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.

1. Organization
   1. Motivate your work; then, present it; and then, explain its physical significance.
   2. Begin each paragraph with a topic sentence.
   3. 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.
   4. 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].
   5. 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.
   6. 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.
   7. 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.
2. Word choice
   1. Use strong, specific words, rather than weak words.
      1. Verbs and nouns are stronger than adjectives and adverbs.
      2. 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”).
   2. 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.
   3. 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.
   4. Use the active voice, rather than the passive voice. Take responsibility for your work. Editors of high-impact scientific journals have endorsed this advice.
   5. 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).
   6. 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.
   7. Avoid dangling modifiers.
   8. 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).
   9. Avoid empty subjects.
   10. Include no unnecessary words.
       1. “So-called” is unnecessary.
       2. “Note that” and “We note that” are unnecessary.
       3. 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.
       4. Never write “is equal to”; “equals” is more concise.
       5. Never write “is able to”; “can” is more concise.
       6. Never write “gives an upper bound to” or “places an upper bound on”; “upper-bounds” is more concise. Analogous statements concern lower bounds.
       7. Never write “a large number of”; “many” is more concise. Never write “a small number of”; “few” is more concise.
       8. Never write “We refer to [symbol] as [name]”; “we call [symbol] [name]” is more concise.
       9. Never write “as long as”; “if” is more concise.
       10. Never write “a single”; “one” is more concise.
       11. The symbol means “greater than”; and , “greater than or equal to.” Don’t translate into “strictly greater than”; the “strictly” is unnecessary. Analogous statements concern and .
   11. Avoid contractions, which are too informal for professional writing.
   12. The possessive is not a contraction and belongs in professional writing. It facilitates conciseness.
   13. 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.
   14. 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 . Evolve it under …”).
   15. The condition defines a regime, not a limit. The conditions and define limits and are inequivalent to .
   16. Write “first,” “second,” “last,” etc., not “firstly,” “secondly,” “lastly,” etc. (I defer in this matter to The Elements of Style.)
   17. Humans can assume, suppose, etc. Mathematical expressions, protocols, etc. can’t.
   18. One multiplies factors together and sums terms. Don’t call factors terms and vice versa.
   19. 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.”
   20. Regarding the words “general” and “generally”:
       1. A general object subsumes every example of that object. If any example behaves unlike a supposedly general object, don’t call the object general.
       2. 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.
       3. 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.”
   21. “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 Then Y.”
       • Examples of appropriate alternatives
         • Define This definition implies Y.
         • If then Y.
         • Define such that Y.
   22. 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].”
   23. Regarding tense:
       1. When describing what you’ve accomplished, use only one tense.
       2. Experiments happened in the past, so describe them in the past tense.
       3. When describing a proof’s steps, use the present tense.
          • Example: We Taylor-approximate the function about . Substituting into Eq. (1) yields [equation].
   24. 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.
   25. “Admit of” means “allow for,” or “permit.” The phrase needs the “of.”
       • Example: The formula admits of the following interpretation.
   26. Use a pronoun only if the reader can identify its antecedent without ambiguity.
   27. “Spin” has many meanings: “a quantum property that resembles angular momentum,” “a particle endowed with that property,” “the observable ,” “the spin quantum number ,” “the observable ” “the magnetic spin quantum number ” and “the observable .” Leave no ambiguity about which object you have in mind.
3. Punctuation
   1. 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.
   2. 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”.)
   3. To write quotation marks in LaTeX, don’t use your keyboard’s quotation-mark key; use the appropriate keys.
   4. Regarding hyphens:
      1. The hyphen (-) feeds into punctuation of three types: the hyphen (-), the en dash (–), and the em dash (—).
      2. The hyphen appears in some compound words, as in “non-negative.”
      3. 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.
      4. 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
      5. Hyphenate compound adjectives.
      6. If an adverb ends in “-ly,” it probably shouldn’t precede a hyphen.
         • Example of inappropriate hyphenation: strongly-coupled systems
      7. Follow a prefix with a hyphen if and only if the Physical Review style guide indicates that you should.
   5. A complete clause must follow any semicolon (unless the semicolon separates items in a list).
