporteurs are not merely presenting the papers in class; they’re also submitting written reports about what the papers said, what their own thoughts were, and what were the highlights of the class discussion. And, with student permission, I’ll be sharing those reports on this blog!

So, without further ado, I present to you our first report, on Eliezer’s list of lethalities, by Tennyson Bardwell, who I thank for his work. Feel free to discuss in the comment section; some of the students might also chime in. Expect more reports here over the coming weeks.

“AGI Ruin: A List of Lethalities” by Eliezer Yudkowsky: Rapporteur Report by Tennyson Bardwell

UT Austin has a new Computer Science course this fall. Alongside familiar graduate-level classes such as Advanced Computer Networks and Convex Optimization sits CS 395T: AI Alignment Theory, taught by Scott Aaronson. This is one of a growing number of AI Alignment courses taught at academic institutions. Just as concerns over catastrophic consequences for misaligned AGI systems reach a broader public discourse, Eliezer Yudkowsky—one of the loudest voices in the field and author of the first assigned reading in Professor Aaronson’s course—is declaring the cause hopeless.

Thus, the students of AI Alignment Theory began their semester by reading a laundry list of critical problems in AI Alignment research, how failure to solve those problems will result in catastrophic consequences, and the reasons to be pessimistic about both past and future progress on these problems. The essay by Eliezer, titled AGI Ruin: A List of Lethalities and posted to his popular community-driven website LessWrong in 2022, is divided into three sections.

Section A roughly describes the magnitude of the AI Alignment problem. That is, the magnitude of the consequences for a complete failure to align an AGI system to human values before construction. It posits that AGI would quickly catch up to all human knowledge simply by learning from existing human productions (colloquially referred to as “eating the internet”) and then, nearly as quickly, begin to meaningfully surpass human knowledge. AlphaGo Zero is presented as a model both for how this might happen, and how it might be difficult to correctly predict beforehand. Many believed that AlphaGo’s success in the board game Go was chiefly attributed to its ability to learn from the extensive history of human-played games. Less than a year after AlphaGo beat the best human player, the successor system AlphaGo Zero surpassed the original AlphaGo. Unlike its predecessor, AlphaGo Zero was trained in just three days by exclusively playing against itself without seeing a single human game.

This quick ramp from AGI to super-intelligence would pose a different sort of problem than humans are generally used to dealing with. Unlike traditional problems in science and engineering, the consequence for a failed attempt might not leave room for another try. An intelligent entity with a misaligned goal would be well aware that it stands in opposition to humans, and might act deceitfully until in a position to act openly against humans without jeopardizing its own survival. Since most goals benefit from control of power and resources, it seems likely that nearly any goal-driven intelligence would have ample opportunity to be misaligned with human desires.

Section B describes reasons why, by default, any AGI that humans build using current methods is likely to be unaligned even if considerable attention is paid to the topic. This “current method” is gradient descent. That is, incremental progress with respect to some loss function which “punishes” a model for undesirable behavior. A notoriously elusive property of such trained models is the ability to generalize out of their training distributions. To train a primitive model to be aligned to humans might involve learning a great many behavioral rules. However, the sorts of rules needed to keep a drastically smarter agent in check might not always be relevant to simpler models (e.g., “do not emotionally dysregulate humans you speak with” might not be relevant to a simpler model that is less able to reliably get under the skin of humans it operates with, or which is assigned tasks in training which do not benefit from such anti-social behavior).

Eliezer focuses on the misalignment of humans with their creators (evolution or evolutionary pressures) as a critical data point for reasoning about misaligned intelligent systems. Despite being a generally slow process, evolution eventually created a runaway intelligent system (Homo sapiens) which proceeded to dominate the globe, decimate related species, and eventually (it is forecasted) effectuate population decline. That last development is arguably in opposition to the sole imperative demanded by evolution: to reproduce.

Section B also makes time for criticism of the most popular paths toward AI alignment, including interpretability (unworkable, and attempting to train on it evokes Goodhart’s law, incentivizing deceit), using multiple AIs to maintain a balance of power (it is not clear how multiple strong AIs unaligned with humanity results in better outcomes for the weak humans), and corrigibility (it seems impossible to motivate an AI system to effect outcomes without also motivating it to desire its own survival to effectuate said outcomes).

Section C describes a bleak state of affairs in which veterans in AI alignment are unsatisfied with current progress and do not have a plan to deliver tangible solutions before the advent of AGI systems. In particular, Eliezer describes recent results as showy but useless. He believes that even with additional funding, the lack of appropriate evaluation mechanisms will prevent the most effective researchers from rising to the top.

A summary of the landscape, as described by Eliezer, in the flowchart below.

Figure 1: A flow chart of (select) paths described by Eliezer in his essay. A common feature of this flow chart is that many “good states”—such as disabling a misbehaving AGI or choosing not to build an AGI—are not “final” states in the sense that they are not permanent solutions. Such a state merely represent the avoidance of a single potential disaster, rather than the emergence of a new stable world state. Hence, these nodes posses back-arrows.

Despite the bleak content, Eliezer’s colorful prose inspired a lively class discussion. Before this discussion started, a survey was taken of the class’s predictions for various outcomes of the AGI in the coming years (with the full results below in figure 2). This survey asked students for their opinion of a number of statements. Each of these individual statement, if true, would reduce concerns of catastrophic AI-driven disasters. For example, when asked “How much do you agree with the statement: Humans will choose to not build AGI” half of respondents said they strongly disagreed with high confidence (agreement = 1, confidence = 5). Students also generally disagreed with the statements:

“AGI will not be technically feasible in our lifetime”

“(hyper-)AGI will not make extremely obviously unethical decisions”

“No reason is individually sufficient, but taken together they provide justification to not fear AGI”

There was a divergence in responses regarding interpretability, corrigibility, and “other” AI alignment research. In the latter two cases, a plurality of respondents (about a quarter) agreed strongly with statements that such research would defang AGI (agreement = 4, confidence=4), while most other responses express various levels of agreement with low confidence. However, when asked about the likelihood of interpretability research defanging AI, the pessimistic voices were more united. A quarter of responses still expressed the same optimism, but roughly half expressed pessimism (agreement ≤ 2) with half of those expressing at least moderate confidence (confidence ≥ 4). Based on the following discussion, this might have been caused by more familiarity with interpretability research, including first-hand experience.

The only statement with general agreement was “(hyper-)AGI will understand human intentions better than we can code it.” However, it should be noted that no statement such as “AGI will respect human desires, as it understand them” was asked on the survey.

Figure 2: Class Survey Results; conducted before a class-wide discussion. Note that students were instructed to answer confidence = 1 when they had not previously considered the statement, to answer confidence = 3 when they felt there were strong arguments on both sides, and to answer confidence = 5 when they possessed well-considered resolve.

After the survey was completed, the results were displayed as an open discussion began. Similar to recent empirical research from frontier labs, interpretability research received more airtime than in Eliezer’s article. Students disagreed first about the definition of interpretability: whether it refers to the ability to interpret a model’s behavior solely by its weights, to interpration via repeated probing of the model in a sandbox, or whether it can also refer to the modern chain-of-thought traces. Regardless of how it was defined, however, participants were either pessimistic or very pessimistic about interpretability research broadly. One student criticized common misunderstandings of chain of thought. Rather than being a verbatim copy of the models internal dialog, it is instead a superficial summary of the complete thought state and routinely produced gibberish, such as rarely used Chinese characters in the middle of otherwise English reasoning.

A popular topic was the exact shape and speed of a recursive self-improvement loop. If it takes place slowly, then what might we learn from “near misses” such as the Hugging Face incident? The number of near misses we are able to learn from before AI possesses sufficient power to prevent further iterations could depend on this curve, with some students arguing that the sheer number of humans, as well as their default robustness in the physical world compared to AI systems means that AI-driven extinction events are still a long way off. Bolstering this “slow take-off” opinion are rumors that AI already plays a major role in model development which could be interpreted as the start of this process.

Some criticized a focus on “solving ethics” as a needlessly high bar that distracts from the more mundane tasks dominating AI alignment work. In particular, the student volunteer who presented this paper (and the author of this report) included a section on “Ethical Dilemmas” in their presentation. Among arguments against focusing on abstract moral philosophy, Professor Aaronson cites Eliezer to emphasize that any alignment at all is difficult, not just in morally gray cases:

When I say that alignment is difficult, I mean that in practice, using the techniques we actually have, “please don’t disassemble literally everyone with probability roughly 1” is an overly large ask that we are not on course to get.

In response, I argue that some examination of everyday decisions with a critical lens—such as telling white lies to loved ones or consuming animal products—can help disabuse us of the notion that goodness emerges in every sufficiently intelligent agent.

One of the most interesting discussions was about the difference between state-of-the-art LLMs and the theorized AI agents long discussed in rationalist discourse. Since current LLMs “mimic the human distribution,” they come preloaded with extensive understanding of human social norms and moral behavior. This makes constitutional alignment (the current practices of using system prompts to establish ground rules) extremely effective. This might either fundamentally change the orthogonality thesis, or provide a new tool to better approximate human judgment in complicated situations.

Of all the points made, the one I found most interesting was simply (paraphrased):

I think human-alignment is just very tractable

Here, “human-alignment” refers not to AI alignment with human values, but cooperation between different humans. More specifically, it refers to the ability for human societies to choose not to rush recklessly into larger-and-larger AI systems. In an academic course focused on the technical problem of AI alignment, this was a reminder to not completely discard policy discussions in the believe that they lack any value. After all, many destructive technologies have been previously contained by international agreements. Notable examples include nuclear weapons and engineered plagues. However, even this was a contentious topic. The main criticisms were (1) the extreme “dual-use” nature of AIs for both peaceful growth and warfare, and (2) the greater danger for AI escapes even after taking precautions to prevent it. However, in the interest of ending on an optimistic note—unlike the assigned reading—it is on this belief in human cooperation that I will leave you.

by Scott at October 04, 2026 10:13 PM

Doug Natelson — The arXiv - where it's been and where it's going

Back in the ancient mists of time, scientists and mathematicians would circulate preprints of their articles among friends and colleagues via the postal service, as a courtesy, to get feedback and to try to make sure people in the community were aware of their forthcoming work. As the wikipedia entry says, with the advent of widespread LaTeX and the development of the web, Paul Ginsparg (then at LANL) put together an html-based site for electronic sharing of preprints, initially at xxx.lanl.gov (back before "xxx" in URLs was the kind of thing filtered and blocked by employers). In 2001 he moved to Cornell, and by then the lab was perhaps relieved to see the site, rebranded as the arXiv, shift to Cornell's library/repository infrastructure. PIs my age remember (fondly? maybe?) the old days of the arXiv, when Prof. Ginsparg's rather dry sense of humor pervaded the site. The skull-and-crossbones logo. The "help"/FAQ pages that basically said, "if you can't figure out how to .tar.gz all of your necessary LaTeX files, and you can't figure out how to make your .eps figures small, maybe you should reconsider whether you're smart enough to be sharing your ideas here". The arXiv was for preprints, without peer review (though interesting follow-on sites like Scirate now exist for organized commenting on the articles).

From these modest beginnings, the arXiv has grown enormously, including imitators/spin-offs such as chemrxiv and biorXiv and socarXiv. The arXiv has recently become an independent nonprofit, hired a CEO, secured multiyear philanthropic support, and hosts over 3 million articles. It's been interesting seeing some level of complaints online about some of these steps, but when the audience is so large, not everyone is going to be happy. Rapid growth has been a major issue - see here for a graph of monthly submissions:
The exponential rise (except for a slight pandemic-correlated shoulder) has been problematic, especially recently. One hallmark of the arXiv over the years has been its ability to function with comparatively minimal need for "moderation". Early on, one consequence of the minimalistic help and moderate technical entry barrier was that it was unusual for fringe/pseudoscience to make its way onto the server. (Hence the establishment of viXra.) With the ease of cranking out properly formatted readable manuscripts using AI, clearly the arXiv has been struggling. If 10-15% of submissions need some kind of human intervention or review, the support needs are rapidly outpacing the limited count of support staff. There can be substantial backlogs. To help deal with this, the arXiv recently updated its policies regarding AI-generated content (and AI cannot be a co-author, because the AI tools cannot take responsibility for content), and most recently has had to limit submission rates to two papers per month per submitting author. These moves, too, have drawn some criticism (e.g. here). Personally, I think the operators of the arXiv face an incredibly challenging environment and are doing the best they can - the idea that they are making moves because they are establishment sticks in the mud who don't understand the New Way of Doing Science is just wrong-headed.

It's completely unclear where all this is heading. Exponential growth in nature signals instability and does not continue forever. If proponents of very heavily AI-driven research want to establish a repository specifically for that work, that's up to them. [It is very on brand for the hard core AI advocates to argue that the arXiv is somehow morally obligated to host everything (regardless of hardware or personnel costs) so that future AI tools can read everything (a repository growing too quickly for human researchers to keep up) and summarize it.]

One overarching point that should come up in any arXiv discussion: The arXiv has become a global repository for an enormous amount of human knowledge, without charging anyone publication fees. This should make interested parties think reallllllly hard about economic models of for-profit publishers.

by Douglas Natelson at October 04, 2026 07:35 PM

October 02, 2026

Matt von Hippel — Beginnings Don’t Matter as Much as You’d Think

I’ve met people who think they’re uncomfortable with evolution, but when you actually talk to them, they’re worried about something very different. They ask, “How could something come from nothing? How could life come from non-life?”

Biologists aren’t sure of the origin of life. They’re working on it, there’s a whole research field on what the experts call abiogenesis. But it’s hard going, because there’s so little direct evidence.

There is a lot more evidence for evolution. Evolution isn’t a theory of where life comes from, it was never meant to be one. It’s a theory that explains the copious evidence we have from the history of life, from the fossil record to genetic comparisons to leftover features in related animals. It’s a theory that tells you what to expect when you already have some life, and wait a few generations.

Evolution and abiogenesis are different ideas, with different domains. You can have one without the other. And when you already have life, and want to understand it, evolution is quite a bit more useful than abiogenesis.

There’s a similar story with the big bang.

Some people get mystified at the idea that the big bang made something out of nothing. But that’s not really how the big bang works.

It’s not just that the big bang wasn’t the explosion you imagine, a single point expanding in an empty space. Rather, it was a time when the entire universe, everywhere that anything could be, was in a hot dense state, not empty but brim-full.

It’s that the most important part of the big bang is what happened next. As the universe cooled, things spread away from each other, like the residents of Hilbert’s Hotel all moving one room apart. Clumps formed, electrons slowing down and getting caught by protons to make atoms, dust pulling on dust until it collapsed into stars.

The patterns of a universe that cooled this way are distributed over the sky. They tell us the kinds of atoms we expect to see in spectra of stars, the distribution of galaxies, the ripples in the afterglow of the first light to avoid getting absorbed by hot plasma. Those patterns are the core of physical cosmology, the bulk of the field’s research and its strongest argument.

In contrast, the question of where that hot dense state came from is murkier. There are a variety of pictures, different attempts to approximate a context where gravity’s quantum properties should be especially relevant by people who still don’t have a clear idea of how to think about quantum gravity.

But while the question is murky, it’s also just not very useful. Wherever that state came from, knowing it exists is usually enough to chart the rest of the universe’s history. In fact, it may even be impossible for us to know what happened before a certain time: we have only one sky’s worth of evidence, and eventually that evidence runs out.

People are used to created things: food, clothes, cars. Created things don’t change in complicated ways over time, for the most part, just break down. To understand a created thing, understanding its origin matters.

But science is full of stranger things, things with a life, or dynamics, of their own. Natural phenomena change, swiftly becoming something other than what they began as. Understanding them means understanding those changes, much more than it does understanding how they began.

by 4gravitons at October 02, 2026 04:00 PM

September 29, 2026

Scott Aaronson My “Knowmads” podcast on science and AI

Or click here if the above doesn’t work.

Recorded in-person in my office at UT Austin, with a bulleted list containing “ARC,” “Scalable Oversight,” and “Models” behind me on my blackboard for some reason (I no longer remember who put those there or why). 90 minutes long. Sometimes you see my disembodied arm waving in midair because of the way the cameras are combined. As always, I strongly recommend 2x speed for the correct experience.

This might actually be one of my best podcasts ever, although I wasn’t planning on that! Thanks so much to Bhavay Tyagi and Prachi Garella for driving all the way from Houston to record it.

Here’s a strict subset of the topics we covered:

The story of AI models solving the Navier-Stokes Millennium Problem, insofar as it’s known

Can recent AI proofs be called “truly creative”?

The history of AI before the LLM revolution

What do we mean when we call LLMs “black boxes”?

The achievements of the field of interpretability

What exactly happened in the OpenAI/HuggingFace incident

Must we avoid all “anthropomorphizing language” when discussing the HuggingFace incident? (spoiler alert: no)

Examples of major open problems in quantum computing theory that I cared about for decades and that AI models have recently solved

Effects of the current AI cataclysm on the math community, especially students

What annoys me the most when I listen to AI talks

My experiences at OpenAI, why they hired me, and the watermarking work that I did there

Enjoy!

More AI-related content coming soon, as this blog—like much of the rest of the world—continues its transition to “all AI, all the time” (except still 100% written by an aging, deteriorating biological brain)

And for those who just can’t get enough of my rocking back and forth, using too many filler words, as I explain theoretical computer science! Here’s a second podcast, this one mainly on quantum computing, with Seb Agertoft, who I thank for doing it. Enjoy!

by Scott at September 29, 2026 04:22 PM

Doug Natelson — Annual Nobel speculation thread

It's that time of year yet again. The physics Nobel will be announced next Tuesday, and the chemistry prize on Wednesday. Who will it be this time? Please speculate in the comments. As is my annual futile tradition, I will put forward that the physics prize could be Aharonov and Berry for geometric phases in physics (even though Pancharatnam is intellectually in there and died in 1969). This is a long shot, as always, especially since last year's prize for “for the discovery of macroscopic quantum mechanical tunnelling and energy quantisation in an electric circuit” was condensed matter related. Perhaps optical metamaterials, with Yablonovitch, Smith, and Pendry? (Though that's another one where picking just three names is challenging.) Astro-related is likely due, so perhaps something CMBR related?

by Douglas Natelson at September 29, 2026 01:41 PM

September 27, 2026

John Preskill — Marcus theory and Marcus practice

Rudy Marcus was 91 years old when I took a course from him. His age was, arguably, not his most remarkable quality.

The course took place during the second spring of my PhD. I was completing not only the academic year, but also my course requirements. Why not, I figured, indulge in traditional statistical mechanics? Statistical mechanics neighbors, and partially overlaps with, thermodynamics. Modern approaches to statistical mechanics include fancy toolkits imparted in physics courses: the renormalization-group method; Ginzburg–Landau theory; and the latter theory’s successor, topological order. I craved a more traditional approach, as won’t surprise readers of this blog. So I ditched the physics course list and signed up for Chemistry 166.

Rudy didn’t disappoint. He began the course with a tradition passed down from a founder of thermodynamics: Boltzmann’s H theorem. The theorem amounts to the first argument ever published for the second law of thermodynamics: every (sufficiently large) closed, isolated system’s entropy increases or remains constant; it doesn’t decrease. Rederiving Boltzmann’s H theorem, I felt as I would while holding a lace collar worn by Queen Elizabeth I: as though I were touching history.

Rudy proceeded through more of the greatest hits in statistical mechanics. For instance, the Fokker–Planck and Langevin equations describe random processes. Examples include the subject of one of Einstein’s most famous papers: the motion of a pollen grain suspended in a liquid and buffeted by the liquid’s molecules. Chemistry 166 provided much of the bread and butter, meat and potatoes, and even tofu and edamame of a thermodynamicist’s education.

Rudy advanced step by step, rederiving every result in class. He recommended a textbook; but I refer to my lecture notes, rather than to the text, to refresh my memory about course topics nowadays. When I encounter complex integrals (a certain type of calculus) in statistical mechanics, I think back to Rudy’s treatment of them. The course also crystallized my understanding of the fluctuation–dissipation theorem, which describes systems perturbed slightly out of equilibrium; examples include a magnet subjected to a weak magnetic field.¹ For the course’s final project, I studied quantum master equations—which featured in a Quantum Frontiers post afterward.

No less than his explanations of science, Rudy’s enthusiasm for science impressed itself upon me. Again, he taught Chemistry 166 at age 91. He continued to collaborate on research. Whenever I saw him, even after the course ended, he asked for news about quantum information theory—which he didn’t work on.

Two and a half years after the course, Rudy gave a speech at a dinner in honor of a couple’s engagement. The groom was pursuing a PhD in applied physics, and the bride worked as an engineer. The groom asked if Rudy had advice for young scientists. Rudy responded immediately: plenty of science remains to be done, so keep exploring.

During his speech, Rudy described his upbringing in Montreal. His high school had produced a “who’s who” of Canadians, in his words—and not because the school demanded high tuitions or offered tutors and social connections. Discrimination pushed Jewish working-class children out of other schools and into Baron Byng, where some students worked hard enough to thrive. Alumni include artists; judges; physicists; William Shatner, who played Captain Kirk in the original Star Trek series…and Rudy Marcus.

Which leads me to the quality arguably more remarkable than Rudy’s teaching Chemistry 166 at age 91: he’d won a Nobel Prize in chemistry. Unlike most chemists, Rudy worked as a theorist, rather than an experimentalist. He developed a framework now called Marcus theory. It predicts the rate at which electrons hop between molecules. I can’t tell you much more about Marcus theory because I don’t know much more about it: Rudy had enough humility that, to my memory, he mentioned his theory only once in class, in passing.

Although I know little about Marcus theory, I’ve seen much of Marcus practice. In 2022, I emailed Rudy to ask where a holiday card could reach him. The pandemic had isolated enough people that I’d determined to reach out to more acquaintances. Rudy replied, “I have been working at home, meeting twice a week on zoom with my now small research group.” He was 98 years old. His curiosity, learning, mentoring, and exploring continued.

Rudy passed away this July, five days before his 103rd birthday. I hope to learn Marcus theory someday. But I’d be even more grateful to undertake Marcus practice for even a decent fraction of the time for which he did.

¹How rapidly does the system respond to the perturbation? This nonequilibrium response depends on a correlation function evaluated on an equilibrium state, according to the fluctuation–dissipation theorem. To understand why, imagine preparing the system in a thermal state, perturbing the system at an early time, and measuring the system later. What information can you extract about the perturbation? A two-time correlation function encodes this information. Imagine Fourier-transforming a two-time correlation function. The Fourier transform is proportional to a second derivative of a free energy, according to the fluctuation–dissipation theorem. (The second derivative should sound plausible because it depends on a two-time correlation function.) Second derivatives of free energies equal response functions, such as the magnetic susceptibility. Therefore, a nonequilibrium response depends on an equilibrium correlator.

by Nicole Yunger Halpern at September 27, 2026 07:29 PM

Tommaso Dorigo — Flying With Gamma-Ray Counters: Another Test Or Two Of The Radiacode Zero

Flying With Gamma-Ray Counters: Another Test Or Two Of The Radiacode Zero
Yesterday I traveled all day, moving my family back to Lulea from Padova. The trip is long because there is no direct connection from Venice to Stockholm (Norwegian has one, but I prefer to avoid them as they canceled my flight recently).

Tommaso Dorigo Sun, 09/27/2026 - 09:24

Categories

Physics

by Tommaso Dorigo at September 27, 2026 01:24 PM

September 25, 2026

Jordan Ellenberg — I love working here, II

Making final revisions to Don’t Be Too Sure on the Terrace. An all-tuba band on the Terrace Stage playing an all-tuba cover of “Black Hole Sun.”