4. Math
   1. 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.
   2. 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.
   3. 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.
   4. If you define a new function, denote it by only one letter. (I defer in this matter to the Physical Review style guide.)
      • Example:
      • Examples of disallowed notation: ,
   5. 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 denote the measurement outcome.
   6. If a variable or constant appears in a superscript, parenthesize it. The parentheses communicate that the superscript isn’t an exponent.
      • Example: Let denote the Pauli- operator of qubit
      • If a superscript is not italicized (stands for a word or phrase), don’t parenthesize it.
   7. Regarding the definition of a symbol :
      1. If you write alone on one side of a defining equation, use \coloneqq or \eqqcolon: \coloneqq [expression], or [expression] \eqqcolon . The symbols \coloneqq and \eqqcolon relate more information than does , encoding directionality.
      2. Use if does not appear alone on its side of the equation: [function of ] [result of replacing with its definition in the equation’s left-hand side].
   8. Refer to the Cartesian axes using the formatting “[italicized letter]-axis.” Don’t include any hat, boldface, or \vec symbol.
      • Example: -axis
   9. Avoid denoting any index by , which means to physicists. Use instead, unless you’re writing for engineers (who denote by ).
   10. Don’t use the lowercase letter l (“ell”) as an index; readers might mistake it for a one. Use (\ell) instead.
   11. 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.”
   12. 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.
   13. Regarding a set-off mathematical expression that extends across multiple lines:
       1. Format the expression as follows by default.
          1. Put an & symbol immediately leftward of the first = sign or analogous symbol (e.g., ).
          2. 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.”)
          3. Suppose that a subsequent line begins with a , –, , or . Find the symbol immediately rightward of the initial = sign. Begin the new line directly below that symbol.
          • Example:
       2. 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):
       3. Suppose a new line begins with a term or factor, such as the in the example under (A). Put the corresponding , –, , or at the beginning of the new line, not at the end of the previous line.
          • Examples of inappropriate placement:
   14. The symbol means “approximately equals”; and ~, “scales as.” Approximations convey more information than scaling relations do.
   15. Use big-O-type notation or ~ symbols, not both; they’re partially redundant.
   16. \ldots, rather than \cdots, should stand in for elements that fit a pattern.
       • Example:
   17. 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: For example, if is odd, then and fit the template.
   18. Parentheses (), square brackets [], and curly braces {} are delimiters. If you nest them, do so in the order dictated by the Physical Review style guide.
   19. 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.
   20. An operator isn’t a matrix; a matrix represents an operator in terms of a particular basis. Therefore, no equals sign should interrelate an and a matrix. An arrow can.
       • Example:
   21. Every real number is complex. Don’t say “complex” if you mean “nonreal.”
   22. 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
       • Examples of appropriate placements
         • the set of real numbers
         • the set of real numbers and
         • Recall the set of real numbers, , in Lemma 1.
5. More mechanics of writing
   1. Use concise sentences, as advocated for in The Elements of Style. The reader can hold only so many ideas in their head at once.
   2. 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)].
   3. The ideal paper title has the structure of a newspaper headline: it presents a claim, containing a subject and a predicate.
   4. Regarding abbreviations:
      1. Don’t abbreviate the first word in any sentence.
      2. Abbreviate “Figure,” “Section,” “Professor,” and “Appendix” if such a word appears partway through a sentence.
      3. Don’t abbreviate “Sections.”
   5. Regarding acronyms:
      1. Write every acronym in capital letters, as per the Physical Review style guide.
      2. Introduce each acronym the first time you use it.
      3. 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.
   6. Every paragraph should contain at least three sentences.
   7. 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.
   8. Never begin a subsection immediately after a section title. Between the two titles, overview the section. Analogous rules govern subsections and subsubsections.
   9. Put the word “only” in the appropriate place.
      • For example, suppose you’ve sampled data at a point in parameter space and sampled data at no other points. “We sampled data only at ” is correct; “We only sampled data at ” is probably not. The latter claim means that (i) you might have sampled data at and (ii) you did nothing to the data apart from sample it: you didn’t analyze the data, discuss the data, etc.
   10. Specify subjects and objects precisely. Often, a physicist will describe an idea in several lines. In the following sentence, the physicist will attempt to encapsulate the idea in the word “this.” The pronoun is too vague to specify the idea. This what? Follow the pronoun with a noun.
6. 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. 

Many freelancers get paid per hour, or per day. Some have standard rates for typical tasks.

Journalism is rarely typical. And usually, we’re paid per word.

I’ve had many friends whose first thought is how easy that must be to exploit. Often they remember a story about a writer like Charles Dickens or Victor Hugo padding their books with digressions to up their pay. Those stories probably didn’t happen, and these days, that method is even less likely to work. When I pitch a piece, the editor and I agree beforehand on a target word count. The only way I get paid more than expected is if the editor agrees that it makes sense.

That said, there definitely are ways to exploit being paid by the word, if you wanted to. They’re just more subtle.

The source of any such exploit is to notice that writing is not actually the hard part of journalism. Research is time-consuming: the more people you talk to, the longer it takes. Writing, once you’re used to it, can go quite quickly. So the length of a piece itself doesn’t determine how much work it takes to write: research does.

In principle, longer pieces are supposed to take more research. Editors will have rough expectations for the number of sources you’ll interview, so this isn’t something you can skimp on, per se. And publications known for their high standards for research usually pay larger rates per-word.

But no matter how much you’re paid, research can go out of hand in the other direction. Journalism is about finding out the truth, and sometimes the truth is more complicated than expected. Sometimes you need more information, and need to fit in one more interview, then another, or run around chasing after missing info. Then you need to figure out how to fit it all into the same word count you started with.

So suppose you’re a freelancer who aspires to consistently pay their bills. You want to earn a certain amount per hour worked, but some projects can take much more work than others. So what do you do?

You look for the simple. You try to find stories that are unlikely to be complicated, that won’t have clashes of opinion that need to be represented or experts with different sides of the story. If you don’t find such a story, you simplify: you write a piece that makes the story sound cohesive, that papers over doubt, even if you’ve run into doubts yourself while writing it. If you’re consistent about it, you can do what your friends with salaries do, and earn a stable living.

Not the kind of exploit you had in mind, I suspect. And not, despite the sound of it, something you can do reliably. I find I’m always tempted by complexity, always drawn to investigate a story that’s a bit too big for the wordcount it’s supposed to fit.

But it is an ever-present incentive, one I suspect people with a larger roster of leads start to feel. If you’re paid by the word, then the best story tends to be a tidy one.

The IAS has opened its application page for its membership program for the upcoming year. This by itself would be unremarkable — the program does this every year — but there is an additional statement included in the announcement:

We realize that applying to positions in the current environment is stressful. To help alleviate this, the School of Mathematics offers applicants the option to submit research statement based only on already conducted research. It would be helpful if such a statement would be focused on a few select projects (in most cases up to three) and not a wide ranging list.

I would expect other departments and institutes advertising positions this year to adopt similar guidance. (See also a recent report from a working group making similar recommendations in the area of theoretical computer science.)

Readers are also invited to post links to other job openings in mathematics in the comments.