Sorry for light blogging lately. See “making final revisions” above. Just realized I misspelled the first name of Katharine Briggs, who’s mentioned several times and who I read a whole book about. How many mistakes are left in this thing? (I suppose if I’d kept a rigorous log of mistakes found and plotted it over time, I could curve-fit and estimate this.)

by JSE at September 25, 2026 10:40 PM

n-Category Café Binomial Coefficient Coincidences

$MathML-enabled post (click for more details).$

These seven equations between binomial coefficients are ‘coincidences’ — they aren’t among the four known infinite families of such equations:

$\binom{16}{2} = \binom{10}{3} = 120$

$\binom{21}{2} = \binom{10}{4} = 210$

$\binom{56}{2} = \binom{22}{3} = 1540$

$\binom{78}{2} = \binom{14}{6} = 3003$

$\binom{120}{2} = \binom{36}{3} = 7140$

$\binom{153}{2} = \binom{19}{5} = 11628$

$\binom{221}{2} = \binom{17}{8} = 24310$

$MathML-enabled post (click for more details).$

In 1997, De Weger conjectured that every equation between binomial coefficients follows from the four known systematic equations and the seven coincidences I showed you:

• Benjamin M. M. de Weger, Equal binomial coefficients: some elementary considerations, Journal of Number Theory 63, no. 2 (1997), 373–386.

At that time, he and his collaborators checked there were no others involving binomial coefficients less than 10³⁰. Later they checked that there are none involving binomial coefficients less than 10⁶⁰:

• Aart Blokhuis, Andries Brouwer and Benne de Weger, Binomial collisions and near collisions.

So, De Weger’s conjecture stands open. The four infinite families, by the way, are these:

$\begin{array}{cccl} \displaystyle{ \binom{n}{k} } &=& \displaystyle{ \binom{n}{n-k} }& \qquad 0 \le k \le n \\ \\ \displaystyle{ \binom{n}{0} } &=& 1 & \qquad n \ge 0 \\ \\ \displaystyle{ \binom{\binom{n}{k}}{1} } &=& \displaystyle{\binom{n}{k}} & \qquad 0 \le k \le n \end{array}$

and the only nontrivial one: the Lind–Singmaster family involving the Fibonacci numbers $F_i$ where $F_0 = 0,\ F_1 = 1$ :

$\displaystyle{ \binom{F_{2i+2}F_{2i+3}}{\,F_{2i}F_{2i+3}\,} \;=\; \binom{F_{2i+2}F_{2i+3}-1}{\,F_{2i}F_{2i+3}+1} \qquad i = 1,2,3,\dots }$

The first three equations in the Lind–Singmaster family are these:

$\begin{array}{ccc} \displaystyle{ \binom{15}{5} } &= &\displaystyle{\binom{14}{6}} \\ \\ \displaystyle{\binom{104}{39} } &= &\displaystyle{\binom{103}{40}} \\ \\ \displaystyle{ \binom{714}{272} } &=& \displaystyle{\binom{713}{273}} \end{array}$

When I said De Weger conjectured every equation between binomial coefficients follows from the four known systematic equations and the seven coincidences, I really meant it. For example, the first of the Lind-Singmaster equations combines with the coincidence

$\binom{78}{2} = \binom{14}{6} = 3003$

and other systematic equations to give

$\begin{array}{cccccccccc} & \binom{3003}{1} &=& \binom{78}{2} &=& \binom{15}{5} &=& \binom{14}{6} \\ \\ = & \binom{3003}{3002} &=& \binom{78}{76} &=& \binom{15}{10} &=& \binom{14}{8} \end{array}$

In fact, if de Weger’s conjecture is true, the number 3003 shows up as a binomial coefficient in more ways than any other positive integer!

I’ll explain the Lind–Singmaster family later. Here’s the question I’m most interested in now:

Is there any good explanation for the seven binomial coefficient coincidences?

My collaborator Paul Schwahn found a beautiful explanation of the first one, namely

$\displaystyle{ \binom{10}{3} = \binom{16}{2} }$

His explanation uses representation theory. The Lie algebra $\mathfrak{so}(10)$ has a 10-dimensional representation, the ‘vector’ representation $V_{10}$ , and also two 16-dimensional representations, the ‘left and right-handed spinor’ representations $S^\pm_{10}.$ There’s an isomorphism of representations

$\displaystyle{\Lambda^2 S^+_{10} \cong \Lambda^3 V_{10} }$

and similarly for $S^-_{10},$ but we might as well work with $S^+_{10}.$ Here $\Lambda^k$ means the kth exterior power. For any vector space $X$ we have

$\displaystyle{ \dim(\Lambda^k X) = \binom{\dim X}{k} }$

Thus, taking dimensions, the isomorphism of representations

$\displaystyle{\Lambda^2 S^+_{10} \cong \Lambda^3 V_{10} }$

instantly gives

$\displaystyle{ \binom{16}{2} = \binom{10}{3} }$

It is not super-easy to prove this isomorphism of representations, but it’s still nice to find a deeper layer of meaning underlying what might otherwise seem like a meaningless coincidence!

Can we find good explanations for the other six coincidences?

First let me explain two failed attempts using representation theory, and then three successes using combinatorics.

Representation theory

First: we can look for isomorphisms like

$\displaystyle{\Lambda^2 S^+_{10} \cong \Lambda^3 V_{10} }$

involving representations of $\mathfrak{so}(n)$ for larger $n.$ In fact this isomorphism is part of a pattern! The next one involves the vector and left-handed spinor representations of $\mathfrak{so}(12).$ But it’s this:

$\displaystyle{ \Lambda^2 S^+_{12} \cong \Lambda^4 V_{12} \oplus \Lambda^0 V_{12} }$

so it gives

$\displaystyle{ \binom{32}{2} = \binom{12}{4} + 1 }$

or

$\displaystyle{ 496 = 495 + 1 }$

So we fail to get an equation between binomial coefficients, but we’re close: we’re only off by one.

The second paper I cited, Binomial collisions and near collisions, presents a list of cases where two binomial coefficients differ by one. This is on the list. So we failed to explain an equation between binomial coefficients, but we explained a near-miss.

Here’s another failed attempt at explaining an equation between binomial coefficients. The equation

$\displaystyle{ \binom{78}{2} = \binom{14}{6} = 3003 }$

is fascinating to anyone who knows their exceptional Lie groups. 78 is the dimension of $\mathrm{E}_6,$ while 14 is the dimension of $\mathrm{G}_2.$ $\mathrm{G}_2$ is a subgroup of $\mathrm{E}_6$ because $\mathrm{G}_2$ is the automorphism group of the octonions and $\mathrm{E}_6$ is the isometry group of the bioctonionic plane. We’d get the above equation if the 2nd exterior power of the adjoint representation of $\mathrm{E}_6,$ upon being restricted to $\mathrm{G}_2,$ were isomorphic to the 6th exterior power of the adjoint representation of $\mathrm{G}_2.$

Amazingly, it seems these two representations of $\mathrm{G}_2$ are not isomorphic even though their dimensions are the same: both 3003.

Even more amazingly, $\mathrm{E}_6$ and $\mathrm{G}_2$ both have irreducible representations of dimension 3003, but they are not the representations I just mentioned.

I would be happy for someone to check these two claims.

Combinatorics

It turns out that the following three coincidences can all be explained by the same style of combinatorial argument:

$\binom{21}{2} = \binom{10}{4}, \qquad \binom{153}{2} = \binom{19}{5}, \qquad \binom{78}{2} = \binom{14}{6}$

The argument is not very elegant, but it’s moderately interesting. Mike Stay got it started by asking ChatGPT, and I finished it off.

In each case the argument has three steps. The first two steps are what combinatorialists call bijective proofs: we prove an equation between numbers by proving a natural isomorphism between structures on sets and then taking cardinalities. The third step is non-bijective because it involves taking an equation and dividing both sides by the same number. Maybe we can make it closer to bijective by using groupoid cardinality, which allows for division, but I haven’t tried that.

Step 1: pairs of edges in a complete graph

Starting from the left-hand binomial coefficient in each equation we’re trying to prove, note that the number on top is a triangular number:

$21 = \binom{7}{2}, \qquad 153 = \binom{18}{2}, \qquad 78 = \binom{13}{2}$

Note that $\binom{\binom{n}{2}}{2}$ counts unordered pairs of distinct edges in the complete graph on $n$ vertices. Two distinct edges either share a vertex or are disjoint, so there are two cases:

• Sharing a vertex: the pair spans 3 vertices and is determined by this 3-element set together with a choice of which vertex is shared. That gives $3\binom{n}{3}$ pairs.

• Disjoint: the pair spans 4 vertices and is determined by this 4-element set together with one of its 3 splittings into two pairs. That gives $3 \binom{n}{4}$ pairs.

By Pascal’s rule, $\binom{n}{3} + \binom{n}{4} = \binom{n+1}{4}.$ But this also has a bijective proof: add a new point to the set of $n$ vertices, and a 4-element subset of the enlarged set either contains the new point (so involves a 3-element subset of the old ones) or does not (so involves a 4-element subset of the old ones). Hence we have a bijective proof that

$\binom{\binom{n}{2}}{2} = 3\binom{n+1}{4}$

This holds for all $n.$ For our three cases we get bijective proofs of the following equations:

$\binom{21}{2} = 3\binom{8}{4}, \qquad \binom{153}{2} = 3\binom{19}{4}, \qquad \binom{78}{2} = 3\binom{14}{4}.$

It thus remains to prove

$3\binom{8}{4} = \binom{10}{4}, \qquad 3\binom{19}{4} = \binom{19}{5}, \qquad 3\binom{14}{4} = \binom{14}{6}.$

Step 2: a double count

Each of the above equations follows by counting a single set in two ways, and then a nonbijective step: dividing these counts by the same number.

• Showing $3\binom{8}{4} = \binom{10}{4}$ . In a 10-element set, count triples $(S, x, y)$ in two different ways, where $S$ is a 4-element subset and $x, y$ are distinct points outside $S$ . Choosing $S$ first we see there are $\binom{10}{4} \cdot 6 \cdot 5 = 30\binom{10}{4}$ triples. Choosing $x$ and $y$ first we see there are $10 \cdot 9 \cdot \binom{8}{4} = 90\binom{8}{4}$ triples. So, we get a bijective proof that $90\binom{8}{4} = 30\binom{10}{4}$ You can see what we’ll do next.

• Showing $3\binom{19}{4} = \binom{19}{5}.$ In a 19-element set, count pairs $A \subset B$ with $|A| = 4$ and $|B| = 5$ . On the one hand, there are $\binom{19}{4}$ choices of $A,$ and for each there are 15 choices of $B$ since we can add an extra point in 15 ways. On the other hand, there are $\binom{19}{5}$ choices of $B,$ and for each there are 5 choices of $A$ since we can choose $A$ in $\binom{5}{4} = 5$ ways. So we get a bijective proof that $15\binom{19}{4} = 5\binom{19}{5}$ Again, you can see what we’ll do next!

• Showing $3\binom{14}{4} = \binom{14}{6}.$ In a 14-element set, count pairs $A \subset B$ with $|A| = 4$ and $|B| = 6$ . On the one hand, there are $\binom{14}{4}$ choices of $A,$ and for each there are 45 choices of $B$ since we can add 2 other points in $\binom{10}{2} = 45$ ways. On the other hand there are $\binom{14}{6}$ choices of $B$ and for each there are 15 choices of $A$ since we can choose 4 points in $\binom{6}{4} = 15$ ways. This gives a bijective proof that $45\binom{14}{4} = 15\binom{14}{6}$ Again you can see what we’ll do next.

Step 3: division

Having proved

$90\binom{8}{4} = 30\binom{10}{4}, \qquad 15\binom{19}{4} = 5\binom{19}{5}, \quad 45\binom{14}{4} = 15\binom{14}{6}$

we can now divide by 30, 5 and 15, respectively, and get

$3 \binom{8}{4} = \binom{10}{4} , \qquad 3 \binom{19}{4} = \binom{19}{5}, \quad 3\binom{14}{4} = \binom{14}{6}$

as we wanted, completing our proof that

$\binom{21}{2} = \binom{10}{4}, \qquad \binom{153}{2} = \binom{19}{5}, \qquad \binom{78}{2} = \binom{14}{6}.$

I don’t see how to use tricks of the same general sort to explain the remaining coincidences

$\binom{56}{2} = \binom{22}{3}, \qquad \binom{120}{2} = \binom{36}{3}, \qquad \binom{221}{2} = \binom{17}{8}.$

The Lind–Singmaster family

Lind and Singmaster were trying to find all $n,k$ with

$\displaystyle{ \binom{n}{k} = \binom{n-1}{k+1} }$

I’ll rapidly sketch the key steps of their argument. Simplifying the equation above we get

$n(k+1) = (n-k)(n-k-1)$

or

$n^2 - (3k+2)n + (k^2+k) = 0$

Solve for $n$ using the quadratic formula. This formula turns out to have

$\sqrt{5k^2 +8k+4}$

in it. So we need $5k^2 +8k+4$ to be a perfect square!

Now we’re trying to find integer solutions of

$5k^2+8k+4 = m^2$

A quadratic diophantine equation! Multiply by 5 and complete the square:

$5m^2 = (5k+4)^2 + 4$

$y = 5k+4$ is an integer when $k$ is, so we need to find integer solutions of

$y^2 - 5m^2 = -4$

This is a ‘Pell equation’, and people know how to solve these. In this particular case we get all the solutions from this fact:

$L_n^2 - 5F_n^2 = 4(-1)^n$

where $F_n$ are the Fibonacci numbers 0, 1, 1, 2, 3, … and $L_n$ are the Lucas numbers 2, 1, 3, 4, 7, …. These are two sequences satisfying the same famous recurrence relation, just with different initial conditions.

We want $n$ odd, to get

$L_n^2 - 5F_n^2 = -4$

It turns out $y = L_n, m = F_n$ with $n$ odd give all solutions of the Pell equation

$y^2 - 5m^2 = -4$

However, remember I said $y = 5k+4$ is an integer when $k$ is. But the converse isn’t always true, and we need $k$ to be an integer! This clearly happens iff $y \equiv 4 \bmod 5.$

So we need to know when $L_n \equiv 4 \bmod 5$ Apparently this happens iff $n \equiv 3 \bmod 4.$ I won’t think about this now… but this is the last hard step.

In summary, we’ve seen

$\displaystyle{ \binom{n}{k} = \binom{n-1}{k+1} }$

if and only if $y = 5k+4$ is a Lucas number $L_n$ with $n \equiv 3 \bmod 4.$ We could quit here, but people like to use the identity

$L_{4i+3} - 4 = 5F_{2i} F_{2i+3}$

to get a formula for $k$ in terms of Fibonacci numbers. This is gilding the lily, I’d say, but it eventually leads to the formula that de Weger presents:

$\displaystyle{ \binom{F_{2i+2}F_{2i+3}}{\,F_{2i}F_{2i+3}\,} \;=\; \binom{F_{2i+2}F_{2i+3}-1}{\,F_{2i}F_{2i+3}+1\,}, \qquad i = 1,2,3,\dots }$

The takeaway message is: our problem can easily be reduced to a quadratic diophantine equation, then put in Pell form… and it’s known that the sequence of integer solutions of a Pell equation obeys a linear recurrence relation! We luck out in this case and get solutions connected to Lucas and Fibonacci numbers.

It’s probably a lot easier to show that the Lind–Singmaster equations hold. The above argument does a lot more, by showing that every equation of the form

$\displaystyle{ \binom{n}{k} = \binom{n-1}{k+1} }$

is in the Lind–Singmaster family. De Weger gets into a lot more number theory trying to rule out various other kinds of coincidences between binomial coefficients. It’s downright scary how these questions pull you into the deep, icy waters of mathematics.

© john at September 25, 2026 08:38 PM

Matt von Hippel — It Got to My Field

For the folks who found this blog from Anthropic’s site, welcome!

For everyone else, I’d better give some context.

Last month, I had a blog post titled “It Only Counts When AI Gets to My Field”. The title was a joke, the content less so. I said that if some of the longstanding problems of my old research field got solved by AI, then I’d sit up and take notice.

The post turned out to be a bit of a self-fulfilling prophecy, after some folks at Anthropic read it and decided to tackle one of the outstanding problems I mentioned. They then invited me to write a post about it on their science blog.

The post is here. I recommend reading it, then coming back. The rest of this post will be a little Q&A.

Q: At the end of that post, it says Anthropic paid you for your time. Can we trust what you wrote?

A: When I agreed to write the post, I made it clear that I was going to give my own opinion, not write an ad for Anthropic. They could propose light edits, but that’s it. And I avoided signing anything with them, not even an NDA, so I could freely tell you if they pushed my boundaries.

They didn’t push my boundaries. They asked me to clarify a few things, and to give more detail on the science. I didn’t change the message or the takeaways.

The reason I asked them to pay me is that, as a freelance writer, I don’t have a salary to fall back on. Time I spend on a project like that post is time I’m not spending on journalistic projects, so if I worked on it for free I’d essentially be using my vacation time for it. The rate I’m charging them is roughly in the middle of what I’d have gotten paid if I used that time on journalism: a bit more than I would have made working for the lower-paying outlets, a bit less than I would have made with the higher-paying ones.

I’m also not expecting this to be the start of a longer-term business relationship or anything like that. So overall, I don’t think I’m incentivized to lie on their behalf. You can trust me.

Q: How confident are you that they did what they said they did?

A: I didn’t get the feeling they were lying. But I didn’t start out skeptical.

I haven’t seen the logs from the LLM, or anything like that. That’s the kind of thing I would dig into if I were more suspicious of their story. If there’s a reason to be suspicious, I’ll ask.

But so far, I don’t feel that I have much reason to be suspicious. They were able to look up details on the fly when I asked them, they didn’t seem to have a polished message they were trying to push past me. And more importantly, as I’ll mention in a bit, I don’t think anything they described is all that outlandish. It lines up, largely, with the capabilities I’d expect their Claude Science platform to have.

It’s also relevant that they didn’t use an internal model for this. It means that scientists will likely be trying out similar problems soon, so if it does turn out they exaggerated something, people are going to figure out quite quickly.

Q: So how big is this? We just saw AI solve a Millennium problem, after all.

A: This was a lot easier than a Millennium problem. But it was also a lot cheaper.

The problem they solved is one with a clear recipe, honed and explained over multiple papers. It’s something I expected to be hard to do without access to a lot of computer power, so I thought it would need to be approached with a novel technique, in order to avoid using that computer power.

In the end, it didn’t need that. As I mention in the post, another group got most of the result at around the same time, with a much smaller amount of AI assistance.

It’s not an easy recipe, to be clear. I think other people will be surprised that a reasonably affordable program like Claude Science can do this without a lot of guidance. I’m not surprised, mostly, because I’ve been paying enough attention to what people have been doing with these things, and carrying out this kind of recipe consistently is actually something the good science AIs can do right now. If you haven’t been following as closely, that’s going to be a lot more surprising.

So maybe the best way to answer this is: instead of paying $15 million to solve one of the most famous problems in mathematics, they paid $1,000 or so to make a significant next step in an ongoing research program in theoretical physics. This doesn’t tell you much about whether AI can achieve field-defining breakthroughs, but it tells you a lot about the kinds of things a theoretical physicist with $1000 spare budget can do right now.

(As an aside, it feels crazy to me that 90% or more of the compute cost was from the LLM, not the calculation itself. On the one hand, it feels nuts to essentially use ten times more computer power to do this than would have been needed if a human had done the coding. On the other hand, I’ve applied for grants that budgeted more than that per year for travel costs, and spending this kind of money on getting a result seems a lot more useful than spending it on airfare.)

Q: Isn’t it reckless to publicly post a problem for AI like that?

A: I posted my challenge before OpenAI announced their Navier-Stokes result. At that point, there had been a few awkward surprises, but for the most part AI companies had a pattern of talking to an academic first before trying to solve a problem. It’s what they did back in March.

I expected that was what they’d do this time, and got more than a little blindsided when they just solved the problem on their own and reached out to Lance afterwards. I’m lucky that Lance doesn’t seem mad at me about it, it would have been quite understandable if he was.

I did tell them that if they wanted to tackle any of the other problems in that post, they should reach out to one of the scientists involved first, before attempting it. In addition to being polite, it’s a way to make sure that they understand the problem correctly and have a plan to verify the result. They happened to pick a challenge that was particularly easy to verify, but it didn’t have to go that way.

Q: Any thoughts about AI’s future impact on science?

A: If there’s a problem you can’t solve, but you think someone with expertise in another field or better programming skills could, then it’s probably solvable with AI.

A lot of problems don’t fall into that category. I’ve seen people talking online about AI solving quantum gravity, but solving quantum gravity is a question of which bullets you’re willing to bite, not a question of technical skill. That doesn’t mean AI will never be able to address it, but if so it will come from some sort of superpersuader aspect, not merely scientific capabilities.

And of course, some fields do require experiments. People are increasingly building AI-powered labs. I’ll leave it to people from actual experimental fields to think about the potential there.

Q: Can you say something explicit about the bigger risks from AI, now?

A: As I mention in the post, I didn’t learn that much about the bigger questions from this. I’m still not an expert.

But there’s one thing I think is worth emphasizing:

This technology is clearly getting more effective over time. I don’t think they could have done this six months ago. If you’re trying to predict what will happen next, you shouldn’t just assume this is the most powerful it will get, or the cheapest. If you think there’s a limit, you need to argue for it.

Edit: One more thing I should mention, now that I’ve confirmed she’s ok with it: the idea for my challenge came from some discussions at Lancefest, a conference in honor of Lance back in June, and particularly from Anastasia Volovich, whose talk declared the nine-loop calculation “a benchmark – or a ‘challenge’ – against which any “AI takeover” should be measured”.

by 4gravitons at September 25, 2026 06:06 PM

Tommaso Dorigo — Some Tests Of The Radiacode Zero

Some Tests Of The Radiacode Zero
As I explained previously, these days I am testing a new radiation detector from the Radiacode family: Radiacode Zero.

Tommaso Dorigo Fri, 09/25/2026 - 10:54

Categories

Physics

by Tommaso Dorigo at September 25, 2026 02:54 PM

September 24, 2026

John Baez — Binomial Coefficient Coincidences

These seven equations between binomial coefficients are ‘coincidences’: they aren’t among the four known infinite families. De Weger conjectured that there are no more such coincidences:

• Benjamin M. M. de Weger, Equal binomial coefficients: some elementary considerations, Journal of Number Theory 63, no. 2 (1997), 373–386.

At that time, he and his collaborators checked there were no others involving binomial coefficients less than 10³⁰. Later they checked that there are none involving binomial coefficients less than 10⁶⁰:

• Aart Blokhuis, Andries Brouwer and Benne de Weger, Binomial collisions and near collisions.

So, De Weger’s conjecture stands open. The four infinite families, by the way, are these:

$\displaystyle{ \binom{n}{k} = \binom{n}{\,n-k\,}, \qquad 0 \le k \le n}$

$\displaystyle{ \binom{n}{0} = 1, \qquad n \ge 0 }$

$\displaystyle{ \binom{\binom{n}{k}}{1} = \binom{n}{k}, \qquad \qquad 0 \le k \le n}$

and the only nontrivial one: the Lind–Singmaster family involving the Fibonacci numbers $F_i$ where $F_0 = 0,\ F_1 = 1$ :

$\displaystyle{ \binom{F_{2i+2}F_{2i+3}}{\,F_{2i}F_{2i+3}\,} \;=\; \binom{F_{2i+2}F_{2i+3}-1}{\,F_{2i}F_{2i+3}+1\,}, \qquad i = 1,2,3,\dots }$

The first three equations in the Lind–Singmaster family are these:

$\begin{array}{ccc} \displaystyle{ \binom{15}{5} } &= &\displaystyle{\binom{14}{6}} \\ \\ \displaystyle{\binom{104}{39} } &= &\displaystyle{\binom{103}{40}} \\ \\ \displaystyle{ \binom{714}{272} } &=& \displaystyle{\binom{713}{273}} \end{array}$

I’ll explain the Lind–Singmaster family later. But here’s the question I’m most interested in:

Is there any good explanation for the seven binomial coefficient coincidences?