Álvaro Lozano-Robledo, Professor of Mathematics at the University of Connecticut and on Youtube

In reply to Who will become the new mathematicians of Math 2.0? by Anonymous

SERIES · As application deadlines approach, Proofs and Prompts is publishing a series of posts from 19 to 23 October for people considering a PhD, doing a PhD or looking for a postdoc: guidelines, discussions and speculation about what to do and what careers in mathematics will look like. This post is one of them. Each is written independently and reflects only its authors’ views. All posts in the series

This is a response to a recent post on Proofs and Prompts titled “Who will become the new mathematicians of Math 2.0?” written by an Anonymous PhD student (who I will refer to as “Anonymous” below). I would urge particularly all tenured and tenure-track mathematicians to read it because there are many students that think the same way, and are planning to leave academia as a consequence.

I recently wrote a guest piece on Terry Tao’s blog (“What should we tell our students?”) where I warned that we are ‘about to lose an entire generation of mathematicians’ so I feel a duty to reply to Anonymous’ post, to try to offer some counterpoints to the arguments expressed therein. More importantly, I feel a duty to talk to my students about these issues, and I have shared Anonymous’ post with them, so that we can have a discussion. I hope my colleagues in the mathematical community reach out to their students as well.

In order to respond to Anonymous, I will quote some key sentences, and then elaborate with my own thoughts.

I recently heard a mathematician say that it is “paradoxical” to not want to study a field that is undergoing a scientific revolution in reference to October 6th. This mindset completely misses the point of why I do mathematics, and I suspect this is true for many other graduate students and even other established mathematicians.

I want to start here because I believe I may be one of the mathematicians that they are referring to. In a video discussing my post on Tao’s blog, I said

“And I think it would be in fact paradoxical that people would not want to join a field of study during a scientific revolution where we’re seeing this incredible acceleration.”

I will come back to the ‘paradoxical’ remark later on, but I want to first mention what I said right after that quote:

“It is somewhat chaotic. It seems somewhat confusing to be a mathematician right now. But we are trying to figure out together what should the math landscape look like in a year and in five years and how to navigate… how we can use LLMs in ethical ways and in a way that is going to construct solid mathematics along the way and in a way that the mathematical field is going to thrive.”

Let me elaborate on that. Historic upheavals of the status quo are usually chaotic while they happen, and in many cases many of those living through such times have been trampled and mistreated (with such dramatic and relevant examples as the industrial revolution). It is the duty of those in positions of power or security to fight for the rights and well-being of those who may be more likely to suffer through a revolution. In our case, we are living through what can be considered a technological revolution that has great potential to accelerate research and progress and, at the same time, it has a terrifying civilization-ending non-zero potential. Unfortunately, trying to pretend that the revolution is not happening, or trying to preserve the status quo, only delays the inevitable, and prevents communities from fully preparing and reinforcing their core values, so that their professions can survive said revolution.

If we limit our conversation to mathematics and the mathematical community, the LLMs have already revolutionized research mathematics, with the proposed resolution of several landmark conjectures, such as Navier-Stokes or the quasi Riemann hypothesis. This is undeniable and, simply put, we must learn how to coexist with this new technology. But they are also an existential threat to the way mathematics works, and the way the mathematical community operates. So it is absolutely normal and expected to be excited about the research advances, and also extremely concerned about the ways it may change our field, as Anonymous very eloquently points out.

In particular, while the OAI drop of October 6th is certainly of “historic” proportions, it clearly also has had a disruptive effect of historic proportions. Whether the global effect of the LLMs will be a net positive will be determined much later in the future. But it is clear that the “arms race” of frontier model companies to develop ever more capable mathematical models is a corporate publicity stunt and, very early on, even before the Fields medalists warned of a misalignment in an open letter, I asked mathematicians not to be complicit in such an arms race.

At the end of my video about the Oct 6 drop, I encouraged people to stay in the field to make changes from within the community. It is up to us in the profession, and more so up to the newer generations, to lead the way to ethical and sustainable research practices that will allow mathematics to prosper. How to do so is to be decided, so all of Anonymous points are well taken. Whatever we do, we need to make sure students do not abandon mathematics because they think it has become unrecognizable. But the key question is: who is in charge of making these changes and laying out a path forward? I claim that no one is in charge, and we all are. Precisely those that see the field driving towards a cliff, have a great role to play within the community to warn about such dangers! It is a chaotic time, the future is uncertain, and this is disconcerting, but I think there are many reasons to be optimistic and to continue in the mathematical profession.

Now let me backtrack in Anonymous’ post to the very first line:

To the mathematicians shaping the future of our profession,

The mathematical community is not monolithic, and this is a feature, not a bug, of the community. There is no single person, group, or entity that dictates what mathematicians do or how they should operate. In particular, Terry Tao is not a spokesperson for all of mathematics and, of course, there are many out there that do not agree with his views and his predictions for the future of mathematics. Personally, I highly value and respect many of the things he has said, and I disagree with others. I single out Tao here because Anonymous does, but we could similarly agree or disagree on Tsimerman’s or Duminil-Copin’s points… or any other non-Fields medalist point of view, from one of the many recent blog posts on the matter of AI.

Right now, there are many different groups trying to pull mathematics in very different directions. For example, while some mathematicians are completely embracing LLMs, other mathematicians completely refuse to use LLMs in their work. Some of us are choosing the middle ground, and we use LLMs with caution for certain tasks, we avoid them for other more intellectual tasks, while respecting the decisions of some of our colleagues (I include students in this category) who cherish them, or do not wish to use them at all.

When students choose graduate programs, and PhD advisors, many of us encourage them to ask lots of questions (to the staff, to faculty members, to students, to friends) about the atmosphere in the department, and their practices, and about the personalities of the potential advisors. Similarly, I would encourage students who are thinking of a career in mathematics to reach out to their current or prospective programs and advisors, and enquire about their stance on AI/LLM use. Does it match the student’s own attitude towards AI, or is it antithetical to their ideals? Since the community is far from monolithic, there are plenty of departments and potential advisors that will be aligned with whatever views the student holds, and it would be wise to choose programs based on that type of alignment, if the issue of AI is key in the decision making.

I love technical work. I love spending days, weeks, months working on a lemma or an idea without success. […] As AI has now become highly capable of performing this kind of work, I see shockingly little from the mathematics community about how this kind of crisis in education will be addressed.

There was a recent summit on “PhD Math Education in the Age of AI” and the recommendations for graduate programs can be found here. TPSE is putting together a panel discussion on the Future of Graduate Education in Mathematics. Many departments are having internal discussions on how to approach these issues. As I said above, the future is uncertain but we are trying to think of the best possibilities in education as well as in research.

What is the goal of a graduate program in mathematics? How do we define success in a PhD program? In my opinion, the PhD is an apprenticeship, and success is measured by whether the candidate has been trained and is proficient in all aspects of the mathematical profession. At the end of their degree, several mathematicians write letters of recommendation. A (good) rec letter or set of letters evaluates the PhD candidate’s aptitude in all of these aspects: research potential, teaching abilities, service to the community, potential as a colleague, etc. And then colleagues in other institutions read these letters and determine whether the candidate will be a good match for the research and/or teaching needs of their department. Many of the conversations surrounding AI concentrate on the research aspects, but there is a lot more to being a mathematician than just research.

On the subject of research aptitude, will letters in the future refer to the fantastic LLM skills of the candidate? Will we compare students and say such and such student is in the top 2% of LLM-prompters that we have ever met? Will we say wonderful things about their slop-paper output in our letters? Will we praise the many slop-papers that they have been able to de-slopify? I do not think so – we will still need to have a way to measure their research potential based on their own capability of thinking deeply about problems, and coming up with innovative ways to attack interesting problems. It is absolutely true, as Anonymous points out, that students need to spend long hours thinking deeply about a problem, and I do not think LLMs should change that. So PhD advisors should encourage students not to use LLMs for certain tasks such as proof generation of key lemmas, because these are the tasks where intellectual development actually happens (see the Specific Recommendations in the section on Research and AI in the recommendations from the Summit). In a recent conference I was at, someone suggested to grad students to always be working on at least one project that is completely LLM-free.