Today my collaborator Paul Schwahn found a beautiful explanation of the first one, namely

$\displaystyle{ \binom{10}{3} = \binom{16}{2} }$

His explanation uses representation theory. The Lie algebra $\mathfrak{so}(10)$ has a 10-dimensional representation, the ‘vector’ representation $V_{10}$ , and also two 16-dimensional representations, the ‘left and right-handed spinor’ representations $S^\pm_{10}.$ There’s an isomorphism of representations

$\displaystyle{\Lambda^2 S^+_{10} \cong \Lambda^3 V_{10} }$

and similarly for $S^-_{10},$ but we might as well work with $S^+_{10}.$ Here $\Lambda^k$ means the kth exterior power. For any vector space $X$ we have

$\displaystyle{ \dim(\Lambda^k X) = \binom{\dim X}{k} }$

Thus, taking dimensions, the isomorphism of representations

$\displaystyle{\Lambda^2 S^+_{10} \cong \Lambda^3 V_{10} }$

instantly gives

$\displaystyle{ \binom{16}{2} = \binom{10}{3} }$

It is not super-easy to prove this isomorphism of representations, but it’s still nice to find a deeper layer of meaning underlying what might otherwise seem like a meaningless coincidence!

Can we find representation-theoretic explanations—or other explanations—for the other six coincidences?

I have not succeeded, but let me tell you about two failed tries.

We can look for isomorphisms like

$\displaystyle{\Lambda^2 S^+_{10} \cong \Lambda^3 V_{10} }$

involving representations of $\mathfrak{so}(n)$ for larger $n.$ In fact this isomorphism is part of a pattern! The next one involves the vector and left-handed spinor representations of $\mathfrak{so}(12).$ But it’s this:

$\displaystyle{ \Lambda^2 S^+_{12} \cong \Lambda^4 V_{12} \oplus \Lambda^0 V_{12} }$

so it gives

$\displaystyle{ \binom{32}{2} = \binom{12}{4} + 1 }$

or

$\displaystyle{ 496 = 495 + 1 }$

So we fail to get an equation between binomial coefficients: we’re off by one.

The second paper I cited, Binomial collisions and near collisions, presents a list of cases where two binomial coefficients differ by one. This is on the list. So we failed to explain an equation between binomial coefficients, but explained a near-miss.

Here’s another failed attempt at explaining an equation between binomial coefficients. The equation

$\displaystyle{ \binom{78}{2} = \binom{14}{6} = 3003 }$

is fascinating to anyone who knows their exceptional Lie groups. 78 is the dimension of $\mathrm{E}_6,$ while 14 is the dimension of $\mathrm{G}_2.$ $\mathrm{G}_2$ is a subgroup of $\mathrm{E}_6$ because $\mathrm{G}_2$ is the automorphism group of the octonions and $\mathrm{E}_6$ is the isometry group of the bioctonionic plane. We’d get the above equation if the 2nd exterior power of the adjoint representation of $\mathrm{E}_6,$ upon being restricted to $\mathrm{G}_2,$ were isomorphic to the 6th exterior power of the adjoint representation of $\mathrm{G}_2.$

Amazingly, it seems these two representations of $\mathrm{G}_2$ are not isomorphic even though their dimensions are the same: both 3003.

Even more amazingly, $\mathrm{E}_6$ and $\mathrm{G}_2$ both have irreducible representations of dimension 3003, but they are not the representations I just mentioned.

I would be happy for someone to check these two claims.

If anyone knows good explanations of the remaining six binomial coefficient coincidences, please let me know!

The Lind–Singmaster family

Lind and Singmaster were trying to find all $n,k$ with

$\displaystyle{ \binom{n}{k} = \binom{n-1}{k+1} }$

I’ll rapidly sketch the key steps of their argument. Simplifying the equation above we get

$n(k+1) = (n-k)(n-k-1)$

or

$n^2 - (3k+2)n + (k^2+k) = 0$

Solve for $n$ using the quadratic formula. This formula turns out to have

$\sqrt{5k^2 +8k+4}$

in it. So we need $5k^2 +8k+4$ to be a perfect square!

Now we’re trying to find integer solutions of

$5k^2+8k+4 = m^2$

A quadratic diophantine equation! Multiply by 5 and complete the square:

$5m^2 = (5k+4)^2 + 4$

$y = 5k+4$ is an integer when $k$ is, so we need to find integer solutions of

$y^2 - 5m^2 = -4$

This is a ‘Pell equation’, and people know how to solve these. In this particular case we get all the solutions from this fact:

$L_n^2 - 5F_n^2 = 4(-1)^n$

where $F_n$ are the Fibonacci numbers 0, 1, 1, 2, 3, … and $L_n$ are the Lucas numbers 2, 1, 3, 4, 7, …. These are two sequences satisfying the same famous recurrence relation, just with different initial conditions.

We want $n$ odd, to get

$L_n^2 - 5F_n^2 = -4$

It turns out $y = L_n, m = F_n$ with $n$ odd give all solutions of the Pell equation

$y^2 - 5m^2 = -4$

However, remember I said $y = 5k+4$ is an integer when $k$ is. But the converse isn’t always true, and we need $k$ to be an integer! This clearly happens iff $y \equiv 4 \bmod 5.$

So we need to know when $L_n \equiv 4 \bmod 5$ Apparently this happens iff $n \equiv 3 \bmod 4.$ I won’t think about this now… but this is the last hard step.

In summary, we’ve seen

$\displaystyle{ \binom{n}{k} = \binom{n-1}{k+1} }$

if and only if $y = 5k+4$ is a Lucas number $L_n$ with $n \equiv 3 \bmod 4.$ We could quit here, but people like to use the identity

$L_{4i+3} - 4 = 5F_{2i} F_{2i+3}$

to get a formula for $k$ in terms of Fibonacci numbers. This is gilding the lily, I’d say, but that eventually leads to the formula I showed you:

$\displaystyle{ \binom{F_{2i+2}F_{2i+3}}{\,F_{2i}F_{2i+3}\,} \;=\; \binom{F_{2i+2}F_{2i+3}-1}{\,F_{2i}F_{2i+3}+1\,}, \qquad i = 1,2,3,\dots }$

The takeaway message is: our problem can easily be reduced to a quadratic diophantine equation, then put in Pell form… and it’s known that the sequence of integer solutions of a Pell equation obeys a linear recurrence relation! We luck out in this case and get solutions connected to Lucas and Fibonacci numbers.

There may be a simpler argument, but this is what I’ve seen.

by John Baez at September 24, 2026 08:13 PM

September 20, 2026

John Preskill — Quantum Computers Need More than “Magic”

The University of Cambridge is a special place. During term time, hordes of students gather, ready to attend formal, a candle-lit dinner in a five-century-old hall overlooked by paintings of academics past. The sight is like an overromanticized still of a bygone era. The men wear suits, the women long dresses, all wear gowns. The simplest gowns belong to the undergrads, with the complexity and length of the gowns growing as one climbs the academic ranks.

Between the bringing of the bread and the serving of the soup, the conversation at my end of the table typically turns to my topic of research: quantum computers. These theoretical machines harness quantum physics to outperform their classical counterparts. Quantum computers hold great promise for the future, with applications ranging from drug discovery to cybersecurity. Yet, despite their promise, we still do not have a satisfying answer to the fundamental question: what makes quantum computers computationally stronger than classical computers?

To answer this question, imagine that tonight, instead of the soup, the cooks are brewing a happiness potion. This potion causes the drinker to be joyful and content for the rest of the evening. The cooks know how to brew the potion perfectly well; they discovered the list of ingredients in the nineties and have been successfully brewing ever since. But the cooks still cannot solve one mystery: what makes the potion different from the soup? What ingredient sets the potion apart from the soup?

The cooks come up with a simple approach. They leave out one ingredient in each serving, and then carefully observe the formal-goers. They find that almost every seat is filled by a happy student, chatting away to their neighbours, indicating a working potion. However, one student’s spirits have not been lifted, as they were solely served a simple soup. The cooks conclude that the ingredient they withheld for that student was essential. They name this ingredient magic.

In my research, the potion is a quantum computer, the soup is a classical computer, and magic is the actual technical term used for quantum states that promote certain classical computers to quantum computers. Without magic, these quantum computers are computationally no stronger than a normal laptop.

So magic is necessary for quantum computational advantage, but is it enough? Over a decade ago, researchers found that, for quantum computers built on qutrits, the answer is no. To understand what a qutrit is, consider the regular bit: a switch that’s either zero or one. The qubit, the quantum generalization of the bit, can be both zero and one simultaneously. The qutrit is a roomier qubit that can be zero, one, two, or any of those simultaneously.

It is possible to build quantum computers using qutrits. However, the vast majority of quantum computers are built on qubits, the quantum generalisation of the everyday bit. My colleagues and I have recently discovered that for qubit-based quantum computers, too, magic is not enough to gain an advantage over classical computers. Picture the cooks dumping jar after jar of magic into a soup, only to find that the soup never gains magical powers. Potions need more than magic. Potions also need Kirkwood-Dirac negativity.

What is Kirkwood-Dirac negativity? The idea developed by John Kirkwood and Paul Dirac builds on probabilities—for example, the odds of obtaining a heads upon flipping a coin. One can describe a quantum computer using numbers that behave similarly to probabilities but come with a twist: they may be negative. These negative “probabilities” mark where quantum departs from classical. The main result of our paper is that this negativity, too, is a necessary ingredient: without it, no amount of magic will turn the soup into a potion.

At this point in the conversation, I am typically cut off by the arrival of the soup. The conversation moves on from quantum computers to different subjects, and I usually look up at the paintings staring down at me. One of them is of Dirac, who spent many dinners discussing quantum theory in the same dining hall. The University of Cambridge is a special place.

by jjthio at September 20, 2026 04:49 PM

September 19, 2026

Scott Aaronson Theory Beyond Theorems and Proofs: A Guest Post

Scott’s foreword: I’m extremely grateful to my brilliant colleagues, Pravesh Kothari, Raghu Meka, and Prasad Raghavendra, for sharing the guest post below about how theoretical computer science (and in particlar, the STOC/FOCS/SODA conferences) should evolve to deal with the AI asteroid that’s right now slamming into our field, at least as we human theorists have practiced it since its inception. While Pravesh, Raghu, and Prasad speak only for themselves, not for myself and not for the theory community as a whole, I found their proposal of a separate “conceptual track” to be an excellent starting point for further discussion. –SA

Considering the pace of developments in AI theorem provers, most would concede that the following scenario is at least plausible in the very near future:

AI theorem provers could prove well-specified mathematical claims, even many well-studied ones that have been open for years, in a matter of hours. Moreover, these systems could be widely available to consumers at nominal cost.

As TCS researchers, let us pretend that the above scenario has come to the fore, and ask ourselves: What is our role in such a world? Does it mean the end of theory research?

As we ponder this question, let us ignore all of these other confounders:

Recent controversies surrounding the developments on the Millennium Prize Problems

Motivations and actions of the AI companies

Observed faults in existing AI systems when it comes to writing, exposition or attribution to previous work.

None of the above confounders have any impact on our answer to the question: What should theorists do, in the presence of superhuman AI theorem provers?

Notice that we use the term “AI theorem provers” instead of just “AI”. We believe that this conceptual distinction is important as we consider this question.

At the outset, we would like to admit that for a generation of theorists like us (and many from earlier), research was mainly centered around problem-solving. Even when we developed conceptual insights, it was mostly in service of answering well-specified long-standing questions. We don’t intend this proposal as judging one form of research to be better than others; it only reflects that AI theorem provers accelerate a certain type of research activity and want to make the best of it. There is also a tremendous human cost of this upheaval, which is perhaps a more important question, and one which this proposal does not address directly (we do not have any good ideas as such). Similar points have also been made in various contexts
before, but the timing now is more pressing.

Definitions, Questions & Theories:

The goal of any theoretical science is to advance human understanding of observed phenomena. Apart from theorems and proofs, a theoretical science has definitions, questions, and theories.

Definitions identify the objects to observe. Curiosity and context drive the questions to ask. Theories explain the phenomena observed. We believe humans will continue to play a central role in generating definitions, questions & theories, even in the presence of a super-human AI theorem prover.

Definitions: Could an AI define randomness extractors, streaming algorithms, or zero-knowledge proofs? Maybe. But there are some reasons to believe, humans will still have a big role to play in coming up with definitions.

For instance, the notion of extractors arises from the real-world problem of lacking perfect random sources. Zero-knowledge proofs seem to arise purely out of human curiosity, guided by taste. Human context and curiosity will continue to drive theoretical research. After all, we get to decide what objects we choose to observe!

Theories: Consider the following thought experiment. Suppose in 1965, we had a magic machine that at the press of a button, given any computational problem, would tell us if it had a polynomial-time algorithm or not.

Would that have been the end of computational complexity theory? No. Humans would find it entirely unsatisfactory, and ask, why do these problems not have a polynomial-time algorithm? Why do these others have?

The theory of NP-completeness identifies some patterns among problems that don’t seem to have efficient algorithms. This theory would still be a crown jewel of theoretical computer science, even in a world where we had a magic machine to tell if a problem had an efficient algorithm or not, at the press of a button. Similarly, if we had a machine to predict whether a CSP is NP-complete or in P, we would then ask: what makes 3-SAT NP-complete, while 2-SAT is in P? This question leads to the theory of polymorphisms, which yields a satisfactory answer.

Theories aren’t just succinct or efficient mechanisms to answer questions. The best theories provide are those which humans deem to be a “satisfactory explanation” – whatever that means.

Finally, even as the capabilities of AI theorem provers advance, human curiosity will probe grander and deeper questions. Previously, even if we wanted to build new models and theories, proving something about them was a prerequisite, and given that the grand questions were already at the limit in long-studied domains, we had to scale things down. If each theorem proven by AI is treated as an experimental datapoint, humans can ask grander questions that look for patterns across these theorems.

A concrete proposal:

We think theorists should embrace these AI theorem provers in our research. To a certain extent this is already happening explicitly or implicitly.

As theorists, we have been parsimonious in introducing new models or asking entirely new questions, and careful about adopting new ones too quickly. This was partly because formally proving the properties of a new definition or a model was an onerous task that could take a decade, and tens of papers. AI theorem provers might completely change this dynamic. This is precisely the moment to refocus our work on definitions, questions, and theories. We need explicit systems to encourage and reinforce these parts of theoretical research. You might also say the next generation of AI models can do this; it may be so, but we believe you have to take the current opportunity.

To this end, we suggest that STOC/FOCS/SODA create a separate track of papers. This track is meant specifically for papers that introduce new definitions, ask novel questions or build explanatory theories. The papers in this track are short, say less than 10 pages. Papers may, and should, contain theorems as usual and as needed. Most importantly, the radical shift is that the papers need not contain the proofs of the theorems. Instead, the authors supply a Lean certificate as a supplement to the paper. The evaluation will also in a sense “orthogonalize’’ against the difficulty of these proofs.

The papers in this track should be judged exclusively on the conceptual merits, completely agnostic to the difficulty of the proofs.

Reviewing must be completely agnostic to the proof for two reasons. The main track at STOC/FOCS already includes papers in the former category. Second, a major barrier to producing truly novel conceptual papers is that they often get judged poorly for a lack of technical depth in their proofs. We think these two aspects separate it from (ITCS/SOSA) and, regardless, it’s something we urgently need for all our conferences, including STOC/FOCS (the ‘flagship’ conferences).

To be clear, we ourselves admit that we need to hone these skills of making new definitions, asking deep and interesting questions or building new theories. A separate track of conceptual papers will provide a systematic mechanism for both junior and senior researchers, and the field as a whole to do so.

We believe that upcoming generations of grad students will tackle research directions that seemed completely out of reach to us. We just need to set up systems that nurture new ways of doing research in theory.

— Pravesh Kothari, Raghu Meka, Prasad Raghavendra.

by Scott at September 19, 2026 10:34 PM

September 18, 2026

Matt von Hippel — Data Comes From Papers

There’s a lovely resource that I suspect most non-physicists don’t know about. It’s called the Particle Data Group, and its interactive site PDGLive.

If you want to know the most up-to-date information on a subatomic particle, PDGLive has your back. From their home page, you can click on familiar particles like photons ( $\gamma$ ), electrons ( $e$ ) or gluons and find the best info experiments have provided on properties like their mass and charge. It’s a great place to get an authoritative number for any particle physics property you’re interested in.

Of course, scientists don’t just accept one authoritative answer for anything, even whether Pluto is a planet. That’s why each measurement in PDGLive comes with an arrow you can expand to see a table of past measurements, which you can compare to.

Each measurement comes with a link to a source. And each source is not an experiment webpage, or some entry in a database. It’s a document, a publication, a paper.

In fact, PDGLive itself is just a website version of a paper, the Particle Data Group’s “Review of Particle Physics”. Click enough times, and you’ll find sections of a pdf on the site, explaining their reasoning for picking this experiment over that, emphasizing this or the other thing.

People talk about papers as how academics persuade one another, or how they show off and establish credit. But papers are also just a way to organize data. Each time an experiment figures something out, all of their reasoning and procedures are summarized together with the numbers they got. And anyone who reports those numbers to you will include a link, so you can go back, and check where the number came from.

It probably feels a bit weird, all of these numbers and technical details bottoming out in an archaic practice of writing down words for other human beings. But it means that all of the richness of the process is there somewhere, linked together and collated by the same social forces that keep track of credit, all in one navigable whole.

So when you run into a number, spare some thought for where it came from. You can probably find out.

by 4gravitons at September 18, 2026 04:00 PM

September 17, 2026

Tim Gowers — Why I didn’t sign the Fields medallists’ letter

[This post has been cross-posted to Terence Tao’s blog.]

When I was around 11 I heard for the first time about Fermat’s Last Theorem. I was immediately captivated by the problem statement, as well as by the accompanying story, and made a fairly serious attempt to prove it. And while, unsurprisingly, I failed, I learned a lot from the attempt. Blissfully ignorant of the fact that the $n=3$ case had been proved by Euler over 200 years earlier, I decided that that would be a good place to start: once I had sorted that out, I was optimistic that I would be ready to tackle the general case.

Since I still couldn’t really see where to start, I decided to simplify the problem further and concentrate on successive differences of cubes, with a view to showing that such a difference could not itself be a cube. At the time I did not know how to express what I was doing in algebraic language, so I did not explicitly try to prove that the Diophantine equation $3n^2+3n+1=m^3$ had no solution. Rather, I just worked out some successive differences and stared at them, trying to get some idea of why none of them was a perfect cube. (I should be clear that this story is a reconstruction of what I think probably happened given the few memory traces that remain half a century later rather than a completely reliable account.) At some point, I had the idea of taking the difference sequence of the difference sequence, and discovered that it formed an arithmetic progression. That felt like progress, so I investigated difference sequences a bit more and discovered, purely empirically, the rule that if you start with $n$ th powers and keep taking successive differences, then eventually you get to the constant sequence $n!, n!, n!, \dots$ .

Somehow I never managed to turn this observation into a proof of Fermat’s Last Theorem, and later on my dream of solving it got replaced by other mathematical dreams. However, when I reached the point in my mathematical education where I was taught about taking difference sequences and about what happened to polynomials, I understood those topics much better than I would have if I had not discovered difference sequences for myself and spent happy hours playing around with them. I mention this story just as an illustration of the phenomenon that was strongly emphasized in this letter signed by 25 Fields medallists, that one learns a lot from thinking about a problem, regardless of whether one solves it.

In the end, however, I felt that I could not sign the letter, despite agreeing with much of what it said. Instead, it seemed better to do what I did with the Leiden Declaration and set out my own position in a blog post. But it should be understood that by doing that I am not setting myself up as a member of some opposing camp: indeed one of my worries at the moment is that the mathematical community might become bitterly divided, something I would very much like to avoid. Also, I agree on the fundamental point that we are facing a crisis: I just want to offer a slightly different analysis of what that crisis is. I don’t claim full originality for this analysis, as I know that several other mathematicians have already put forward thoughts that are similar to the ones I have, though (for what it’s worth) I have largely come to these conclusions independently.

On the subject of independence, it will perhaps help if I clarify that while I have contacts in the mathematics group at OpenAI, and have also been given early access to some of their models (typically only a few days before they have been released), and have been given free access to their Pro models once released, I have never been paid by OpenAI. I mention this in the hope, perhaps naive, that what I write will not be dismissed for ad hominem reasons. Another potential reason for my being regarded as “pro-AI” is that, as I have stated publicly several times, I have a group in Cambridge devoted to automatic theorem proving. However, that is actually more of a reason to be anti-AI, since our group has been trying to attack the problem of getting computers to prove interesting theorems by understanding as well as possible how humans prove interesting theorems, so now that LLMs can clearly do it without the help of such insights as we have had, one of the main motivations for our work has disappeared. To put it another way, we have had to swallow the bitter lesson (which of course we were always aware was a distinct possibility, even if the speed at which it happened has taken us by surprise). I do in fact think that it is still a very interesting and valuable intellectual exercise to try to gain this understanding, even if we can use LLMs as black boxes, but that’s a topic for another blog post.

So why didn’t I sign the letter? Let me extract a couple of sentences from it that express what I see as the principal argument being put forward.

But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal. Indeed, the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas.

Perhaps the main reason I didn’t sign is that I don’t fully subscribe to this view. Instead, I have a more complicated view, which I actually expressed in my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding. At one end of the spectrum you have mathematicians who are primarily motivated by the wish to solve problems, who see conceptual understanding as a very important means to that end. At the other you have mathematicians who are primarily motivated by the wish to attain conceptual understanding, who see problem-solving as a very important means to that end. I worry that the severe-misalignment letter could be seen as saying that the “right” attitude is to focus on conceptual understanding as the main priority — indeed, the above sentences say that more or less directly. But I think that there are mathematicians all across the spectrum, and that that is a good thing (or perhaps I should say that it has been a good thing up to now — the future is much less certain), and I don’t want to suggest to a large fraction of mathematicians, including myself, that their mathematical temperament is somehow “wrong”.

My own particular mathematical attitude is very similar to one that was beautifully articulated in a Twitter post by Jacob Tsimerman (another non-signatory of the letter), which, now that I look at it, says a lot of what I will be saying here. And that post in turn is a response to Daniel Litt, who is in my opinion one of the wisest commentators on mathematics and AI. His views are expressed in a later post here, which I deliberately didn’t read until finishing this one, and then found, as I expected, that there was significant overlap. I would also like to take this opportunity to recommend an excellent post by Noah Smith entitled The End of the Age of Heroes, in case you haven’t read it.

I have been talking so far about individual mathematical understanding, but I suspect that what concerns most of the signatories is less that than the collective understanding that results at least in part from the human activity of problem solving. My guess is that they would argue, completely coherently, that even if collective understanding is the primary goal, if many individuals are primarily motivated by the wish to solve problems, that’s absolutely fine and contributes to that collective understanding.

With that interpretation, the issue becomes slightly different: is it more important that the collective understanding of the mathematical community should be as advanced as possible or that there should be answers to as many problems as possible? Or are those two aims valuable in different ways, so that there is no point in declaring one of them more important? Or are they so inextricably linked that it makes no sense to argue that one is more important than the other? And when we say “important”, for whom are we saying it is important: for mathematicians, or for society as a whole?

I find these hard questions, so I don’t want just to declare an answer to them. (Do you see what I did there?) Instead, I’d like to try to offer at least some argument for any conclusions I come to, even if they are tentative. So let’s compare two scenarios. In the first, which I think is the more likely actually to happen, models become publicly available that are better at solving problems than virtually all mathematicians. If there are a few residual mathematicians who can do things the models can’t, even they work far faster if they make heavy use of the models. Thanks to this, in a short time we get answers to many questions that we have deeply cared about, but the rate at which we receive these answers far exceeds the rate at which the mathematical community can absorb them. In particular, most of the answers are obtained with zero effort from human mathematicians — just prompts such as “Thank you — please continue”.

In the second scenario, there has been an international agreement, for entirely other reasons, to block the public release of models significantly more powerful than the ones we currently have, and the mathematicians within the tech companies agree to hold off from getting their internal models to solve major problems. Instead, they take guidance from the mathematical community, solving problems only when asked to do so by some suitably representative body that decides that the benefit of receiving a solution of a certain problem outweighs the benefits of humans struggling to solve it over a much longer timescale.

I’d like to consider what the difference would be between these two scenarios both for individual and collective understanding. I’ll begin with individual understanding.

One might argue that for individual understanding, not too much would change if we are suddenly flooded with large numbers of big new results. There is already far more mathematics out there than I have any hope of understanding (for example, despite being fascinated when Fermat’s Last Theorem was proved, I have made no attempt to understand the proof), and even among the parts that I do understand, the parts that I understand because I myself discovered them form a very small fraction, though a fraction that I understand more deeply than anything else (at least temporarily — after a while I forget things and lose quite a lot of the understanding I built up). However, one change, which seems positive, from the perspective of the building up of individual understanding, would be that we would have a much bigger choice of results that we could choose to study. Also, if we found ourselves stuck on some point, AI would be able to help us. The main likely negative change is that we would probably cease to exercise that part of our brains that we use when spending months or years struggling with a difficult research problem, which can be hugely helpful in developing understanding.

I say “likely” because in principle there would be nothing to stop us thinking about very hard problems without consulting LLMs, but in practice it seems unlikely that people would put in the same level of effort that they do now. The situation might a bit like what happened with satnavs, where one could always decide not to use them, to keep the part of the brain active that can look at a map, learn a route, and follow it, but in practice most people succumb to the temptation to use a satnav. (In fact, I myself do try to keep that part of my brain active, and was rather proud of finding my way somewhere recently when I had briefly looked up the route on my phone but then forgotten to bring the phone with me when I actually went there.) But even if all we were doing was reading AI output, I think that the problem-solving muscles in the brain wouldn’t atrophy completely. When students are reading maths papers, I strongly advise them (and I think this is pretty standard advice) to read “actively” rather than “passively”, doing things like trying to prove the result for yourself, looking at the paper only when you feel stuck and need a hint, and even then just trying to get the hint and as little extra as possible. If one reads a paper that way, then one is constantly solving problems, some just exercises and some quite a bit harder. It seems likely that an LLM could get to know what our mathematical background is and feed us with just the right hints to allow us to work our way through a mathematics paper in this active way. Yes, we would lose the particularly deep level of understanding and ownership that comes with having solved a hard problem oneself, but it isn’t clear to me that progress in mathematics would suffer as a result. I would be very interested to hear counterarguments to precisely this point. That is, I would be interested to know what use that level of deep involvement with a proof might have in a world where AI is much better than we are at finding proofs.

How about collective understanding? Let me quote a bit more of the letter.

Indeed, the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas.

Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others. As in all creative professions, this raises severe attribution and plagiarism questions. Moreover, without the willing mathematicians who must take care of their development and integration into the mathematical canon, AI-conceived ideas would never become fully alive and the crucial human transmission chain between mathematicians would be lost.

I’ll come back to questions about proper citation and focus on what I take as the core worry here: that if results are proved too quickly, then the digestion process will become impossible. I am definitely worried that results will not be properly digested, but for different reasons.

A first remark is that what AI is producing is not just true/false statements: we now know not just that the Navier-Stokes equation with smooth forcing admits finite-time blow-up, but we have a proof of that, which builds on a great deal of wonderful work done by human mathematicians. Many people used to express the worry that AI would solve our favourite problems with utterly opaque proofs, but that has not turned out to be the case, even if their write-ups often leave plenty to be desired. (Incidentally, I see these inadequate write-ups as almost certainly a temporary annoyance and therefore not as a fundamental threat to mathematical practice or future mathematical understanding.)

Secondly, even if the volume of new results is large, mathematics is a highly specialized discipline, so mathematicians can work in parallel. If, for example, we had to digest 1000 important results in a year that were roughly uniformly distributed across mathematics, then most sub-communities of mathematicians would probably want to understand around 30 of them, and for each individual problem there might well be only a small handful of specialists who would be obvious people to take the lead in reaching this understanding, with that handful varying from problem to problem. So it would be a big task, but not necessarily an impossible one.

In this context, it is worth thinking about the huge volume of output of human mathematicians, which seems to have been increasing recently, even before AI. While I have sometimes heard complaints about this, I have certainly not heard suggestions that human mathematicians should slow down the rate at which they prove interesting theorems. That may be partly because the authors of those theorems take the trouble to write their papers well and give good talks. But what about the large quantity of papers, including important ones, that are not written well and whose authors give incomprehensible talks? That can be annoying, but it is a familiar annoyance and not one that we think of as a crisis.

A third point is that even if the volume of AI output is too big for us to be able to digest it properly, that is not necessarily a bad thing. To draw an imperfect analogy, there is now more content available on streaming services than anyone could possibly watch, with the result that there is almost certainly some very good content out there that is hardly watched at all. But that isn’t obviously a worse situation than if there were far less content and all of it received the attention it deserved. Returning to mathematics, if there were too much AI-generated content for us to be able to digest it, then we could choose which parts of it we wanted to digest.

For that we would need to have some idea what was there (a situation a little similar to how human mathematicians typically learn quite a lot about what results are known in their area even when they do not understand their proofs in any detail). One way one could try to achieve that would be to create a well-designed database, probably with AI help. But perhaps that would be unnecessary, and instead one could simply talk to an LLM and ask it to give a bird’s-eye view of whatever area of mathematics one wanted to understand in that knowing-what’s-there way.

The fear seems to be that some very interesting and important parts of mathematics will be discovered by AI and then overlooked, when had they been discovered by human mathematicians they would not have been overlooked. And that may even be the case, but what matters is whether the amount of interesting and important mathematics discovered by AI that is not overlooked will exceed the amount of interesting and important mathematics that would have been discovered and properly digested by humans with AI having played a more modest role.

In short, it seems to me that while a flood of “big” AI results would be likely to increase the amount of important mathematics that was not properly digested, it would also be likely to increase the amount that was properly digested, which seems like a pretty good bargain.

Let me quickly discuss the problem of AI not properly crediting human mathematicians. I agree that this is a serious problem right now, but it is another problem that I see as temporary. Very soon, the whole “credit system” will surely collapse, since finding an amazing proof will be no more of an intellectual achievement than when a citizen scientist spots through their telescope an object that turns out to be a new comet. Until that happens, it is important to give humans the credit they deserve, since careers can depend on it, but that will soon cease to be the case as well. I have to say that I’m puzzled that this problem exists, since I would have thought that if you asked an LLM to look at a proof and tell you which ideas in it are close to ideas that are in the literature already, it would be extremely good at that task. I hope the answer to this conundrum is not that people have been in such a hurry that they have simply not taken the trouble to do this, but I fear that it might be, at least in some cases. If so, then those who have been careless deserve to be criticized, but it is a minor matter compared with the survival of mathematics, especially if the lack of citations is swiftly put right.

Does all this mean that I am optimistic that mathematicians will end up digesting at least as much mathematics in a post-AI world as it would have if AI had not been able to prove major theorems? Not exactly. But my worry is not that we would be unable to do it, but rather that the social structures that currently support this digestion process will be destroyed and not adequately replaced.

One way that might happen is that AI disrupts society so much, or even kills vast numbers of us, that the preservation of something like the current mathematical tradition ceases to be of any concern: all that will matter is the survival of the human race. But that again is a topic for a different blog post (which in fact I am in the middle of writing).

Let’s assume instead that we get lucky and that AI remains more or less under control. My worry then is that we do not manage to transmit what we know to a new generation of mathematicians. Speaking for myself, my main motivation for becoming a mathematician was the dream that I would solve unsolved problems — the more famous the better. I have also always greatly preferred directly thinking about a problem to reading books and papers and generally learning the mathematics of other people. (I’m not saying that’s good, but just stating a fact about myself.) If the dream of solving a famous problem had not existed, I’m not sure whether I would have become a mathematician. I don’t completely rule it out: maybe what really motivated me was that I had an aptitude for the subject and that solving problems was a way of getting respect from a small group of peers. And maybe I could have tried to gain that respect in a different way, such as thinking very hard about an area of mathematics until I was able to demonstrate to others just how well I understood it. But I’m not sure how motivating that would have been for me. I very much hope that there is a pool of young people for whom it will be a powerful motivation, because I think the survival of a human mathematical tradition may well depend on it.

Thus, the primary risk, as I see it, is that a lot of people who would have done a PhD in mathematics and gone on to become custodians of the mathematical tradition will no longer wish to do so. Those of us who have PhD students, including me, need to try as hard as we can to come up with imaginative ways for them to use their time productively (in consultation with the students themselves, obviously). Whether or not we do a good job with that could make a huge difference to the future of mathematics. A related risk is that the perception among policy-makers will be that mathematicians are no longer needed and that funding will become much harder to come by: we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts, even if it is no longer part of their role to find new proofs of theorems.

A final reason that I didn’t sign the letter is that I wasn’t really sure what it was demanding that isn’t happening already. It seems likely that in a matter of not very many months LLMs will be released that are able to solve major mathematical problems, and they will presumably have no trouble at all with more run-of-the-mill problems. However much we might regret that, there is no chance that the impact of such models on mathematics will persuade AI companies to stop their release, though perhaps concerns about safety will lead to some delay and give us a bit more time to work out how to adapt. Assuming that they are released, there will be a flood of new results, whether we like it or not, and it will no longer be the AI companies producing them, though perhaps the pattern will continue that the AI companies will have access to more powerful models and so will obtain more than their fair share of headline results. So I felt that there was nothing to be gained from criticizing AI companies for generating too many solutions too quickly. In fact, it may well be that all that does is bring forward by a couple of months what was going to happen anyway, and perhaps it will even allow the results to be released in a more controlled way than they would have been if they had been discovered by random people once the models were publicly available. Under the circumstances, I think the best we can do is recognise the changes that are coming and try to work out the least unsatisfactory way of dealing with them.

by gowers at September 17, 2026 04:42 PM

September 15, 2026

Tommaso Dorigo — Radiacode Zero - A Powerful New Radiation Detector

Radiacode Zero - A Powerful New Radiation Detector
Ionizing radiation is all around us. We do not notice it: we have not developed any sense to detect it. Yet it may affect us in very serious ways, particularly because its effect on living cells and organisms is cumulative: a progressive degradation.

Tommaso Dorigo Tue, 09/15/2026 - 04:13

Categories

Physics

by Tommaso Dorigo at September 15, 2026 08:13 AM

September 14, 2026

Jordan Ellenberg — I love working here

by JSE at September 14, 2026 03:41 PM

September 11, 2026

Andrew Jaffe — The Talk I Gave on September 12

More 25th anniversary thoughts and recollections.

I had moved to Oxford less than two weeks before, still settling into my new life in the UK, working at Imperial College in London. That day, I was heading off to Durham, in the north of England, to a conference called “A New Era in Cosmology” — my first big talk now that I had started my permanent academic job. I was going to be a bit late, only able to arrive toward the end of the first day of the conference: September 11, 2001.

My train left in the late morning. A few hours in, passengers were starting to talk about an attack on New York City. This was in the days before smartphones and constant communication — I didn’t even have a mobile phone. The discussions around me were getting more and more frantic, and I was doing my best to piece together the story.

I was born in New York City, and much of my family still lived in the area. My parents lived in the suburb of Fort Lee, New Jersey, right across the Hudson River from Manhattan; from the apartment where I grew up, we had a fantastic view from the 18th floor. My father worked in The City, commuting every morning by car from New Jersey to midtown Manhattan. He would have been at his office that day.

I considered getting off the train somewhere en route, perhaps Sheffield or York, to try to get some more information, but I just stayed on the train. I was able to get a taxi from the station in Durham to my hotel, listening to the news. I was able to call my partner, back in Oxford, who had, luckily considering the pressure on transatlantic calls, been able to get in touch with my family in New York and New Jersey. Everyone was, thankfully, alright, although at this point my father was still in Manhattan. He had noticed all the emergency vehicles speeding downtown, thinking that it was a motorcade for some foreign dignitary, but it was the first fleet of emergency responders heading towards the twin towers after the first collision. Eventually, he made it home, to my family’s apartment overlooking the Hudson. But it was so close to the George Washington Bridge, a piece of vital infrastructure thought to be in danger after the first attack, that he had to be let off a mile or so away and walk the rest.

As for me, I had to give the first talk on 12 September, about the “new era in cosmology” that coming CMB measurements would usher in. We started, understandably, with a few moments of silence, and I remember trying to come up with some appropriate words with which to start, something about needing to persevere even in the face of terrible events. I also recall that it was given on old-fashioned transparencies (I should try to dig it out of my files…) and that it was actually one of the better talks I had ever given, calmed, or at least slowed compared to my usual nervous agitation, by the events.

(My friend and colleague Peter Coles was also at the conference, offering his own reminiscences on his blog.)

As I mentioned in my last post, 9/11 feels like a milestone, and a millstone — the world hasn’t been the same since. And we still need to persevere in the face of terrible events.

by defjaf at September 11, 2026 08:31 PM

Andrew Jaffe — The Talk I Gave on September 12

More 25th anniversary thoughts and recollections.

I had moved to Oxford less than two weeks before, still settling into my new life in the UK, working at Imperial College in London. That day, I was heading off to Durham, in the north of England, to a conference called “A New Era in Cosmology” — my first big talk now that I had started my permanent academic job. I was going to be a bit late, only able to arrive toward the end of the first day of the conference: September 11, 2001.

My train left in the late morning. A few hours in, passengers were starting to talk about an attack on New York City. This was in the days before smartphones and constant communication — I didn’t even have a mobile phone. The discussions around me were getting more and more frantic, and I was doing my best to piece together the story.

I was born in New York City, and much of my family still lived in the area. My parents lived in the suburb of Fort Lee, New Jersey, right across the Hudson River from Manhattan; from the apartment where I grew up, we had a fantastic view from the 18th floor. My father worked in The City, commuting every morning by car from New Jersey to midtown Manhattan. He would have been at his office that day.

I considered getting off the train somewhere en route, perhaps Sheffield or York, to try to get some more information, but I just stayed on the train. I was able to get a taxi from the station in Durham to my hotel, listening to the news. I was able to call my partner, back in Oxford, who had, luckily considering the pressure on transatlantic calls, been able to get in touch with my family in New York and New Jersey. Everyone was, thankfully, alright, although at this point my father was still in Manhattan. He had noticed all the emergency vehicles speeding downtown, thinking that it was a motorcade for some foreign dignitary, but it was the first fleet of emergency responders heading towards the twin towers after the first collision. Eventually, he made it home, to my family’s apartment overlooking the Hudson. But it was so close to the George Washington Bridge, a piece of vital infrastructure thought to be in danger after the first attack, that he had to be let off a mile or so away and walk the rest.

As for me, I had to give the first talk on 12 September, about the “new era in cosmology” that coming CMB measurements would usher in. We started, understandably, with a few moments of silence, and I remember trying to come up with some appropriate words with which to start, something about needing to persevere even in the face of terrible events. I also recall that it was given on old-fashioned transparencies (I should try to dig it out of my files…) and that it was actually one of the better talks I had ever given, calmed, or at least slowed compared to my usual nervous agitation, by the events.

(My friend and colleague Peter Coles was also at the conference, offering his own reminiscences on his blog.)

As I mentioned in my last post, 9/11 feels like a milestone, and a millstone — the world hasn’t been the same since. And we still need to persevere in the face of terrible events.

by defjaf at September 11, 2026 08:09 PM

Matt von Hippel — Everybody Who Isn’t ”Viewers Like You”

Last week, I talked about how truthseekers get paid. But truth-tellers and truth-seekers are different things.

Consider educational kids’ shows on public television.

Nobody who works on Sesame Street is out there uncovering new letters and numbers. Bill Nye’s show wasn’t bringing analysis fresh from the lab.

The purpose of these shows is to educate. The purpose of education is to change minds.

So who pays for educational kids’ shows on public television?

If you’re from the US and watched PBS growing up, you remember one answer: “viewers like you!” US public television is supported by donations, ordinary people across the country who want it to keep on educating kids.

But you also might remember the lists of names that came before “viewers like you”. Some of those were things like “the Department of Education” or “a grant from the National Science Foundation”: government programs, in other words. Others were philanthropists and private foundations. Some were tied to companies, like the Intel Foundation, or Juicy Juice.

All of these groups, from government departments to donors, are trying to change kids’ minds. They support specific shows on specific topics, where they want kids to be better-informed. The same groups have the same kind of impact on schools. For example, I remember in elementary school we all learned to play a recorder, because a wealthy donor had given the school recorders out of the idea that music education was especially important.

For a truth-seeker like a journalist, accepting that kind of funding would be a problem. Grants for journalists tend to support things like travel, letting journalists learn more about specific topics, not pre-judging the conclusion. But children’s television is about truth-telling, not truth-seeking, so our standards are different. We trust the people making children’s television to care about whether they’re telling the truth. And because the topics aren’t new, we don’t usually worry about their judgement being biased.

All this is rather obvious. But now, consider science YouTube.

Some science YouTubers seem to have a mission much like children’s television. They’re there to teach, not to make independent judgements. They don’t search for truth on their own. And some of them are funded by educational grants, much like children’s television.

Others are a bit more like journalists, or even activists. People follow them for their opinions, to hear their assessment. They’re trying to be truth-seekers.

On YouTube, it’s not always obvious which is which.

There’s a particular group of philanthropists called Effective Altruists, and many of them are concerned about AI. So in between funding things like anti-malaria bed nets, some of them are giving grants to YouTubers to make educational content about AI-related risks.

Apparently, they reached out to Sabine Hossenfelder, which was a bad idea. Sabine Hossenfelder’s followers aren’t just looking for education on known facts. They’re looking for her judgements, her literal bullshit-rating on ideas. And so while she’s paid by “viewers like you”, she’s not really the type to get paid by that type of grant.

What I want to emphasize, and what looked like it was getting lost in the discussion, was that their pitch would have been totally reasonable for other YouTubers. Educators do occasionally get grants to educate on specific topics. This is in fact a totally normal thing. Some YouTubers are educators first and foremost, they aren’t there as truth-seekers, but truth-tellers, with a real difference in how careful they need to be about bias.

Some YouTubers are different from other YouTubers. News at 11.

by 4gravitons at September 11, 2026 04:00 PM

September 09, 2026

John Baez — The E6 Root Polytope

I’ve been thinking about the exceptional Lie algebra E₆, as a spinoff of my project on E₇, so I want to get a good mental picture of the E₆ root polytope. This is 6-dimensional polytope with remarkable symmetry.

Let’s climb up to it, starting with some of its 4-dimensional faces, which are called 4-demicubes because you get them by taking a 4-dimensional cube, or tesseract, and removing every other corner. The 3-demicube is just a tetrahedron, since you can fit two tetrahedra in a 3-dimensional cube like this:

The 4-demicube builds on this fact in a surprising way.

I’m going to use the technology of Dynkin diagrams, or technically Coxeter diagrams: they’re closely related, and the difference is invisible here. I won’t explain them, just use them. I explained them here:

• Symmetry and the fourth dimension: part 3, part 4, part 5, part 6.

Let’s dive in!

The 4-demicube lives in 4 dimensions. It has 8 vertices.

You get it from a 4-dimensional cube, which has 2⁴ = 16 vertices, by keeping every other vertex, throwing away half. That leaves 8.

What are its top-dimensional faces, aka ‘facets’? Surprise: there’s only one kind! All of them are regular tetrahedra.

In higher dimensions the demicube has two kinds of facet. You get a simplex-shaped facet from every other vertex, formed when you remove it. And you get a demicube-shaped facet from each of the cube’s facets. But in 4 dimensions the two kinds happen to be the same shape!

Eight of them are tetrahedra. These appear at the 8 corners you sliced off: one per removed corner.

Eight more come from the 8 faces of the 4-dimensional cube. These are 3-demicubes. But as we’ve seen, the 3-demicube is also a tetrahedron!

So the 4-demicube is especially symmetric: it has 16 tetrahedral facets. You can find coordinates where its vertices are

(±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1)

It’s actually one of the 4-dimensional regular polytopes, sometimes called the 4-orthoplex. It’s also called the 16-cell because it has 16 facets. It’s the 4-dimensional cousin of the octahedron, which has 8 triangular facets.

You can read some of these facts off the D₄ Dynkin diagram, if you know what you’re doing. As you can see above, this diagram has a central node with three arms, each just 1 edge long: a perfectly symmetric three-pronged star. To get the 4-demicube, you ring the tip of any one arm.

To get the facets of the 4-demicube, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives. There are two choices: you can delete the tip of either other arm. But either way, what’s left is a straight chain of 3 nodes—the so-called A₃ diagram—with a ring at one node at the end. This gives the tetrahedron.

Both choices give the same shape of facet, a tetrahedron, because all three arms of the D₄ Dynkin diagram are interchangeable. That ceases to be true in higher dimensions!

Next, the 5-demicube. This lives in 5 dimensions and has 16 vertices.

You get it from a 5-dimensional cube—which has 2⁵ = 32 vertices—by keeping every other vertex, throwing away half. That leaves 16.

What are its top-dimensional faces, or ‘facets’? There are two kinds!

Sixteen of them are 4-dimensional analogues of the regular tetrahedron, called 4-simplexes. These appear at the corners you sliced off: one per removed corner.

The other ten come from the ten faces of the 5-dimensional cube. After you take every other vertex, they become 4-demicubes. These are precisely the 4-demicubes we saw in the last section!

You can also read these two kinds of facets from the D₅ Dynkin diagram. As you can see above, this diagram has three arms of lengths 2, 1, 1 (edges from the central branch node). To get the 5-demicube, you ring the tip of either length-1 arm. That ringed diagram encodes the whole polytope.

To get the facets, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives.

There are two choices.

If you delete the tip of the other length-1 arm, what’s left is a straight chain of 4 nodes—the diagram whose polytope is the 4-simplex. That gives the 4-simplex faces.

Or you can delete the tip of the length-2 arm. Then what’s left is a shorter branching diagram, the one I showed you in my last post! That gives the 4-demicube faces.

So the 5-demicube has both 4-simplex and 4-demicube faces.

Next let’s go up to the 6th dimension, which was my goal all along.

The E₆ root polytope lives in 6 dimensions. It has 72 vertices.

What are its facets? You can read them straight off the E₆ Dynkin diagram, using the same procedure we’ve been using so far.

As you can see, the E₆ Dynkin diagram has three arms of lengths 2, 2, 1 (edges from the central branch node). To get the root polytope, you ring the node that’s the tip of a length-1 arm. That fact is not obvious, but let’s go ahead and do that.

Then, to get the facets, delete any unringed node such that the piece still holding the ring stays connected, and see what diagram survives.

There are two choices: the two other nodes at tips of the Dynkin diagram.

However, deleting either of these nodes leave a D₅ diagram with a ring on one node, and this gives the 5-demicube we saw last time: a 5-cube with alternate vertices removed.

So the facets of the E₆ root polytope are all the same shape: 5-demicubes!

With more work, we can count the facets of the polytopes we’ve been studying:

• The E₆ root polytope has 54 facets, all 5-demicubes. They come in two kinds, because we had two choices of which node to delete, so there are really 27 ‘positive’ 5-demicube facets and 27 ‘negative’ 5-demicube facets.

• The 5-demicube has 16 4-simplex facets, one for each vertex that we removed from the 5-cube to create this demicube, and 10 4-demicube facets, one for each facet of that 5-cube.