As a result, I do not think anyone should or will rob the students from their long hours working on a lemma by “pure thought” because most of us recognize the irreplaceable value of the time spent on that activity. It is precisely the outcome of those thinking sessions that sets students apart at the time of evaluating their intellectual and research ability in research letters.

On the other hand, if an advisor demands results with a quick turnaround, and expects that the student will access an LLM for help to be able to move on – against the wishes of the student – then it is time to choose a different advisor. The ideal scenario is that the student will ask the advisor for help or a hint on how to proceed, even if an LLM could provide such ideas, because those conversations are actually also part of the apprenticeship: students learn how to collaborate with other mathematicians through those interactions. Plus the advisor may have a much better intuition than the LLM does anyway.

Many celebrate the possibility that AI will relieve them of the burden of difficult calculations, technical lemmas, and other supposedly tedious aspects of mathematical research.

I do not think that is a common sentiment. For the most part, we all value the time spent in understanding the technical difficulties of key lemmas. On the other hand, LLMs can be fantastic tools in helping us write code and run numerical experiments. I am not great at coding, and I find that the LLMs are allowing me to run computations that I only dreamed possible. It would have been very tedious for me to learn how to write code in Python, html, javascript, etc., but now I am able to create code in all those languages, run experiments, and later verify the results mathematically, all in hours instead of weeks (or instead of never!).

Terence Tao has proposed a vision of a new way of doing mathematics, Math 2.0, where the emphasis moves away from solving open problems and toward a more holistic approach. […] But I feel like I am asking the obvious question that has not been addressed: What if I do not want to spend my professional life doing these activities?

Two points here. One (1), some of us have always had a holistic approach to a PhD: I have always emphasized to my students that I expect them to become well-rounded mathematicians, not just great researchers, and that means that I try to push them not just to write a great thesis, but to be better instructors, better members of the community, better at networking, better colleagues, better at giving talks at all levels, etc. There is nothing wrong with a holistic approach, and in fact it makes candidates more appealing to a broader range of colleges and universities at the time of applying for jobs.

And two (2), many do not agree with Tao’s vision either, and that is ok. In my research, I want to solve problems. The coolest the problem the better. I do want to think about a problem and come up with an ingenious solution, and then tell the world about it. I do not want to inherit a slop-solution to a problem that a company dropped in a repository, and try to make sense of it. I would not want to do that, and I would not expect a student to want to do that either, so I would not ask a student to do that as part of their duties to complete a PhD. I simply do not think we will move in the direction of “moving away from solving problems.”

However, I have thought of problems that an LLM could solve, prompted them, and now I have LLM-prepared drafts on problems I care about. Instead of posting these papers directly to Hexagonmath.org, say, my current strategy is to introduce students to research by explaining the problem I am interested in, explaining why others are also interested in the same problem, discussing the idea that led me to prompt an LLM, and sharing the paper with the student. Then use that paper and that topic as a thread for an independent study on the particular area, so that the student learns enough background to understand the motivation for the paper, and the techniques that are used to solve the problem. Later, we will read together the LLM draft, understand the proof, and rewrite it in a way we understand and can make sense of it. While we think about the proofs, I hope we come up with ways to extend the arguments to strengthen the result, or apply the techniques to other similar problems. And finally polish the paper so that we can submit it to a journal, so that the student has an early experience in writing a research paper, and so that they have published a paper early on to talk about during research conferences. A paper they understand and they can feel is their own, because they have contributed to make it their own.

Will this strategy work? I do not know! I am just trying things out, honestly, and I am listening to any other mathematicians who are trying out different strategies to coexist with the new technology.

I do not expect to follow this early-LLM-draft protocol for a thesis problem, however. It would only be appropriate just as some first steps into research.