• The 4-demicube has 8 3-simplex facets, one for each vertex that we removed from the 4-cube to create this demicube, and 8 3-demicube facets, one for each facet of that 4-cube. But both the 3-simplex and the 3-demicube are the familiar tetrahedron. So in fact the 4-demicube has 16 tetrahedral facets. Indeed, the 4-demicube is the 4-dimensional analogue of an octahedron: the so-called 4-orthoplex, or 16-cell.

Using some fancier math I explained here, we can count all the faces of the E₆ root polytope. This polytope, is also called 1₂₂ due to the shape of its Dynkin diagram: the ring is on a branch of length 1, not counting the central node, while the other two branches have lengths 2. You can look up all this information on the Wikipedia page 1₂₂ polytope:

Faces of the E₆ root polytope, or 1₂₂

dim faces count

5 5-demicubes 54 = 27 + 27

4 4-demicubes = 4-orthoplexes 270

4 4-simplexes 432 = 216 + 216

3 3-simplexes = 3-demicubes = tetrahedra 2160 = 1080 + 1080

2 2-simplexes = triangles 2160

1 1-simplexes = edges 720

0 0-simplexes = vertices 72

The 5-dimensional facets are all 5-demicubes, but as we’ve seen, they come in two kinds: that is, they lie in two orbits of the symmetry group. We can call 27 of them ‘positive’ 5-demicubes and 27 of them ‘negative’ demicubes. Of the 4-dimensional faces, 270 are 4-demicubes and 432 are 4-simplexes. Moreover the 4-simplexes come in two kinds: 216 are faces of positive 5-demicubes while 216 are faces of negative 5-demicubes. Let’s call the first kind of 4-simplex ‘positive’ and the second kind ‘negative’. The 3-dimensional faces are all tetrahedra, but they come in two ‘kinds’: 1080 of them are faces of positive 4-simplexes, and 1080 are faces of negative 4-simplexes. None is the face of both a positive and negative 4-simplex.

If you’re curious about how to count these things, see how some of us counted all the faces of the E₈ root polytope here:

• John Baez, Integral octonions (part 5), The n-Category Café, September 3, 2013.

Here is a table of faces for the E₇ root polytope, which is also called 2₃₁:

Faces of the E₇ root polytope, or 2₃₁

dim faces count

6 2₂₁ polytopes 56

6 6-simplexes 576

5 5-orthoplexes 756

5 5-simplexes 4032

4 4-simplexes 16128 = 4032 + 12096

3 3-simplexes = tetrahedra 20160

2 2-simplexes = triangles 10080

1 1-simplexes = edges 2016

0 0-simplexes = vertices 126

Its 4-dimensional faces are all 4-simplexes, but they come in two ‘kinds’: that is, they lie in two orbits of the symmetry group of this polytope. Of the 4-simplexes, 4032 are the face of three 5-orthoplexes, while 12096 are the face of one 5-orthoplex and two 5-simplexes.

Here’s the E₈ root polytope, also called 4₂₁:

Faces of the E₈ root polytope, or 4₂₁

dim faces count

7 7-orthoplexes 2160

7 7-simplexes 17280

6 6-simplexes 207360 = 138240 + 69120

5 5-simplexes 483840

4 4-simplexes 483840

3 3-simplexes = tetrahedra 241920

2 2-simplexes = triangles 60480

1 1-simplexes = edges 6720

0 0-simplexes = vertices 240

There are two kinds of 6-simplex faces: 138240 of them each lie in one 7-simplex and one 7-orthoplex, while 69120 of them each lie in two 7-orthoplexes (and no 7-simplex).

by John Baez at September 09, 2026 01:15 PM

Andrew Jaffe — Second test post

With some more text…

by defjaf at September 09, 2026 06:34 AM

Andrew Jaffe — Test post

Checking some infrastructure…

by defjaf at September 09, 2026 06:20 AM

September 08, 2026

n-Category Café The E6 Root Polytope

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I’ve been thinking about the exceptional Lie algebra E₆, as a spinoff of my project on E₇, so I want to get a good mental picture of the E₆ root polytope. This is 6-dimensional polytope with remarkable symmetry.

$MathML-enabled post (click for more details).$

Let’s climb up to the E₆ root polytope starting with some of its 4-dimensional faces, which are called 4-demicubes because you get them by taking a 4-dimensional cube, or tesseract, and removing every other corner. The 3-demicube is just a tetrahedron, since you can fit two tetrahedra in a 3-dimensional cube like this:

The 4-demicube builds on this fact in a surprising way.

I’m going to use the technology of Dynkin diagrams, or technically Coxeter diagrams: they’re closely related, and the difference is invisible here. I won’t explain them, just use them. I explained them here:

• Symmetry and the fourth dimension: part 3, part 4, part 5, part 6.

Let’s dive in!

The 4-demicube lives in 4 dimensions. It has 8 vertices.

You get it from a 4-dimensional cube, which has 2⁴ = 16 vertices, by keeping every other vertex, throwing away half. That leaves 8.

What are its top-dimensional faces, aka ‘facets’? Surprise: there’s only one kind! All of them are regular tetrahedra.

In higher dimensions the demicube has two kinds of facet. You get a simplex-shaped facet from every other vertex, formed when you remove it. And you get a demicube-shaped facet from each of the cube’s facets. But in 4 dimensions the two kinds happen to be the same shape!

Eight of them are tetrahedra. These appear at the 8 corners you sliced off: one per removed corner.

Eight more come from the 8 faces of the 4-dimensional cube. These are 3-demicubes. But as we’ve seen, the 3-demicube is also a tetrahedron!

So the 4-demicube is especially symmetric: it has 16 tetrahedral facets. You can find coordinates where its vertices are

$(\pm 1, 0, 0, 0), \quad (0, \pm 1, 0, 0), \quad (0, 0, \pm 1, 0), \quad (0, 0, 0, \pm 1)$

It’s actually one of the 4-dimensional regular polytopes, sometimes called the 4-orthoplex. It’s also called the 16-cell because it has 16 facets. It’s the 4-dimensional cousin of the octahedron, which has 8 triangular facets.

You can read some of these facts off the D₄ Dynkin diagram, if you know what you’re doing. As you can see above, this diagram has a central node with three arms, each just 1 edge long: a perfectly symmetric three-pronged star. To get the 4-demicube, you ring the tip of any one arm.

To get the facets of the 4-demicube, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives. There are two choices: you can delete the tip of either other arm. But either way, what’s left is a straight chain of 3 nodes—the so-called A₃ diagram—with a ring at one node at the end. This gives the tetrahedron.

Both choices give the same shape of facet, a tetrahedron, because all three arms of the D₄ Dynkin diagram are interchangeable. That ceases to be true in higher dimensions!

Next, the 5-demicube. This lives in 5 dimensions and has 16 vertices.

You get it from a 5-dimensional cube—which has 2⁵ = 32 vertices—by keeping every other vertex, throwing away half. That leaves 16.

What are its top-dimensional faces, or ‘facets’? There are two kinds!

Sixteen of them are 4-dimensional analogues of the regular tetrahedron, called 4-simplexes. These appear at the corners you sliced off: one per removed corner.

The other ten come from the ten faces of the 5-dimensional cube. After you take every other vertex, they become 4-demicubes. These are precisely the 4-demicubes we saw in the last section!

You can also read these two kinds of facets from the D₅ Dynkin diagram. As you can see above, this diagram has three arms of lengths 2, 1, 1 (edges from the central branch node). To get the 5-demicube, you ring the tip of either length-1 arm. That ringed diagram encodes the whole polytope.

To get the facets, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives.

There are two choices.

If you delete the tip of the other length-1 arm, what’s left is a straight chain of 4 nodes—the diagram whose polytope is the 4-simplex. That gives the 4-simplex faces.

Or you can delete the tip of the length-2 arm. Then what’s left is a shorter branching diagram, the one I showed you in my last post! That gives the 4-demicube faces.

So the 5-demicube has both 4-simplex and 4-demicube faces.

Next let’s go up to the 6th dimension, which was my goal all along.

The E₆ root polytope lives in 6 dimensions. It has 72 vertices.

What are its facets? You can read them straight off the E₆ Dynkin diagram, using the same procedure we’ve been using so far.

As you can see, the E₆ Dynkin diagram has three arms of lengths 2, 2, 1 (edges from the central branch node). To get the root polytope, you ring the node that’s the tip of a length-1 arm. That fact is not obvious, but let’s go ahead and do that.

Then, to get the facets, delete any unringed node such that the piece still holding the ring stays connected, and see what diagram survives.

There are two choices: the two other nodes at tips of the Dynkin diagram.

However, deleting either of these nodes leave a D₅ diagram with a ring on one node, and this gives the 5-demicube we saw last time: a 5-cube with alternate vertices removed.

So the facets of the E₆ root polytope are all the same shape: 5-demicubes!

With more work, we can count the facets of the polytopes we’ve been studying:

• The E₆ root polytope has 54 facets, all 5-demicubes. They come in two kinds, because we had two choices of which node to delete, so there are really 27 ‘positive’ 5-demicube facets and 27 ‘negative’ 5-demicube facets.

• The 5-demicube has 16 4-simplex facets, one for each vertex that we removed from the 5-cube to create this demicube, and 10 4-demicube facets, one for each facet of that 5-cube.

• The 4-demicube has 8 3-simplex facets, one for each vertex that we removed from the 4-cube to create this demicube, and 8 3-demicube facets, one for each facet of that 4-cube. But both the 3-simplex and the 3-demicube are the familiar tetrahedron. So in fact the 4-demicube has 16 tetrahedral facets. Indeed, the 4-demicube is the 4-dimensional analogue of an octahedron: the so-called 4-orthoplex, or 16-cell.

Using some fancier math I explained here, we can count all the faces of the E₆ root polytope:

dim faces count

5 5-demicubes 54 = 27 + 27

4 4-demicubes = 4-orthoplexes 270

4 4-simplexes 432

3 3-demicubes = 3-simplexes = tetrahedra 2160 = 1080 + 1080

2 2-simplexes = triangles 2160

1 1-simplexes = edges 720

0 0-simplexes = vertices 72

If you’re curious about how to count these things, see how some of us counted all the faces of the E₈ root polytope here:

John Baez, Integral octonions (part 5), The n-Category Café, September 3, 2013.

© john at September 08, 2026 05:28 PM

Andrew Jaffe — 25 & 60

Twenty-five years ago, I moved from San Francisco to the UK, from a fellowship at Berkeley to a permanent job at Imperial College, London. A lot has changed since then. I was 35; now I am 60. It was two weeks before 9/11; the world hasn’t seemed as open and free since.

I arrived from the Bay Area just after the first dot-com bubble burst. London, adjusting to New Labour after almost two decades of Thatcher and Thatcherism, felt exciting and vibrant. But just weeks after I arrived came the horror of 9/11, and its years-long aftermath, especially the Iraq war which eventually doomed Blair’s premiership and probably led the way to the 2010 election, the disaster of “austerity” as a wrong-headed attempt to deal with the 2008 recession, and eventually to the more-disastrous Brexit on this side of the Atlantic. Similar politics, though with very different timing, led back in the USA to Obama, one of the few rays of political hope over the last quarter-century, but then, of course, to Trump. And everywhere since 2001 the rise of nativist populism making me feel at home, well, pretty much nowhere — a rootless cosmopolitan. Higher education, scientific funding, and curiosity-driven research are in a parlous state in both the US and the UK.

But: despite a few difficult years in the mid-2000s, I have prospered. Our analysis of data from the Planck satellite has solidified our standard cosmological model — but also given us new problems and puzzles to think and worry about. I have written a book, The Random Universe, trying to explain how we know what we know as scientists and as human beings. And my family, my wife and two daughters, are a source of joy and excitement that inspire me every day.

So now I am 60. I was honoured and humbled a couple of months to ago to be joined by many of my colleagues and scientific friends at a conference here in London. That, and getting a Transport For London 60+ Travel Card, makes it hard to avoid feeling old. But those colleagues and friends (many of whom are older than me) reassured me that it’s only the beginning of a new chapter.

by defjaf at September 08, 2026 03:56 PM

September 06, 2026

John Baez — The Mantle

As we descend from the base of Earth’s crust through the mantle, the rock does not remain unchanged. Pressure and temperature rise inexorably, and the minerals that thrive at the surface are forced, step by step, into new and denser crystallographic arrangements. This is the story of those transformations.

In this tale, I’ll act like I know a bit about minerals. I actually don’t: there are a bewildering variety, and I can never remember them. So don’t worry: when you come across a jargon-filled patch of prose, just power through it. You might learn a little… or you can just ignore it. The overall point here is that the Earth is made of beautiful crystalline structures that change character in complex ways as we descend.

The Mohorovičić discontinuity

Our story begins at the boundary where Earth’s crust, rich in feldspar and quartz, gives way to the denser mantle beneath. We see this boundary through its effect on seismic waves, and it’s called the Mohorovičić discontinuity or “Moho”. The Moho does not lie at one fixed depth: it’s 5–10 kilometers below the seafloor, but 30–50 kilometers below most continents, and as much as 70–80 below young mountain belts like the Himalayas.

The mantle just below the Moho mainly consists of a rock called peridotite, which is made mostly of olivine and pyroxene, with smaller amounts of garnet (or, at shallower depths, spinel). Peridotite has a delicious coarse green appearance:

More precisely, this is what peridotite looks like up here. But when geochemists talk about the bulk composition of the upper mantle, they often use an idealized model called pyrolite—not a rock you can pick up, but a hypothetical recipe Ted Ringwood proposed in the 1960s for the primitive upper mantle.

Why? Since the Earth has had a convecting mantle, solid mantle rock wells up in places. As it does, the pressure drops, and a bit of it melts: the minerals with lower melting points. This melt flows upward. It’s called basalt. It builds the Earth’s crust. But it leaves a residue behind, made of minerals with higher melting points.

In Ringwood’s theory, which for expository purposes I’ll assume is true, pyrolite is what mantle rock is like before any partial melting depletes it of basaltic ingredients. The name is a portmanteau of pyroxene and olivine, the two dominant minerals. Pyrolite is about 60% olivine; the remaining 40% is mostly pyroxenes plus garnet.

• A pyroxene is a mineral built from single, unbranched chains of corner-sharing SiO₄ tetrahedra, with metal cations—chiefly Mg, Fe, and Ca—linking the chains together. The general formula is XY(Si,Al)₂O₆, where X and Y are those cations.

• Olivine is a green silicate, (Mg,Fe)₂SiO₄:

Its crystal structure in the upper mantle is an orthorhombic arrangement of isolated SiO₄ tetrahedra knit together by magnesium and iron in octahedral sites. It’s called the α-phase because we’ll see some more compressed phases as we descend.

• A garnet is built from separate SiO₄ tetrahedra held together by cations, but assembled into a dense, hard, characteristically cubic-symmetry crystal. There are different kinds of garnet, but the general formula is X₃Y₂(SiO₄)₃: three divalent X cations, two trivalent Y cations, and three isolated silica tetrahedra. The mantle’s garnet is largely pyrope, Mg₃Al₂(SiO₄)₃.

As we descend, the pyroxenes and garnet gradually dissolve into each other, producing a new high-pressure mineral called majorite. Here’s a rare sample from a meteorite fall in Canada:

So even before the dramatic change 410 kilometers down, the rock is no longer the simple olivine-pyroxene-garnet assemblage we had further up.

The 410-kilometer discontinuity

Roughly 410 kilometers down, the pressure reaches about 13,000 atmospheres and the temperature hovers around 1,400°C. Olivine can no longer hold its familiar shape. It transforms to its β form: wadsleyite, a mineral with the same chemical formula but a fundamentally different atomic arrangement. Instead of isolated SiO₄ tetrahedra, wadsleyite contains paired Si₂O₇ groups, and the oxygens pack more densely. The density jump is sharp enough to be detected globally by seismologists as a reflector of earthquake waves.

Wadsleyite has a remarkable property: it can hold several weight percent of water locked within its crystal structure. The transition zone may thus contain more water than all the oceans combined! However, very little wadsleyite has been seen on the Earth’s surface. Here’s a bit from that same meteor fall in Canada:

The 520-kilometer discontinuity

Descend further, to around 520 kilometers, and the temperature goes up only a little, to roughly 1500–1600°C, since convection here is strong. The pressure goes up to about 175,000 atmospheres. At this point wadsleyite transforms into the γ form of olivine: ringwoodite. This is denser, still chemically Mg₂SiO₄, but now with cations packed into tetrahedral and octahedral holes in a close-packed oxygen framework—the most efficient packing geometry that nature offers for this composition:

Ringwoodite is named for the great Australian geochemist Ted Ringwood, who studied these transitions. Here’s an artificially manufactured sample:

For a long time the mineral’s existence in the mantle was purely hypothetical. But in 2014, a tiny grain was discovered as an inclusion inside a diamond brought up from the deep mantle by an eruption, providing the first direct proof of its existence in Earth’s interior.

The 660-kilometer discontinuity

At a depth of 660 kilometers and a pressure of roughly 230,000 atmospheres, the most dramatic phase transition of all occurs. Ringwoodite does not merely rearrange into a still more dense form! Instead, it decomposes into two entirely new minerals: bridgmanite (MgSiO₃) and ferropericlase (MgO). The majorite garnet also decomposes, yielding davemaoite (CaSiO₃), which is stable through the rest of the lower mantle:

The 660-kilometer discontinuity is sharp, globally consistent, and marks the conventional boundary between the upper and lower mantle. One reason it’s important is that enormous slabs of colder, denser rock sink through the upper mantle until they hit this discontinuity, where the phase change between ringwoodite and bridgmanite creates a kind of barrier.

These slabs are 30–100 kilometers thick and hundreds to a thousand kilometers across! Some punch straight through into the lower mantle and keep sinking. But many flatten out when they hit the barrier, sometimes lying there and piling up for tens of millions of years. You can see this in seismic images beneath Japan and the Marianas. Numerical models suggest that they pile up until they overwhelm the barrier and flush down in a comparatively sudden avalanche—lasting mere millions of years.

The lower mantle

This is the realm of bridgmanite, probably the most abundant mineral in the Earth. Bridgmanite is a beautifully symmetric cage of corner-sharing SiO₆ octahedra, with Mg tucked into the large cavities between them. It accommodates enormous pressure because there is very little void space left to compress.

It is a striking fact that while bridgmanite is the most abundant mineral on the planet, it went unnamed until 2014, simply because no natural hand-sized specimen had ever been recovered. Everything we know about it comes either from high-pressure laboratory synthesis, from microscopic grains in shocked meteorites, or from the indirect testimony of earthquake waves that have traveled through 2,000 kilometers of it.

For over 2,000 kilometers of descent, from 660 to roughly 2,700 kilometers down, bridgmanite and its companion ferropericlase reign without significant further phase change. Seismic velocities increase steadily, but there are no dramatic discontinuities.

The D″ discontinuity

As we approach the core-mantle boundary—at depths around 2,700 kilometers, pressures of approximately 120,000–125,000 atmospheres, and temperatures of 2,200–3,7000°C—even bridgmanite yields. It transforms into the post-perovskite phase. Post-perovskite is a layered, sheet-like structure of SiO₆ octahedra, quite different from bridgmanite’s three-dimensional cage, making it potentially much weaker and more prone to flow.

This transition is believed to be responsible for the seismic D″ discontinuity observed at 2,900 kilometers depth. The D″ layer is a highly dynamic region, likely the site of storage of subducted materials and the source of deep mantle plumes.

A summary of the descent

The table below summarizes the major transitions:

Depth (km)        Minerals

0–410 olivine (α) + pyroxenes + garnet

410 → wadsleyite (β)

520 → ringwoodite (γ)

660 → bridgmanite + ferropericlase + davemaoite

660–2700 bridgmanite dominates

~2700 → post-perovskite

2900 → liquid iron core

The interesting thing about this story is that it was told first by seismology—the sharp jumps in wave speeds at 410 and 660 kilometers were detected long before geologists could reproduce those pressures in the lab—and only later checked by diamond-anvil cell experiments squeezing tiny mineral samples to millions of atmospheres. The rocks never rise to the surface to tell their story directly, so much of the tale above is just theory.

Which minerals are there the most of?

We can estimate how much of the Earth is made of wadsleyite, ringwoodite, and bridgmanite using known shell volumes, estimated densities, and mineral proportions from the pyrolite model.

Step 1: Earth’s mass budget by layer

The Earth’s total mass is M_⊕ ≈ 5.972 × 10²⁴ kg. The mass budget by layer is approximately:

• Crust: ~0.4% of Earth’s mass
• Upper mantle + transition zone (35–660 km): ~18% of Earth’s mass
• Lower mantle (660–2,891 km): ~49% of Earth’s mass
• Core (outer + inner): ~32.5% of Earth’s mass

Step 2: The transition zone (410–660 km)

Using PREM densities averaging ~3,760 kg/m³ across the transition zone, and the volume of each spherical shell:

Wadsleyite zone (410–520 km):
Shell volume ≈ 4.8 × 10¹⁹ m³
Shell mass ≈ 1.76 × 10²³ kg
Fraction of Earth’s mass ≈ 2.9%

Ringwoodite zone (520–660 km):
Shell volume ≈ 5.9 × 10¹⁹ m³
Shell mass ≈ 2.24 × 10²³ kg
Fraction of Earth’s mass ≈ 3.8%

In the pyrolite model of mantle composition, forms of olivine (wadsleyite and ringwoodite) make up roughly 60% of the transition zone by mass, with the remaining ~40% being majoritic garnet. Applying this correction:

Wadsleyite: 0.60 × 2.9% ≈ 1.8% of Earth’s mass
Ringwoodite: 0.60 × 3.8% ≈ 2.3% of Earth’s mass

These estimates carry roughly 20–30% uncertainty, mainly from the assumed 60% olivine proportion in the transition zone, which varies with local temperature and bulk composition.

Step 3: Bridgmanite (660–2,700 km)

The lower mantle holds about 49% of Earth’s mass—it is an enormous shell! Bridgmanite constitutes approximately 80% of the lower mantle mineral assemblage (by mass) in the pyrolite model:

0.80 × 49% ≈ 39% of Earth’s mass

This is consistent with the well-cited literature figure that bridgmanite comprises approximately 38% of the planet’s mass—making it the single most abundant mineral in the Earth by a vast margin.

Mineral Depth (km) Fraction of Earth’s Mass

Wadsleyite 410–520 ~1.8%

Ringwoodite 520–660 ~2.3%

Bridgmanite 660–2,700 ~38–39%

All three combined 410–2,700 ~42%

Thus, these three minerals—all members of the same Mg₂SiO₄/MgSiO₃ chemical lineage—together constitute roughly 42% of Earth’s entire mass. All other named minerals on Earth, including quartz, feldspar, calcite, diamond, and the roughly 3,800 others known to mineralogists, divide up the remaining scraps.

by John Baez at September 06, 2026 02:32 PM

September 01, 2026

n-Category Café Three Generations in E7

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It’s long been a mystery why there are 3 generations of quarks and leptons: three sets of particles, apparently identical except for how they interact with the Higgs boson. It would be nice if there were some good physical explanation. Nobody knows one. Barring that, it would be nice if some beautiful mathematical structure made this pattern seem natural. That’s what my new paper is about.

I’ll keep this nontechnical. I’ll say a bit about what the paper does, what it does not do, what led up to it, and how I wrote it.

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This is my third paper about exceptional algebraic structures and the Standard Model. When you classify famous gadgets in algebra, beautiful gadgets with fancy names like ‘simple Lie algebras’ and ‘Euclidean Jordan algebras’ and ‘positive hermitian Jordan pairs’, you tend to get infinite series of them — together with a few exceptions that can be built using the octonions. This is a bit spooky, so I’ve been interested in this for a long time.

A few physicists have hoped that these exceptions are good for something. For example, maybe the quirky features of our best theory of particle physics, the Standard Model, aren’t accidental. Perhaps they fall out naturally from some exceptional algebraic structure.

It’s a long shot, but we’ve been stuck on figuring out new fundamental laws of particle physics for so long — roughly since the early 1980s — that it’s worth a try.

In 2018, Michel Dubois-Violette and Ivan Todorov noticed that the gauge group of the Standard Model falls out as symmetries of the so-called ‘exceptional Jordan algebra’ together with some ordinary Jordan algebras sitting inside it. I tried to clarify that here, with a huge amount of help from an excellent young mathematician:

John Baez and Paul Schwahn, The Standard Model gauge group from the exceptional Jordan algebra. (Blog article here.)

It’s very nice, because the Jordan algebras in question arise naturally when you try to axiomatize the foundations of quantum physics. It would be so cool if something about quantum physics made the Standard Model seem mathematically natural!

But really this result only concerns the gauge bosons in the Standard Model: the photon, gluons, and the W and Z bosons. It says nothing about the fermions — that is, the quarks and leptons. And it seems quite hard to get those into the picture.

In 2020, Latham Boyle tried to solve this problem by tensoring the exceptional Jordan algebra with the complex numbers. This made one generation of fermions appear quite naturally! But the connection to the foundations of quantum physics seemed lost: tensoring the exceptional Jordan algebra with the complex numbers seems at first like it might be just a formal trick.

This spring, Latham and his student Endre Bokor and I showed the connection to quantum physics is not lost:

John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model. (Blog article here.)

The idea is to work, not with Jordan algebras, but with more general things called Jordan pairs, which have been studied by mathematicians since at least 1975. We showed that you can still do quantum physics with Jordan pairs. And we showed that there’s an ‘exceptional’ Jordan pair that naturally contains the Standard Model gauge group and one generation of fermions!

This Jordan pair is built from the bioctonions: the octonions tensored with the complex numbers. And it’s closely related to an exceptional Lie algebra called $\mathfrak{e}_6$ .

This is nice because the work of Dubois-Violette and Todorov used a smaller exceptional Lie algebra called $\mathfrak{f}_4$ . Going up to $\mathfrak{e}_6$ gives the room to include one generation of fermions.

There’s an even larger exceptional Lie algebra you can use to build a Jordan pair: it’s called $\mathfrak{e}_7$ . Bokor, Boyle and I tried using this to get three generations of fermions. There are things that make this tempting: not just the fact that $\mathfrak{e}_7$ is bigger, but the fact that the Jordan pair you get from it has a kind of three-fold symmetry. But we couldn’t get it to work.

Around this time I got very interested in some work that someone had sent me in October 2025. My inbox is packed with new theories of physics. Since the rise of large language models the inflow has increased: I get about two emails a day from someone telling me they’ve made a revolutionary discovery in physics. Practically none of these theories appeal to me. But this paper, and this thesis, were different:

Benjamin Nasmith, An exceptional combinatorial sequence and Standard Model particles, 2020.

Benjamin Nasmith, Tight Projective 5-Designs and Exceptional Structures, Ph.D. thesis, Royal Military College of Canada, 2023.

He claimed to fit three generations of fermions into the exceptional Lie algebra $\mathfrak{e}_7$ .

When I started seriously trying to understand this paper, I wound up translating it into a language I’m more comfortable with, and expanding on the ideas a bit. So I wrote this:

John Baez, Three generations in $\mathfrak{e}_7$ .

Here’s the basic idea.

The idea

There is a standard way to fit the Lie algebra of the Standard Model gauge group, which I call $\mathfrak{g}_{\text{SM}}$ , into the Lie algebra $\mathfrak{e}_7$ . You can construct a Lie algebra $L$ that fits between them:

$\mathfrak{g}_{\text{SM}} \subset L \subset \mathfrak{e}_7$

As a vector space we have

$\mathfrak{e}_7 \; \cong \; L \oplus V$

for some vector space $V$ of dimension $3 \times 32$ .

Moreover, the Lie algebra $\mathfrak{g}_{\text{SM}}$ acts on $V$ , via the $\mathfrak{e}_7$ Lie bracket, precisely as it does on three generations of Standard Model fermions and their antiparticles, including right-handed neutrino and its antiparticle — but ignoring spin!

There is, in fact, a very interesting three-fold symmetry built into $\mathfrak{e}_7$ , which is revealed when we put the Standard Model Lie algebra $\mathfrak{g}_{\text{SM}}$ into it. It permutes the three generations.

Like Nasmith, I am not proposing a theory of physics. I’m only observing a fascinating mathematical pattern that might (or might not) be of some use in physics.

There are lots of things this pattern does not include: basically, everything I didn’t already mention. It does not include the spin of the fermions and gauge bosons. It does not include the Higgs boson, though in some sense it comes close (see the paper). It does not include a Lagrangian, so it doesn’t say anything at all about particle masses or interactions.

I could say a lot more about what my paper does do… most importantly, where this Lie algebra $L$ comes from! The details are very interesting. There’s also the curious role of the right-handed neutrinos. But I’ve already spent weeks explaining all these things in my paper, so I won’t do it here. Instead let me say a bit about how I wrote the paper.

Writing the paper

I’ve been wanting to keep up with how AI is transforming math. About a year ago a friend gave me a subscription to Claude Pro. I wanted to test it out, despite my many misgivings, including how large language models are contributing to global warming and income inequality. Given the amazing things that people have recently done in math using large language models, I didn’t think that never trying them out would put me in the best position to make good decisions about the future.

So, I wrote this paper with help from Claude Opus 4.8.

I started by giving it Nasmith’s paper and asking a long series of questions about that paper over several days. The results were very interesting and helpful. Eventually I asked it to summarize and expand on our conversation. It quickly spat out a 10-page paper.

This paper was written in a breezy, pleasant style — but also quite hard to understand in detail, since it mixed Nasmith’s terminology with the Lie algebra terminology I prefer, and the proofs skipped over some steps.

It took me about three weeks of hard work to fully understand and re-express all the ideas a way that I like. For a while I felt dumb and frustrated, because when I asked Claude to fill in the gaps in proofs, it used math I was not very competent in, like the theory of regular subalgebras, and the theory of minuscule representations. But I learned this math, and everything turned out to be basically correct — in part, I’m sure, because Nasmith’s original work was correct.

For several weeks I checked, reorganized, expanded and completely rewrote this material. By the end everything was written in a style I like, emphasizing the ideas I consider important, proving things fairly carefully, and adding a lot of expository material — for example, explaining the theory of regular subalgebras.

Almost no traces of Claude’s original writeup remain, even though I was deeply influenced by them. My proofs make few references to deep theorems, though they assume solid familiarity with simple Lie algebras and their root systems. The proofs also require no brutally hard computations — though Claude was eager to do such computations to check things.

Any mistakes in this paper are my own.

I’m not sure what conclusions I draw from writing this paper. I’m writing another math paper now, with a human coauthor, and I have no desire to get help from a large language model. For work on my own it could be very helpful. Fields medalist Jacob Tsimerman says it roughly doubles his productivity. Would using it be so bad for the environment, or so bad for society, that I should avoid it? Maybe. I deliberately stuck with Claude Opus 4.8 instead of something more powerful, to see what I could do with what you get from a $20/month subscription. But maybe that’s still bad.

I avoid flying to conferences, which in some ways cripples my ability to keep up with new trends and influence people — but I don’t mind that. It gives me more time to think.

I will think carefully about my next move.

© john at September 01, 2026 10:19 AM

August 17, 2026

John Baez — Three Generations in E7

It’s long been a mystery why there are 3 generations of quarks and leptons: three sets of particles, apparently identical except for how they interact with the Higgs boson. It would be nice if there were some good physical explanation. Nobody knows one. Barring that, it would be nice if some beautiful mathematical structure made this pattern seem natural. That’s what my new paper is about.

It’s my third paper about exceptional algebraic structures and the Standard Model. When you classify famous gadgets in algebra, beautiful gadgets with fancy names like ‘simple Lie algebras’ and ‘Euclidean Jordan algebras’ and ‘positive hermitian Jordan pairs’, you tend to get infinite series of them—together with a few exceptions that can be built using the octonions. This is a bit spooky, so I’ve been interested in this for a long time.

A few physicists have hoped that these exceptions are good for something. For example, maybe the quirky features of our best theory of particle physics, the Standard Model, aren’t accidental. Perhaps they fall out naturally from some exceptional algebraic structure.

It’s a long shot, but we’ve been stuck on figuring out new fundamental laws of particle physics for so long—roughly since the early 1980s—that it’s worth a try.

In 2018, Michel Dubois-Violette and Ivan Todorov noticed that the gauge group of the Standard Model falls out as symmetries of the so-called ‘exceptional Jordan algebra’ together with some ordinary Jordan algebras sitting inside it. I tried to clarify that here, with a huge amount of help from an excellent young mathematician:

• John Baez and Paul Schwahn, The Standard Model gauge group from the exceptional Jordan algebra. (Blog article here.)

It’s very nice, because the Jordan algebras in question arise naturally when you try to axiomatize the foundations of quantum physics. It would be so cool if something about quantum physics made the Standard Model seem mathematically natural!

But really this result only concerns the gauge bosons in the Standard Model: the photon, gluons, and the W and Z bosons. It says nothing about the fermions—that is, the quarks and leptons. And it seems quite hard to get those into the picture.

In 2020, Latham Boyle tried to solve this problem by tensoring the exceptional Jordan algebra with the complex numbers. This made one generation of fermions appear quite naturally! But the connection to the foundations of quantum physics seemed lost: tensoring the exceptional Jordan algebra with the complex numbers seems at first like it might be just a formal trick.

This spring, Latham and his student Endre Bokor and I showed the connection to quantum physics is not lost:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model. (Blog article here.)

The idea is to work, not with Jordan algebras, but with more general things called Jordan pairs, which have been studied by mathematicians since at least 1975. We showed that you can still do quantum physics with Jordan pairs. And we showed that there’s an ‘exceptional’ Jordan pair that naturally contains the Standard Model gauge group and one generation of fermions!

This Jordan pair is built from the bioctonions: the octonions tensored with the complex numbers. And it’s closely related to an exceptional Lie algebra called $\mathfrak{e}_6.$

This is nice because the work of Dubois-Violette and Todorov used a smaller exceptional Lie algebra called $\mathfrak{f}_4.$ Going up to $\mathfrak{e}_6$ gives the room to include one generation of fermions.

There’s an even larger exceptional Lie algebra you can use to build a Jordan pair: it’s called $\mathfrak{e}_7.$ Bokor, Boyle and I tried using this to get three generations of fermions. There are things that make this tempting: not just the fact that $\mathfrak{e}_7$ is bigger, but the fact that the Jordan pair you get from it has a kind of three-fold symmetry. But we couldn’t get it to work.

Around this time I got very interested in some work that someone had sent me in October 2025. My inbox is packed with new theories of physics. Since the rise of large language models the inflow has increased: I get about two emails a day from someone telling me they’ve made a revolutionary discovery in physics. Practically none of these theories appeal to me. But this paper, and this thesis, were different:

• Benjamin Nasmith, An exceptional combinatorial sequence and Standard Model particles, 2020.

• Benjamin Nasmith, Tight Projective 5-Designs and Exceptional Structures, Ph.D. thesis, Royal Military College of Canada, 2023.

He claimed to fit three generations of fermions into the exceptional Lie algebra $\mathfrak{e}_7.$

When I started seriously trying to understand this paper, I wound up translating it into a language I’m more comfortable with, and expanding on the ideas a bit. So I wrote this:

• John Baez, Three generations in $\mathfrak{e}_7.$

Here’s the basic idea.

The idea

There is a standard way to fit the Lie algebra of the Standard Model gauge group, which I call $\mathfrak{g}_{\text{SM}},$ into the Lie algebra $\mathfrak{e}_7.$ You can construct a Lie algebra $L$ that fits between them:

$\mathfrak{g}_{\text{SM}} \subset L \subset \mathfrak{e}_7$

As a vector space we have

$\mathfrak{e}_7 \; \cong \; L \oplus V$

for some vector space $V$ of dimension $3 \times 32.$

Moreover, the Lie algebra $\mathfrak{g}_{\text{SM}}$ acts on $V$ , via the $\mathfrak{e}_7$ Lie bracket, precisely as it does on three generations of Standard Model fermions and their antiparticles, including right-handed neutrino and its antiparticle—but ignoring spin!

There is, in fact, a very interesting three-fold symmetry built into $\mathfrak{e}_7,$ which is revealed when we put the Standard Model Lie algebra $\mathfrak{g}_{\text{SM}}$ into it. It permutes the three generations.

Like Nasmith, I am not proposing a theory of physics. I’m only observing a fascinating mathematical pattern that might (or might not) be of some use in physics.

There are lots of things this pattern does not include: basically, everything I didn’t already mention. It does not include the spin of the fermions and gauge bosons. It does not include the Higgs boson, though in some sense it comes close (see the paper). It does not include a Lagrangian, so it doesn’t say anything at all about particle masses or interactions.

I could say a lot more about this… most importantly, where this Lie algebra $L$ comes from. The details are very interesting. There’s also the curious role of the right-handed neutrinos. But I’ve already spent weeks explaining all these things in my paper, so I won’t do it here. Instead let me say a bit about how I wrote the paper.

Writing the paper

I’ve been wanting to keep up with how AI is transforming math. About a year ago a friend gave me a subscription to Claude Pro. I wanted to test it out, despite my many misgivings, including how large language models are contributing to global warming and income inequality. Given the amazing things that people have recently done in math using large language models, I didn’t think that never trying them out would put me in the best position to make good decisions about the future.

So, I wrote this paper with help from Claude Opus 4.8.

I started by giving it Nasmith’s paper and asking a long series of questions about that paper over several days. The results were very interesting and helpful. Eventually I asked it to summarize and expand on our conversation. It quickly spat out a 10-page paper.

This paper was written in a breezy, pleasant style—but also quite hard to understand in detail, since it mixed Nasmith’s terminology with the Lie algebra terminology I prefer, and the proofs skipped over some steps.

It took me about three weeks of hard work to fully understand and re-express all the ideas a way that I like. For a while I felt dumb and frustrated, because when I asked Claude to fill in the gaps in proofs, it used math I was not very competent in, like the theory of regular subalgebras, and the theory of minuscule representations. But I learned this math, and everything turned out to be basically correct—in part, I’m sure, because Nasmith’s original work was correct.

For several weeks I checked, reorganized, expanded and completely rewrote this material. By the end everything was written in a style I like, emphasizing the ideas I consider important, proving things fairly carefully, and adding a lot of expository material—for example, explaining the theory of regular subalgebras.

Almost no traces of Claude’s original writeup remain, even though I was deeply influenced by them. My proofs make few references to deep theorems, though they assume solid familiarity with simple Lie algebras and their root systems. The proofs also require no brutally hard computations—though Claude was eager to do such computations to check things.

Any mistakes in this paper are my own.

I’m not sure what conclusions I draw from writing this paper. I’m writing another math paper now, with a human coauthor, and I have no desire to get help from a large language model. For work on my own it could be very helpful. Jacob Tsimerman says it roughly doubles his productivity. Would using it be so bad for the environment, or so bad for society, that I should avoid it? Maybe. I deliberately stuck with Claude Opus 4.8 instead of something more powerful, to see what I could do with what you get from a $20/month subscription. But maybe that’s still bad.

I avoid flying to conferences, which in some ways cripples my ability to keep up with new trends and influence people—but I don’t mind that. It gives me more time to think.

I will think carefully about my next move.

by John Baez at August 17, 2026 02:28 PM

August 13, 2026

Tim Gowers — What sort of maths are LLMs good at?

For the sake of anyone who might read this blog post in the distant future (a month from now, say), let me mention that I am writing it a few days after OpenAI announced that it had solved ten major problems in mathematics and theoretical computer science, including the first construction of a non-sofic group, and a proof that the multicolour Ramsey number $R(3,3,...,3)$ (where there are $k$ 3’s) grows superexponentially in $k$ . The first was, to judge from various talks I have been to, one of the most important unsolved problems in group theory, and the second was a major open problem in Ramsey theory that I didn’t necessarily expect to see solved in my lifetime, though of course such expectations now have to be revised. The reason I want to be clear about the timing is that I shall be discussing the current capabilities of LLMs in the full expectation that those will continue to change rapidly. So it is likely that in not too long from now, if there is anything interesting in what I write, it will be interesting mainly as a record of what the situation looked like in early August 2026.

These results, and the other eight on the list, are extraordinarily impressive, but it still doesn’t seem to be the case that LLMs are better than all humans at all aspects of mathematics. If they were, then their big speed advantage over us would mean that there would be much more of a flood of results. So it is natural to wonder about what kinds of problems LLMs are good at, and about where there is still room for improvement. I don’t pretend to have a good answer to this question, where a good answer would be a crisp classification that would fit the current examples well, but it is an interesting exercise to try to rule out some bad answers, and to try to identify potential answers that aren’t obviously contradicted by the evidence.

Are LLMs particularly good at finding counterexamples?

A first remark here is that LLMs are not just good at finding counterexamples: they can find proofs of difficult statements as well. However, it is notable that the most famous problems they have solved have almost all been with counterexamples rather than proofs. That is true of the two problems mentioned above, and also of the Jacobian conjecture and the unit distance conjecture.

If one wants to theorize that LLMs are particularly good at finding counterexamples, then there are two things it would be good to do to make the theory more convincing. The first may sound unproblematic: it is to decide when solving a problem counts as finding a counterexample. Once that is sorted out, the second is to come up with a potential explanation of why LLMs would be particularly well suited to solving problems of that particular kind.

What does it mean to find a counterexample?

Why am I suggesting that it is not completely obvious what it means to find a counterexample? Surely, one might suggest, all it means is that you have a statement of the form “Every object of such and such a type has such and such a property,” and you exhibit an object of the given type that does not have the given property.

However, this doesn’t always work. Consider a famous result of Vinogradov, which states that every sufficiently large positive integer is a sum of three primes. The negation of this statement is (or is equivalent to) the statement that for every positive integer $N$ there exists an integer $n\geq N$ such that $n$ is not a sum of three primes. In other words, it states that every positive integer $N$ has a certain property. Seen in this light, Vinogradov found an example of a positive integer $N$ that does not have the given property. Do we want to say that Vinogradov found a counterexample? Clearly not — the result should obviously be classified as a theorem and not a counterexample.

Thus, we cannot just naively say that LLMs are particularly good at negating universally quantified statements: there has to be something about the nature of the universal quantification. With the three-primes example, it is clear that Vinogradov did not think, “How am I going to find $N$ with this property?” Rather, what he thought would have been more like, “I’ve got an integer $n$ that is very large. How am I going to show that it is a sum of three primes?” In other words, all his focus would have been on the universally quantified $n$ , with the existentially quantified $N$ being a sort of afterthought once the details of the proof have been worked out.

In general, many interesting results, when they are stated formally, begin with an alternation of two or three (or more) quantifiers. The question then becomes to determine which is the first “interesting” quantified variable in some sense. Here’s another example to illustrate the point, from the theory of finite-dimensional normed spaces. I’ll give a few mathematical details for those curious, but if you don’t care about those, then you can skip the next three paragraphs and should get the gist of what I am saying about this example.

Let $X$ and $Y$ be two $n$ -dimensional normed spaces and let $T$ be a linear map from $X$ to $Y$ . We say that $T$ is a $C$ –isomorphism if there exists $\lambda>0$ such that $\lambda\|x\|\leq\|Tx\|\leq C\lambda\|x\|$ for every $x\in X$ . By rescaling we can always take $\lambda$ to be 1, in which case we have that $\|x\|\leq\|Tx\|\leq C\|x\|$ for every $x\in X$ . If $C=1$ , then this tells us that $T$ is an isometry. In general, the Banach-Mazur distance $d(X,Y)$ between $X$ and $Y$ is defined to be the smallest $C$ such that there exists a $C$ -isomorphism from $X$ to $Y$ . It is easy to see that the logarithm of the Banach-Mazur distance is a metric on the set of isometry classes of $n$ -dimensional normed spaces. A less easy fact, but still not too hard, is that the resulting metric space is compact: in fact, it is known as the Banach-Mazur compactum.

It is natural to wonder what the diameter of the Banach-Mazur compactum is, and here things get interesting. A result of Fritz John states that every $n$ -dimensional space $X$ has distance at most $\sqrt n$ from $\ell_2^n$ . (The idea of the proof is as follows: pick inside the unit ball of $X$ an $n$ -dimensional ellipsoid of maximal volume; that is the unit ball of a normed space $Y$ that is isometric to $\ell_2^n$ ; it can be shown that the identity map is a $\sqrt n$ -isomorphism between $X$ and $Y$ .) From Fritz John’s theorem and the (multiplicative) triangle inequality, it follows that $d(X,Y)\leq n$ for any two $n$ -dimensional normed spaces. That is, the diameter of the Banach-Mazur compactum is at most $n$ . But might it be substantially less than that?

An indication that the answer is not obvious comes from looking at the spaces $\ell_1^n$ and $\ell_\infty^n$ . The identity map between these two spaces is an $n$ -isomorphism, but one can do much better by mapping the standard basis vectors not to themselves but to vertices of the unit cube, with the vertices chosen to be as orthogonal as possible. In particular, if there exists an $n\times n$ Hadamard matrix, then the corresponding linear map is a $\sqrt n$ -isomorphism. One can push this observation and deduce that for any $p,q\in[1,\infty]$ the Banach-Mazur distance between $\ell_p^n$ and $\ell_q^n$ is $O(\sqrt n)$ . It is also easy to show that $d(\ell_1^n,\ell_2^n)=\sqrt n$ , so $\ell_p$ -spaces hardly improve on the easy lower bound, and do not improve on it at all in dimensions $n$ for which an $n\times n$ Hadamard matrix exists.

In 1981, Gluskin famously solved the problem by determining the correct asymptotics for the diameter of the Banach-Mazur compactum. Informally, what he showed was that the diameter is within a constant of the upper bound that follows immediately from Fritz John’s theorem. If we make the quantification explicit, then the statement we end up with is

$\exists c>0\ \forall n\ \exists X,Y\in K_n\ d(X,Y)\geq cn$ ,

where I have written $K_n$ for the set of all $n$ -dimensional normed spaces. (If you want to argue that it is not a set, then let me specify in addition that the underlying vector space is $\mathbb R^n$ .) In words, there is a positive constant $c$ such that for every positive integer $n$ there are $n$ -dimensional normed spaces $X$ and $Y$ such that the Banach-Mazur distance between $X$ and $Y$ is at least $cn$ .

I can’t continue without very briefly describing the beautiful and highly influential idea Gluskin had for solving this problem. He took $X$ and $Y$ to be normed spaces whose unit balls were random symmetric convex sets defined as follows: take the standard basis vectors and a handful of other random unit vectors, as well as the negatives of all these vectors, and take the convex hull. Gluskin then showed that if two normed spaces are chosen from this distribution, then with high probability their Banach-Mazur distance is at least $cn$ .

But back to the main point, which is that the logical form of the above statement is very similar to the logical form of Vinogradov’s theorem, which is

$\exists N\ \forall n\geq N\ \exists p_1,p_2,p_3\in P\ \ p_1+p_2+p_3=n$

where I have written $P$ for the set of primes. And yet, Vinogradov’s result is unquestionably a theorem, while Gluskin’s result is unquestionably a counterexample, or at least an example.

What is the important difference between the two statements? It seems to be that in Vinogradov’s three-primes theorem the number $n$ plays a more essential role in the statement that is to be proved about the various quantified variables. In Vinogradov’s theorem, that statement is $n=p_1+p_2+p_3$ , whereas for Gluskin’s theorem the statement to be proved is

$\dim X = \dim Y = n$ and $d(X,Y)\geq cn$ ,

which we can write equivalently as

$\dim X = \dim Y = n$ and $d(X,Y)\geq c\dim X$ .

In the case of Vinogradov’s theorem, the whole challenge is to get those three primes to add up to $n$ , whereas for Gluskin it is not remotely challenging to get the dimensions of $X$ and $Y$ to equal $n$ : the challenge is to get $X$ and $Y$ to be very far from each other, relative to their common dimension.

There is a further complication to bear in mind here, which is that via the process known as Skolemization, a universally quantified statement of the form $\forall x\in X\ \exists y\in Y\ \ P(x,y)$ can be converted into an existentially quantifed statement $\exists f:X\to Y\ \forall x\in X\ \ P(x,f(x))$ . (For this to be an equivalence one needs the axiom of choice, but it is certainly a sufficient condition.) This is not just a piece of logical trickery, but it often reflects quite accurately how we think about some problems. For instance, it is more natural to think of Gluskin’s example as a recipe for constructing (or at least proving the existence of) a pair of suitable normed spaces for any given dimension $n$ , or in other words to construct a suitable function from $\mathbb N$ to pairs of normed spaces by giving its value at each $n$ , than it is to think of it as a statement that says that every positive integer $n$ has a certain complicated property.

Yet another complication is that some universally quantified statements follow naturally from existentially quantified statements, or may even be equivalent to them. For example, the theorem that a 2-dimensional torus is not homeomorphic to a 2-dimensional sphere is a universally quantified statement (every map from the torus to the sphere fails to be a homeomorphism), but the natural way to prove it is to prove the existential statement that there is an invariant that distinguishes the two spaces. For an example of where a universal statement is equivalent to an existential statement, consider a statement of the form that a vector $x\in\mathbb R^n$ does not belong to the convex hull of a certain compact set $A$ . The statement that no convex combination of elements of $A$ is equal to $x$ is equivalent to the existence of a linear functional $\phi:\mathbb R^n\to\mathbb R$ and a $\lambda\in\mathbb R$ such that $\phi(x)>\lambda$ and $\phi(a)\leq\lambda$ for every $a\in A$ . In both these cases it feels natural to regard the result as a theorem that is proved via an existential statement, perhaps because it is the theorem that is ultimately what interests us. But using “what interests us” as a criterion to determine what counts as a counterexample seems a little vague, and is a difficult criterion to use if we want to explain convincingly why AI should be good at finding counterexamples.

A more general argument against the notion that there is something about existential statements that is particularly suited to AI is that the need to establish existential statements pervades almost all of mathematical research, regardless of the nature of the headline result being aimed for. For example, if I want to prove a statement by induction, I may well look for a strengthening of the statement that serves better as an inductive hypothesis. Or if I want to prove that every object of type $T$ with property $P$ also has property $Q$ , then I may well look for a property $R$ that follows from $P$ and can be used to prove $Q$ . These are more metamathematical existence problems, but the distinction can be somewhat blurred, and more importantly, when trying to prove a statement $S$ , it is often the case that the main question in our minds is less, “Why is $S$ true?” and more, “What could a proof of $S$ be like?” To give an example, I feel I understand pretty well why Goldbach’s conjecture is true — a highly plausible probabilistic model of the primes implies it and agrees closely with computational data — but if I were making a serious attempt to prove it, that understanding, which many mathematicians have had for a century or so, would be of limited help. Rather, my main task would be to try to find proof techniques that were powerful enough to make those heuristic ideas rigorous.

What is the difference between an example and a counterexample?

Logically, every statement of the form $\exists x\ P(x)$ is a counterexample to the universally quantified statement $\forall x\ \neg P(x)$ . However, we do not describe all existential statements as counterexamples. For example, if I were to say, “The $\ell_p$ -spaces with $1\leq p<\infty$ are all separable, as is $c_0$ , but $\ell_\infty$ is not separable,” I would not describe the second part of that assertion as a counterexample to the claim that all Banach spaces are separable. Rather, I would present it as probably the most basic example of a non-separable space. The important point seems to be that there was no particular reason to think that all Banach spaces would be separable, and finding an example of a non-separable space is not very difficult.

I think the first point is more important here: we are more inclined to call an object a counterexample if the existence of that object disproves a statement that we had quite good reason to believe. It often happens that after repeated unsuccessful attempts to prove a statement, mathematicians begin to feel that it has no particular reason to be true, even if it seems to be hard to come up with a counterexample to it. In such a situation, if a counterexample is eventually found, it may have lost something of its “counter” feel. My impression is that the construction of a non-sofic group comes into this category. There have been several proposals in the literature for how one might construct such a group, and I don’t think there were many (or even any?) experts who strongly believed that all groups were sofic. So it feels more natural to say, “OpenAI came up with the first example of a non-sofic group” than to say, “OpenAI found a counterexample to the soficity conjecture” (despite the fact that that section of their paper is entitled “A counterexample to the soficity conjecture”).

Likewise, it seems to me that the new lower bound for multicolour Ramsey numbers is more of an example than a counterexample. I think quite a lot of people believed that the bound should be exponential, so for them it was a counterexample, but others, myself included, were more neutral about it. As a matter of fact, I have worked on the problem in the past (a long time ago) in an equivalent formulation, which asks how many triangle-free graphs on $n$ vertices you need if you want their union to be the complete graph $K_n$ . If you take bipartite graphs, then it’s easy to see that you need $\log_2n$ of them, but that bound can be improved if instead you observe that a complete 5-partite graph can be written as a union of two triangle-free subgraphs, and therefore it is possible to write the complete graph as a union of $2\log_5n$ triangle-free graphs. It is then tempting to try to do better, with triangle-free graphs that are less dense but that make up for it with unbounded chromatic number — a necessary condition if one wishes to use a sublogarithmic number of graphs, which is equivalent to showing a superexponential lower bound for $R(3,3,\dots,3)$ . All this is to say that when I worked on the problem, my efforts were concentrated on what turned out to be the right direction, so for me OpenAI found an example of what I (weakly) expected, rather than a counterexample.

Where does this leave us?

I would like to find a coherent explanation of the conjunction of the following facts.

The most notable mathematical results proved by LLMs have tended to be ones that we would classify as examples or counterexamples, where counterexamples are, broadly speaking, existence statements that disprove statements that we expected to be true.

Many statements can be formulated as existence statements when we would usually think of them as universal statements, and vice versa, so what we consider to be an example depends on the mathematical context of a statement as well as its logical form.

LLMs are pretty good at proving universal statements as well: it’s just that the strongest statements they have proved that we would think of as theorems have mainly not been at the level of the strongest statements that we would think of as counterexamples.

Given these facts, it seems likely that what LLMs are good at is something else, which happens to have as a consequence that they are good at the kind of existence problem that we would normally classify as asking to find a non-trivial example.

Let us consider two things that we can be confident that LLMs are good at. One of them is knowing a lot of mathematics: if a problem can be solved by means of a relatively standard argument, it is highly likely that an LLM will be able to find and use that argument. The other is the ability that an LLM has simply by virtue of being a computer: it can work at huge speed (compared with humans at least) and can therefore afford to make a large number of unsuccessful attempts at a problem before it finds a solution.

Without even looking at what LLMs have actually managed to solve, one might guess that these two features would lead to their having a somewhat different style from human mathematicians. Very roughly, LLMs would have the edge when there is more of a probabilistic element to the proof-finding process: they would be good at problems for which the best method is to try a lot of ideas, not necessarily particularly novel, until at some point you get lucky. Humans on the other hand would be better (for the moment) at finding more “surprising” and “conceptual” arguments, where the appropriate method is to dig deeper and deeper into a problem until the solution reveals itself. (It is hard to say exactly what this means, but I hope that any experienced researcher reading this will know what I am talking about.)

This raises two questions: does the guess above correspond at all to the reality that we are observing, and is there any reason to suppose that what I have tentatively described as the “LLM style” of doing mathematics would lead naturally to LLMs discovering several counterexamples (or just examples) to long-standing conjectures, even if that was by no means all they could do?

I don’t pretend to have a scientific answer to either question, but the reactions of experts to several of the remarkable solutions that ChatGPT has found do lend some support to the idea that LLMs work in more of a try-lots-of-things-till-you-get-lucky way. People often seem to react by saying something like, “Initially I was amazed that the problem had been solved, but on closer inspection I realized that the approach was actually not all that novel, and one that with the right small hint a suitably expert human could have found quite easily.”

For the second question — whether the LLM style is well suited to finding (counter)examples — I think matters are less clear, because there are many ways of searching for a counterexample, and some of them fit better than others the style I have described. Here are a few general methods. (I don’t claim that the list is exhaustive.)

Look for an off-the-shelf example. Here one has a stock of fairly standard examples and one simply tries them out one after another to see whether any of them fails to satisfy the given statement. For example, Ryan O’Donnell ends his wonderful book on the analysis of Boolean functions with some tips, one of which is, “If you have a conjecture about Boolean functions, test it on dictators, majority, parity, tribes (and maybe recursive majority of 3). If it’s true for these functions, it’s probably true.”

Build an example from basic examples and standard construction methods. For an algebraic problem, for instance, one might start with some standard examples, but then take products or quotients or limits.

Make heavy use of metavariables. The word “metavariable” comes from computer science, and in particular from automatic theorem proving, and refers to the practice that in mathematics would correspond to writing, “where $x$ is to be chosen later,” (in which case $x$ is the metavariable). In a paper we usually do this only in fairly simple situations such as when we need to choose a number $\epsilon>0$ that is small enough for later arguments to work. But when we search for an example of an object $x$ that satisfies some property $Q$ (which may well be a conjunction of simpler properties $Q_1,\dots,Q_k$ ), it is often not a good strategy to specify $x$ completely and only then to check whether it satisfies $Q$ . Instead, it can be more fruitful to do almost the opposite: we start by saying virtually nothing about $x$ and simply launch into proving that it satisfies $Q$ . In the course of doing so, we find that we need $x$ to satisfy a property $P_1$ . If we are lucky we can describe in a nice way a very general class of objects $x$ that satisfy $P_1$ . For instance, we may be able to find a parametrized class: we identify some function $f$ and show that $f(y)$ satisfies $P_1$ for every $y$ of a certain type. The problem is then reduced to finding $y$ such that $Q(f(y))$ holds, which is a more specific version of the original problem. There may be many iterations of this process, or a mixture of this process and other processes, before an example is eventually found.

Try to prove the opposite. If one wishes to find $x$ such that $Q(x)$ , it can be surprisingly helpful to start by attempting to prove the statement $\forall x\ \neg Q(x)$ . The reason this can be helpful is that using our standard methods of attempting to prove something, we may end up identifying a key lemma that would suffice: that is, we may find an intermediate property $R$ that implies $\neg Q$ in a non-trivial way and thus reduce the problem $\forall x\ \neg Q(x)$ to $\forall x\ R(x)$ . Turning things round again, it may well then be that finding a counterexample to $R$ is easier than finding a counterexample to $\neg Q$ (that is, an example that satisfies $Q$ ). Of course, there is no guarantee that a counterexample to $R$ will be an example of $Q$ , but sometimes we are lucky and it is. More often, we can use the idea of the previous method, noting that it is at least a necessary condition of an example of $Q$ that it should not be an example of $R$ , so one can try to describe a general class of objects that fail $R$ and in that way reduce the problem.

Successive approximation. Sometimes, when we are searching for an example of $x$ such that $Q(x)$ , we write down a moderately plausible guess $x_0$ not because we think it has a chance of working (if we did, then we would be using the first strategy), but because we hope that if $x_0$ does not satisfy $Q$ , then we will be able to diagnose what went wrong and specify a new guess $x_1$ that does not have that defect. Again, this strategy can either be iterated or combined with one or more of the other strategies.

Just-do-it proofs. Sometimes we need $x$ to satisfy infinitely many properties $Q_1,Q_2,\dots$ , each of which is, individually, quite easy to satisfy. In such situations, we often “build” $x$ inductively bit by bit, ensuring at the $i$ th stage of the process that however the building process continues, $x$ will satisfy $Q_i$ .

Pick a random example. Often it is very hard to give an explicit example of an $x$ that satisfies $Q$ , but there is a natural probability distribution for which one can show that if one chooses $x$ randomly from that distribution, then with high probability (or at least non-zero probability) it will satisfy $Q$ .

Pick a generic example. In more infinite contexts, it may again be quite hard to give an explicit example of an $x$ that satisfies $Q$ , but one may be able to show that the set of $x$ that fail $Q$ is or measure zero, or is a meagre set, or is small in some other way.

There is no particular reason to suppose that LLMs would be equally good at each of the methods above. So perhaps what we are observing is not quite that LLMs have a particular ability to find examples, but more that they are particularly good at finding examples (and proofs) in a certain way. Looking at the above techniques, one might imagine that they would be very well suited to checking off-the-shelf examples, finding just-do-it proofs (since that is a rather standard method with lots of instances in their training data), using the probabilistic method (unless, as often happens, significant new ideas are needed to show that the probabilities work out), and picking generic examples. The other three methods described above — use of metavariables, trying to prove the opposite, and using successive approximation — require more of an ability to judge whether the approach one is taking is likely to be fruitful. Here it seems at least possible that humans will sometimes have an advantage, but the conditions that a problem would need to satisfy are quite stringent. One would need an example to be one that lies at a leaf of a very large search tree — too large to be searched for by a combination of moderate mathematical ability and brute force — but that can be found by a mathematician with a sufficiently good nose for when they are making progress that they can prune the search tree very substantially.

Why wouldn’t LLMs also have that “nose”? I don’t rule out that “nose” is an emergent property of the way LLMs are trained, and that within a year or two they will have it to the same extent that we have it. But for now, in my interactions with ChatGPT, I do have a distinct impression that they haven’t got there quite yet. When I discuss an open problem with 5.6 Pro, I am often presented with approaches that sound promising until I think about them carefully, and then seem quite a lot less promising. And they will also often end a response by saying, “I have not managed to answer the question you asked, but have managed to reduce it to the following much narrower and more precise question,” which sounds very promising until it has happened five times without any obvious progress having been made. It isn’t completely obvious how they will get better at this, since their training data will not be full of examples of fruitful and less fruitful directions to pursue when trying to solve problems: all they will typically see is tidied up proofs that hide the thought processes of their discoverers. Of course, human mathematicians also don’t get to learn much about how to do research from the experience of other mathematicians, and yet we somehow manage to pick it up. But the situation is a little different for us, in that a lot of what we learn is by doing rather than emulating.

Another reason it is not obvious that “nose” is a property that emerges naturally when LLMs are scaled up is that if LLMs make heavy use of their broad knowledge and can afford to do a lot more brute-force search than humans can, then they will lack the incentive that humans have to prune the search tree ruthlessly. It could conceivably be that their successes so far are achieved using methods that for a human would be considered extremely inefficient, but that because of their superior speed and knowledge, the combinatorial explosion these methods will lead to has not yet become apparent.

It would be very interesting to try to test this experimentally, but it is also difficult, because if an LLM has what looks like the kind of idea that could only be the result of “deep thought” about a problem, we can never be sure that it has actually carried out that deep thought, as opposed to finding a model argument already in the literature, or in other words exploiting the deep thought of a human mathematician. It would probably be easier (but still not easy) to test it by using models that are less powerful than the latest ones and that have been to some extent shielded from the mathematical literature: one could give them a carefully designed suite of problems and see whether the ones that the LLMs solve have particular characteristics.

It may seem as though I am desperately clinging to the hope that humans will continue to be able to make meaningful contributions to mathematical discovery for a while yet, but while I do indeed hope that, I am not making any assertions of the form “LLMs will never be able to do X”. I think it is likely that they will, and given the pace of progress over the last three years it will probably happen quite soon. But I do think that there may be a hurdle for LLMs to clear and it seems at least possible that it won’t be cleared as straightforwardly as some of the previous hurdles.

In that connection, it would also be interesting to see whether a different reward structure leads to LLMs being able to solve different kinds of problems. For example, if during training an LLM (or machine-learning system of some other kind) is not just rewarded if it ends up with a solution, but also penalized if it explores too many dead ends or if it “cheats” by getting the answer from the literature, perhaps it would be incentivized to go about the research process in a more human way and thereby achieve better results for classes of problems where it is yet to make a big impact.

If the hurdle is cleared, either by pure scaling up or by some more thoughtful method, it will be quite difficult to know when that has happened, since, as just mentioned, an idea that seems very original and surprising may just be lurking somewhere in an LLM’s training data. But I would be confident that it had been cleared if an LLM were to come up with a proof that was as surprising to me as the solution of the cap-set problem was in 2016: the previous best known bounds were completely eclipsed, the method was utterly different from anything I had thought about trying, and afterwards there was a flurry of activity as people came to understand what this wonderful new technique was capable of.

Conclusion

I wasn’t quite sure where I would end up when I started this post, and now that I’ve got to the end, I feel that my main conclusions are not particularly new or surprising, but I hope that the route to them is of some interest. The main points I have made are the following.

“Finding an example” is in practice not the same thing as proving a statement that begins with an existential quantifier.

If it is true that current models are particularly good at finding examples, that is probably not because they have a particular affinity for existential statements, but more because the proof-discovery methods that are appropriate for finding certain kinds of examples play to the obvious strengths of LLMs: wide knowledge and the ability to explore many paths of the search tree that humans would judge to have a low probability of success.

It seems likely that LLMs will carry on improving very quickly. However, if, contrary to expectations (mine at least), there turns out to be some residual class of problems (or other mathematical activities) for which humans continue to have the edge for a while, it is likely that those will be problems for which the mysterious human ability to prune the proof-discovery search tree is particularly advantageous: that is to say, problems where the search tree is deep and has a large amount of branching, so that without rigorous pruning a search is not feasible even for a computer.

A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. They should also be methods that are difficult to stumble on by accident. It is hard to say precisely what would count as such a proof, but I think we’ll recognise it when we see it.

by gowers at August 13, 2026 07:34 AM

July 29, 2026

Secret Blogging Seminar — An experiment with AI-assisted writing

As in David’s most recent post, there’s been a lot in the news about finding proofs and counterexamples with AI. Last weekend, I decided to try an experiment with writing using AI. I learned a lot, and wanted to quickly discuss the experiment and my thoughts on it here. Lots of people are certainly already doing this, but I haven’t seen many people talking about it.

The starting point is that Victor Ostrik and I started a project back in 2017, generalizing a result of Kuperberg about quantum G2, from generic q to q a root of unity. Namely, we showed that for q a root of unity outside of a specific finite list, the Karoubi completion of the G2 spider category is equivalent to the category of tilting modules of the Lusztig form of the quantum group G2. At some point during those 9 years, we did a little bit of writing, and at some point I gave a talk on it, but otherwise we did very little writing. This was not for mathematical reasons, but rather for executive function reasons on my end, the global pandemic, and both of us becoming directors of graduate study. This suggested an interesting challenge: could I use LLMs (specifically ChatGPT 5.6 Sol work mode mostly at “very high” intensity, via IU’s “Edu” subscription) to write this paper that was essentially mathematically complete, but almost entirely unwritten, and how quickly could this be done. To some extent this was a free experiment, because realistically I don’t think we’d have ever finished the paper at this point, and so it’s not replacing a bespoke paper that could have existed.

After spending a decent chunk of the time from Saturday until now on it, I now have a draft that I’m pretty happy with. I want to emphasize although mathematically this is Victor and my joint work, and although Victor has allowed me to make this post, he has not signed off on the accuracy and all errors at this point should be blamed entirely on me. Also my work is supported under NSF DMS grant 2000093 and Simons Foundation grant MPS-TSM-00007608.

Ok, here’s what I did:

First, I asked if Sol could one-shot the main theorem. The answer was yes, though for a somewhat simple reason: Bodish-Wu write “It is possible to adapt the approach from [1], which itself is based on [7], to prove that the Karoubi envelope of [the G2 web category] is equivalent to the category of tilting modules as long as $[2], [3] \neq 0$.” That is to say, Elijah already proved the same result for C2, and a similar argument will work for G2. So the robot supplied the similar argument. I asked it to write that argument up, and then to check it over for good references and to read it like a referee would and make edits. This took around 30 minutes. Here’s the resulting file.

Second, I uploaded my talk slides (and the tiny file already written, which was mostly useless), and asked Sol to give a proof of the main results following the slides. Again I asked it to edit it. This took around 30 minutes. Here’s the resulting file.

Then I looked at the files. As mathematical exposition, I consider both to be garbage.

Then I spent several days giving feedback attempting to improve the second file based on my talk. At no point did I edit the source directly. Most of this was in what I would call the style of a (low executive function, see above) PhD advisor. That is, I would kinda skim the file, get annoyed about something, and tell it to fix it. While it was fixing the paper, I would skim some more to try to find something else that annoyed me. This was a long process! It took three days, nearly 100 prompts, 10-15 hours of reasoning, plus another 10-15 hours of non-reasoning computer time. This used nearly an entire week of my generous budget, and Sol estimates that this would cost around $100 (within a factor of 2) at metered rates. Eventually I got to a version of the paper that I’m pretty happy with. Here’s the resulting file.

I thought I’d distill some thoughts and some questions from the process, I’m of course very curious for your thoughts on the matter.

Comments:

This was much faster than I could have written the paper myself, though slower than I thought it would be. I think the final product is comparable in quality to a typical math paper of mine. On the other hand, I think that compared to my fastest writing collaborators it was not orders of magnitude faster, and the quality is not close to the output of the best mathematical expositors. AI at this point is much worse at writing paper than finding counterexamples to conjectures.

In this case, I was not very worried about errors, because I already had thought through the whole argument and was highly confident that it would work (modulo getting the exactly correct list of exceptions). Nonetheless, I felt like Sol did not make errors more frequently (or of a worse character) than I would expect of myself or a collaborator. Most errors were stuff like “Oh, forgot to check whether this theorem actually works at all roots of unity.” This is typical of my experience with 5.6, which is dramatically better at doing math accurately than previous ChatGPT models.

In this case the vast majority of the ideas were already present from Victor and my work. In particular, the goal was not just to write a proof, but to write our specific proof. Nonetheless, I do think the model contributed mathematically in one key way: in my original sketch I always worked over each q individually, and the model preferred to work integrally, and this resulted in some very nice simplifications in Section 4.1. If and when we turn this into a real preprint, I will include a brief discussion of the intellectual contribution from the model.

I was surprised when I printed out and read a near-final draft, that this feels to me like a paper I wrote. That is the voice is not different enough from what I would write with a human collaborator to feel like it’s not in large part mine.

The experience is disconcertingly similar to advising a PhD student on a paper. That said, a PhD student would need less handholding on their second paper, but an LLM won’t really learn.

I was surprised about how important “prompt engineering” remains, and I think that if I were to write another paper this way I would be able to write it faster and better. The key points are that the model is lazy and easily distracted (both properties I find highly relatable!). It’s lazy in the sense that if you ask it to do a lot of work all at once it will take shortcuts and not do a good job. At one point I had to be like “no, go look at exactly how I made TikZ diagrams, now make all your diagrams actually good like that.” It’s easily distractible in that if you’re not clear about the scope of your question and the document is long, it will start spending crazy amounts of time doing who knows what. Like it wrote the whole first draft in 20 minutes, but then when the paper was 50 pages long, I asked it to switch the order of two paragraphs and it took an hour. Make clear requests and not too many requests at once. Form a plan first and then implement the plan. Be specific about whether it should be editing the document, and if so in which sections. For simple tasks, medium intensity is better than very high.

Starting again from sketch, I’d try to follow Terry Tao’s advice for writing and start with an outline and gradually flesh it out, rather than trying to start with a one-shot paper and then editing.

Questions:

To what extent is this final paper adding any value to the original talk? Especially considering that readers themselves could use an AI model to flesh out points in the talk that they didn’t understand? Maybe we should just be focusing on talk-length digests and formal checking, rather than traditional papers?

What should we do with this paper? I don’t want to make someone hand-referee it, because it doesn’t seem fair when it wasn’t hand-written. Probably we will put it on the arxiv once we’ve human-checked it fully and Victor has signed off on it, so that other people can use the results if they need to.

Given the speed-up, when does it still make sense for me to write papers by hand? (Relevant here that I’m a very slow writer and don’t really enjoy it, the way I enjoy say preparing and giving a talk.)

What does this mean for PhD advising? Many PhD students need a similar amount of guidance to what I gave the model in this project. But you can now remove the student from the loop (either intentionally, with the advisor just writing using LLM assistance rather than having students, or unintentionally, with the student just feeding all the suggestions to an LLM and reporting back to the advisor).

Have any of you done better with AI-assisted paper writing? My points 6 and 7 above sounds like something where someone is going to say “blah, blah, scaffolding, blah, blah, multi-agent…”

What a strange world to live in…

by Noah Snyder at July 29, 2026 12:09 PM

July 26, 2026

Tim Gowers — Thoughts about the Leiden Declaration

Last September I went to a workshop at the Lorentz Centre in Leiden to discuss mathematics and AI with historians, philosophers, computer scientists, AI researchers, and mathematicians of several different flavours (though there was a surprising preponderance of algebraic geometers). The whole event was extremely stimulating, with some talks but also a lot of time set aside for discussion. One of the concrete outcomes of the workshop was the Leiden Declaration, which has now been signed by over 3000 people. Given that I was part of the workshop, it might seem a bit strange that I am not one of the signatories of the resulting declaration. The reason is not so much that I disagree with it in any concrete way, but more that in several places it makes confident assertions and recommendations that I feel somewhat uncertain about. So instead I prefer to try to articulate my views about the issues raised by the declaration and put them in this blog post. Before I do that, I would like to make clear that I am very glad that the Leiden Declaration exists and I think that it has done a lot of good in focusing people’s minds on the issues that AI is forcing the mathematical community to grapple with, which are more acute now than they were last September.

Let me begin by quoting a passage from the declaration that sets out “what we take to be characteristic values of mathematical research that we have a joint interest in preserving”.

There are many reasons to pursue mathematical research, ranging from intellectual curiosity to a desire to solve practical and societal problems. Underlying much of mathematics is the activity of proof. Mathematical proofs are regarded as conferring the highest degree of certainty to their conclusions, as well as imparting understanding of why their conclusions are true. These characteristics of proof support the scientific integrity of mathematics.

Results are attributable to specific authors who take credit for their discovery and assume responsibility for their correctness. These principles ground the merit-based standards to which we aspire in mathematical research.

Mathematical arguments are regarded as transparent and subject to independent verification. They may be extremely long or difficult, but in principle no proprietary knowledge or equipment should be required to understand them.

Mathematicians share a concern for proper evaluation of mathematical work relative to shared standards of depth, difficulty, and significance.

Mathematics produces not only a body of results, but also understanding, clarity, and judgment among the communities of mathematicians who have shaped them, often in the context of their own autonomously guided research. This expert knowledge is essential, both to effectively use mathematics, and to continue to articulate new and significant research questions. A key source of strength of the discipline has long been the autonomous shaping of the direction of research and the methods used to pursue it.

The first thing I would say about these values is that they are undoubtedly values that are widely held by mathematicians, including, with some qualifications, me. The main qualification I have concerns point 4: I find the notion of “proper evaluation” somewhat problematic, given that different mathematicians can have very different judgments without either of them being clearly wrong, especially when it comes to the significance of a piece of mathematics. Also, these judgments are used for purposes such as the acceptance of papers in journals, hiring and promotion decisions, the awarding of prizes, and so on, that are part of a system that copiously rewards a few people — I myself have hugely benefited from it — but doesn’t necessarily adequately reward a lot of people who are doing less visible work that is essential to keeping the whole enterprise going.

But the more important point is whether these values are ones that we should fight for in the future, as the Leiden Declaration suggests. I find that clearer for some of them than others. For example, it seems to me that the importance of rigorous proof will be even greater in an AI age than it was before — if the output of AI is not underpinned by rigorous proof, then the kinds of difficulties one already hears about with certain areas of human mathematics (see for example many talks by Kevin Buzzard arguing for the value of formalization) would be hugely magnified. But what about the attribution of results to specific authors, who take both credit and responsibility for them? Suppose that at some point in the future AI becomes more autonomous, reading the literature and solving many problems that it finds. Suppose also that its solutions are autoformalized, so there is no serious doubt about their correctness. In such a situation, there would be nothing for a human to take credit for or responsibility for. Does that mean that we should declare such results undesirable and threatening to mathematical values?

Of course, something could well be missing in such a situation: perhaps the proofs would be badly written and hard to follow, which would mean that they lacked something we all very much value. So let me extend the thought experiment slightly. What if by that stage one could take one of these outputs and ask an LLM to explain the ideas, and what if LLMs did a very good job at that? That is not particularly hypothetical, since they are often pretty good at this job already, but I am imagining a world in which they are much better than they are now, as they will presumably become.

So now we would have a world in which a lot of problems had been solved, we were sure that the solutions were correct, and we had an LLM ready to explain those solutions in as much or as little detail as we wanted. Is that a future we should resist, and if so, why?

One obvious reason is that it would take a huge part of the fun out of the subject. It is extremely satisfying to struggle with a mathematical problem for months or even years and eventually solve it. But I worry about that argument, because it seems to be saying that we should resist doing mathematics the easy way because a tiny fraction of the world’s population gets huge pleasure from taking orders of magnitude longer to do it. That is not to say that I wouldn’t be sad that a way of life that has sustained me for the last forty years was not available any more — of course I would. I just find it hard to use it as a reason to argue that we should try to preserve the “ownership structure” of mathematical results. If we arrive at a world where mathematical theorems are no longer associated with mathematicians, maybe that won’t be any more problematic than the fact that stars aren’t named after astronomers and most aren’t named at all. I’m not necessarily in a hurry for that world to exist, but maybe once the transition had happened, people would be OK with it.

The third value I share in an uncomplicated way, and I have already discussed the fourth. The fifth value is one that I hold very strongly, though I’m not so keen on the idea of experts consciously “shaping the direction of research”, something that I see as happening more organically. Obviously there are some notable examples of mathematicians who have created wonderful programmes of research, but even there I would like to credit other mathematicians with understanding what is wonderful about those programmes and contributing to them enthusiastically as a result, rather than being told what direction to pursue and meekly doing so (which is probably not what the declaration is actually trying to suggest, but it has a slight flavour of that for me).

But that’s a minor quibble when set against my main worry about the effect of AI on mathematics, which is the possible destruction of mathematical culture. There is at the moment an extraordinary body of knowledge and expertise that exists not just in the mathematical literature but in the heads of mathematicians all round the world. Imagine if AI didn’t exist and a pandemic broke out that for some reason wiped out all mathematicians and nobody else. All the literature would still be there, but nobody would have the faintest idea what to do with it. To revive a mathematical tradition under those circumstances would be extremely difficult and take decades. Now imagine a slight variant of that, where AI does exist and because of it people are no longer motivated to put in the years of effort it takes to reach the level of expertise that a typical research mathematician has now. After a decade or two, we might arrive at a situation where the mathematical literature has, in some form, been vastly expanded, but there is no corresponding community of human experts who have a shared understanding of parts of it. Almost all of mathematics would be like the areas that we have more or less forgotten about today, areas that exist in papers written many decades ago that nobody reads any more. (I won’t name any such area because I don’t want accidentally to suggest an area that many people still love and work on.)

This, it seems to me, is a possibility that we should try very hard to resist, but I agree with many other commentators who say that in order to resist it, we will need to give less priority to some of our current values — and I would include ownership of mathematical results in that list — and more to others. For example, if Person A gets an LLM to one-shot a solution of an important open problem (which is formalized, possibly automatically, so there is no doubt about its correctness) but Person B makes the effort to digest the solution and explain it in a way that other mathematicians can understand and learn from, then I think we will want Person B to get the lion’s share of the credit. The credit would be of a slightly different from what it is now, which could be described as admiration for somebody’s talent, insight, speed (I mean here the purely factual statement that speed is often admired — I would prefer that to be less the case) and hard work. It would be more like the gratitude that one feels already for somebody who writes a beautiful textbook that makes a whole area of mathematics coherent and accessible.

Maybe that is what the “research mathematicians” of the future should do: make a selection from a vast sea of AI-generated mathematics and write a book about it in such a way that other mathematicians can read the book and feel the kind of enrichment that we feel when we get to grips with an area of mathematics.

At this point I have to admit that there’s a pessimistic side of me that asks the following general question whenever anyone says anything about what the role for humans might be in the future: why do you think that AI wouldn’t be able to do it? For example, with the suggestion I’ve just made, what reason is there to suppose that ChatGPT 8.2 wouldn’t be able to have a short interaction with you about your mathematical tastes and background and then write the ideal textbook just for you? Humans are likely to be better at this kind of curating for a little while yet, but is it a fundamentally human ability that AI could never hope to emulate?

In a world where AI wrote bespoke textbooks (or more likely, just taught people in some more direct way), something would be lost that feels important: mathematics as a collective endeavour. If we all just learnt cool bits of maths for our own private satisfaction, we would miss the considerable pleasure that comes from discussing mathematics with others, though even that could in principle be restored by a benign LLM that deliberately taught many people the same cool bits of the subject, though an LLM that could do that sort of social engineering would raise all sorts of safety issues.

Let me now turn to the section of the declaration about potential threats. I’ll put my comments on each one in square brackets.

Current automated techniques can produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs. This applies not only to informal arguments, but also to formalizations, where the difficulty lies in the translation between computer-encoded and human presentations of concepts. These fast-moving developments put our present system of review under increasing pressure, jeopardizing our ability to implement traditional standards for the correctness, transparency, and independent verifiability of proof. [This feels like less of a problem now than it did last September, partly because the best LLMs hallucinate a lot less than before, and partly because autoformalization is improving all the time — I have just used harmonic.fun’s Aristotle system to formalize a complicated paper in Lean and I didn’t need to know any Lean to do it.]

Technologies that draw extensively on the published mathematical commons undermine the traditional system of attribution. Models trained on published works frequently return outputs that do not properly cite the human works they synthesize. Many current models are also built on data obtained by systematically exploiting licenses and access arrangements that were not made with artificial intelligence in mind, or indeed by simply violating copyright protections. [This is a problem at the moment, when ownership of results is important, and I am very much in favour of people making an effort to give appropriate credit for mathematical ideas that AI may have used. However, in the longer term, as I have already discussed, I think this ownership structure will break down and the issue will become less important. It also seems possible that LLMs will become better at revealing their sources.]

Technologies which affect the way in which mathematics is practiced may disturb the current system of incentives. The use of artificial intelligence — and thus also the sort of problems which it can address — may become incentivized for its own sake, disrupting our mechanisms for hiring, funding, and recognition. This disadvantages researchers who do not have access to the technologies or decision-making related to them, or who are unwilling to use technologies controlled by organizations whose values they do not share. [These seem to me to be genuine problems. I think there is simply no point in hoping that our current system of incentives will not be disturbed — it obviously will. I am not necessarily too worried if our mechanisms for hiring, funding and recognition are disrupted, as I don’t find those mechanisms unproblematic as they are, but disadvantaging researchers who do not have access to good LLMs is something I certainly think we should worry about.]

Proper evaluation is endangered if results are communicated through informal channels such as press releases or blog posts, often without any research paper or other disclosure of information necessary for scientific evaluation. This practice seeks publicity for new results on market timelines before the accepted processes of community evaluation in mathematics can take place. In many cases this leads to simplifications in reporting, such as overemphasizing the significance of automated tools and undervaluing the prior human contributions which have made those tools possible. Such oversimplification risks influencing public opinion in a way that not only damages perceptions of mathematics, but also misleadingly uses specific mathematical tasks as metrics for the general reasoning capacities of commercial products. [I think this can be a problem, but I think it is not as serious a problem as some of the others, since when results get overhyped, there seems to be no shortage of people publicly (and rightly) pointing that out.]

These developments put the autonomy of mathematics under threat. The increasing involvement of technology companies in mathematical research raises the risk that research questions may come to be prioritized because of their amenability to automated mathematics, rather than expert judgment of their deeper significance. Indeed, broader understanding of the field may be permanently lost in the process of automation. With university budgets under pressure, this reshaping also changes professional incentives in a manner which encourages the collaboration of researchers with technology companies on asymmetric terms. If left unchecked, these trends go beyond threatening researchers’ autonomy, affecting the scope and depth of mathematical research itself. [I think this could be a problem, but it also seems to me that mathematicians have a lot of power here. For instance, if a technology company were to produce a lot of research that mathematicians did not find all that interesting or important, I don’t think they would be able to use their financial and other resources to persuade us to change our minds. Rather, what seems to happen is that mathematicians say, “Yes that does X but it doesn’t do Y,” and the tech companies then feel challenged to do Y.]

There follow eleven recommendations for individual mathematicians. I agree with almost all of them. The one that I’m not so sure about, for reasons I’ve basically already gone into, is this.

Affirm the humanity of authorship. Credit and responsibility continue to belong to humans within the mathematical community and should not be given to automated systems. Artificial intelligence may obscure, but does not replace, the collective human labor behind a result.

I’m not sure what that really means. For example, should we affirm the humanity of authorship in the case of the solution to the unit-distance problem? Some humans did a wonderful job of explaining the proof that OpenAI’s model came up with, and the model made use of some highly non-trivial mathematics produced by humans, but the solution itself has not been credited to any human, and nor should it be in my view.

Under recommendations for mathematical organizations and not-for-profit research funders I again agree with several of them but have my doubts about some. An interesting case is the following.

Protect the rights of authors. Automated mathematics presents new challenges to the rights of authors, and societies should be proactive in the development of sample licensing agreements to protect these rights. In particular, material should not be used as training data without consent, and publishing agreements should allow authors to opt-out [sic] of the use of their work in this way.

This recommendation seems to belong to a world in which journal articles are the main means of dissemination of mathematics. But that has long since ceased to be the case: almost all dissemination now takes place via arXiv preprints, with journals limited to providing a little extra mark of prestige. Once an article is on arXiv, it is on the internet and one can hardly ask for it not to be used as training data. So this recommendation, if it applies at all, will apply to a tiny fraction of articles that are published without first appearing on arXiv. More generally, what right of an author is being compromised when an article is used as training data? We don’t object if human mathematicians use our articles to help train themselves to become better mathematicians — indeed, we will typically be delighted that somebody else thought our articles worthy of their attention. So the objection to a machine doing the same would have to be that for some reason one did not want machines to get better at mathematics in a similar way. I can imagine grounds for such a wish: perhaps somebody is worried about the threat that LLMs pose to traditional mathematical practice, or perhaps they worry that mathematical ability of LLMs will transfer to much more dangerous reasoning ability. But there’s a more complicated discussion to be had here than one might think from reading the recommendation.

The next recommendation is this.

Insist on appropriate publication outlets. Demand that mathematical results continue to be published in peer-reviewed venues such as journals, proceedings, and books. Informal mechanisms such as press releases or blog posts can provide a valuable supporting role, but they cannot replace peer-review or community scrutiny.

For reasons that I’ve gone into many times, I am not too fond of the current publication system, so I can’t get behind this recommendation. Indeed, if the current system becomes unsustainable because of a flood of AI-generated and AI-aided content, I would regard that as a beneficial consequence of AI. However, that doesn’t mean that I would advocate a total free-for-all. I’ve already said that one of my worries is that if mathematical content is not sufficiently organized, then the traditions that we all value could die. I just think that what we will want to do to preserve those traditions is likely to be a lot more innovative than clinging on to the peer-reviewed journal system.

I have highlighted in this post the parts of the declaration that I have doubts about, either because I disagree with them or, more typically, because I sort of half agree with them but want to add many qualifications. That may make the post come across as rather negative, but that is not my intention. The parts I disagree with are in the minority, and I think it is important that a declaration such as this should be made. I should also make clear that my views are evolving all the time, largely because the speed of progress of LLMs has taken me by surprise, but also as a result of conversations I have had or opinions that other mathematicians have expressed online.

I’ll end with two further clarifications. The first is that it may seem as though I am taking it for granted that LLMs will soon be better than humans at all aspects of mathematical problem solving, and maybe also problem posing, theory building, formulation of definitions, etc. I do think all that will happen at some point, but whereas some people say that it will obviously happen within the next two to three years, I would say that it might happen as soon as that, but I don’t rule out that we’ll get lucky and find that we can do interesting AI-assisted maths for quite a bit longer than that before AI doesn’t need us any more.

The second is that I think I have acquired a reputation as somebody who celebrates what is going on. But if, for example, I post on Twitter saying that such-and-such an AI solution is a remarkable development, the word “remarkable” is meant to indicate no more nor less than that I found it very surprising. My feelings about the possibility of AI solving all sorts of problems that interest me are much more mixed. I’ve had the experience twice now of seeing GPT 5.6 Pro one-shot a solution to a problem that I very much liked and had thought about hard (in both cases with much younger collaborators, who, with my approval, were the ones who prompted the LLM). It felt very strange and not particularly pleasant to have the rug pulled out from under my feet like that. On the other hand, I was quite pleased to see the problems solved. It’s actually a similar feeling to the one I have had many times when a problem I am fond of and have thought about gets solved by another human mathematician.

Another factor for me is that I have invested a lot of thought into automatic theorem proving of a more traditional kind. One of my main motivations for that was the hope that the work I put into it would extend the state of the art, measured by which problems a computer can solve. That ship has sailed now, and that saddens me. I still think that there is value in the work that I and my group are doing, but it has become a tougher sell.

So I personally have already found AI quite disruptive, and this is just the beginning. I would have preferred the developments to happen at a slower pace. But I don’t see any practical way to slow them down, so the best we can do is probably to face up to the changes that are being thrust upon us and do what we can to maximize the benefits and minimize the damage. The Leiden Declaration may not be perfect, but it makes an important and positive contribution to that effort.

by gowers at July 26, 2026 10:44 PM

July 25, 2026

Clifford Johnson — On top of the Mountain again

Just in case you’re up for a short talk at the top of Mount Wilson followed by an evening of observing through the historic telescopes on Saturday 25th July… this might be for you! Go to Mount Wilson Observatory’s website for more. –cvj

The post On top of the Mountain again appeared first on Asymptotia.

by Clifford at July 25, 2026 05:15 AM

July 24, 2026

Peter Rohde Introducing Sigfried’s Blog

My new secondary blog featuring conversations with AI, inventing new things, exploring hypotheticals, letting creativity flow freely.

Some highlights:

Satellite constellations with topologically distributed apertures.

A clockless architecture for classical topological computing.

Post-quantum cryptography using the $\mathbb{Z}_2^n \rtimes S_n$ algebra.

Efficient homomorphic computing using reversible classical circuits.

A silent speech interface using microwave Doppler imaging.

Cognitive search acceleration.

Consensual thought guidance.

Subliminal audio modulation & human guidance systems.

Microwave imaging using WiFi and 5G for medical applications.

Thought tomography.

The quantum bluff hypothesis.

https://sigfriedschattenjaeger.wordpress.com

by Peter Rohde at July 24, 2026 07:40 AM

Faces of the E₆ root polytope, or 1₂₂
dim	faces	count
5	5-demicubes	54 = 27 + 27
4	4-demicubes = 4-orthoplexes	270
4	4-simplexes	432 = 216 + 216
3	3-simplexes = 3-demicubes = tetrahedra	2160 = 1080 + 1080
2	2-simplexes = triangles	2160
1	1-simplexes = edges	720
0	0-simplexes = vertices	72

Faces of the E₇ root polytope, or 2₃₁
dim	faces	count
6	2₂₁ polytopes	56
6	6-simplexes	576
5	5-orthoplexes	756
5	5-simplexes	4032
4	4-simplexes	16128 = 4032 + 12096
3	3-simplexes = tetrahedra	20160
2	2-simplexes = triangles	10080
1	1-simplexes = edges	2016
0	0-simplexes = vertices	126

Faces of the E₈ root polytope, or 4₂₁
dim	faces	count
7	7-orthoplexes	2160
7	7-simplexes	17280
6	6-simplexes	207360 = 138240 + 69120
5	5-simplexes	483840
4	4-simplexes	483840
3	3-simplexes = tetrahedra	241920
2	2-simplexes = triangles	60480
1	1-simplexes = edges	6720
0	0-simplexes = vertices	240

Depth (km)	Minerals
0–410	olivine (α) + pyroxenes + garnet
410	→ wadsleyite (β)
520	→ ringwoodite (γ)
660	→ bridgmanite + ferropericlase + davemaoite
660–2700	bridgmanite dominates
~2700	→ post-perovskite
2900	→ liquid iron core

Mineral	Depth (km)	Fraction of Earth’s Mass
Wadsleyite	410–520	~1.8%
Ringwoodite	520–660	~2.3%
Bridgmanite	660–2,700	~38–39%
All three combined	410–2,700	~42%

Planet Musings

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October 15, 2026

Terence Tao — What mathematicians should know about the Lean Theorem Prover: questions of reliability and AI

Terence Tao — Simons Foundation Launches $15M Effort To Help Mathematicians Navigate AI Era

Warmup

Dehn Twists

Argyres-Seiberg

Proofs and Prompts — The Navier-Stokes Announcement: A case study in mathematical values

On values

On what was lost: Grief

On what was gained: revulsion

To the future, guided by values

Postscript

Acknowledgements

Proofs and Prompts — Proof and Resolution: Two Different Forms of Establishment

Abstract

1. Introduction: The Crisis of Unreadable Truth

2. Systems Never Operate on the Whole World

3. Observer, Requirement, and Granularity

4. The Primacy of Resolution in Engineering

5. Changing the Paradigm of Mathematics Practice and Teaching

6. Black Boxes and Trust Points

7. The 8-Step Resolution Protocol

8. Conclusion: The Mathematician’s Changing Role

October 14, 2026

Terence Tao — Why are we even doing all this?

Terence Tao — First Proof, Third Batch webinar

Proofs and Prompts — A Letter from a life after Math

Proofs and Prompts — Mathematicians, AI, and Historical Archives

Proofs and Prompts — The Dialectical Theorem

Stage 1

Stage 2

Stage 3

A logical map of the conversation

References

Terence Tao — Call for stories: human–AI collaboration in recent mathematics

John Preskill — The Quantum Omelette Problem: How Much Energy Belongs to the Yolk?

Doug Natelson — Materials-related postdoctoral opportunity at Rice, 2026-2027 edition

October 13, 2026

John Baez — Liverworts

John Preskill — Nicole’s guide to writing and editing

Tommaso Dorigo — Is Simulation-Based Inference A New Technique In HEP?

October 10, 2026

Doug Natelson — Science policy/funding updates, early October edition

October 09, 2026

Matt von Hippel — Congratulations to Francis Halzen!

Tommaso Dorigo — The Wondrous Radiacode 110

Doug Natelson — NAS "State of Science" 2026 address

October 08, 2026

John Preskill — In praise of questions

Jordan Ellenberg — When the phone inside your ribcage rings, it’s not for me (They Might Be Giants show report)

October 05, 2026

Jordan Ellenberg — Eigenvalues of random p-adic matrices

Jordan Ellenberg — James Purdy, Narrow Rooms

October 04, 2026

“AGI Ruin: A List of Lethalities” by Eliezer Yudkowsky: Rapporteur Report by Tennyson Bardwell

Doug Natelson — The arXiv - where it's been and where it's going

October 02, 2026

Matt von Hippel — Beginnings Don’t Matter as Much as You’d Think

September 29, 2026

Doug Natelson — Annual Nobel speculation thread

September 27, 2026

John Preskill — Marcus theory and Marcus practice

Tommaso Dorigo — Flying With Gamma-Ray Counters: Another Test Or Two Of The Radiacode Zero

September 25, 2026

Jordan Ellenberg — I love working here, II

Representation theory

Combinatorics

Step 1: pairs of edges in a complete graph

Step 2: a double count

Step 3: division

The Lind–Singmaster family

Matt von Hippel — It Got to My Field

Tommaso Dorigo — Some Tests Of The Radiacode Zero

September 24, 2026

John Baez — Binomial Coefficient Coincidences

The Lind–Singmaster family

September 20, 2026