There is a problem with the proposed future of mathematics, and it is more concerning than my personal dissatisfaction. How exactly do you expect to train the next generation of mathematicians? You know how to formulate new research directions. Where did these skills come from? I doubt they were developed by reading successful proofs or listening to an explanation of an idea.

As a matter of fact, I do not think intuition and mathematical taste is learned by thinking deeply about a problem for hours on end. I have learned much more attending lectures and conferences, and talking to other mathematicians during lunches and dinners. I have had my best ideas reading other people’s papers and proofs. As an anecdote, early on in my career I signed up to write brief reviews for MathSciNet. I am not sure exactly why, but I was assigned to read and review several articles by Bo-Hae Im (some solo papers, and some co-authored by Im and others), and after reading the third or fourth paper by Im dealing with a conjecture of Larsen, I had an idea of how to prove a related result. I wrote to Im and we ended up collaborating on a research article that eventually was published by the Journal of the London Mathematical Society (and it had a significant impact in getting tenure).

As I mentioned above, we should make an explicit effort to allow students to think deeply about problems, but we should also make it clear that students should be reading papers and proofs. And whether we like it or not, some of the proofs dropped by OAI contain some interesting arguments that we should absolutely study and think about! A particularly valuable example is the proof of the non-existence of Siegel zeros, which several mathematicians (Banks, Kerr, Green, Barreto) have improved on, and now it seems the papers have converged to a remarkably simple argument that many of us would highly benefit from understanding.

Back to the question on how to train the next generation of mathematicians… I am not sure we have the answers yet, but I do not think it will look dramatically different from what we do now. There is a new tool in town, yes, but the guiding principles should be the same: train students to be well-rounded mathematicians.

If the future consists of interpreting AI generated results, organizing knowledge, and directing automated research, why should I remain in academia?

The way this question is phrased, it assumes that the worst, most dystopian possible scenario will come to pass, where AI can solve any single problem humans could solve, and that we would reach a point where only machines do math research. As I mentioned in my piece in Tao’s blog, the future of LLM capabilities is completely unclear. There are many ways in which LLMs could hit a limit in mathematics, for example from the theoretical or feasibility points of view. The frontier model companies are driving the narrative and trying to convince us all that their products are already super human when this is far from the truth. I wish mathematicians had a multi-million PR team behind them to build praise and support for our fantastic research record!

The reality is that many of us have started using LLMs to aid our research, as a tool or as a collaborator, and it turns out that it can be very exciting and fulfilling to do so. As one of the best recent examples of LLM use I know of, one of my colleagues (Kyu-Hwan Lee) joined a team of mathematicians to prove that the sporadic simple group M_23 is Galois over Q. Their team succeeded with the help of LLMs in a task that many had tried and failed, and made some really interesting progress in the inverse Galois problem. Here is an excerpt from their declaration of AI use:

“No text in this article was written by AI. Our work relies on longstanding domain knowledge and techniques for computing curves of positive genus [KMSV14]. The key strategic decisions and the conceptual analysis for this project were made by the human team.

AI served as a tool that accelerated a large part of our work, particularly the computations. We used Claude Fable 5 and 5.1, Claude Opus 4.8, and ChatGPT 5.6 Sol for searching the literature, generating code, checking results, and proofreading. The workflow that eventually succeeded was a tightly controlled collaboration in which the computational work was much faster with AI, while the human collaborators continued to direct the research and verify the results. AI systems increased our confidence in tackling difficult problems and helped us communicate more efficiently across our different areas of expertise.”

I find this result fantastic (the paper is a gem by the way), and I think it exactly illustrates the way senior mathematicians (Lee, Poonen, Pries) can use LLMs to accelerate research and understanding, and include graduate students (Shaowu Zhang) and postdocs (Xiaoyu Huang) in the process!

Mathematics is never going to be the same and I recognize that the way I did research and found fulfillment from it are no longer here.

Mathematics will be different, yes, but I firmly believe it will be similar to what we do now, in all the best ways. As a final example, very recently three mathematicians (Elkies, Klagsbrun, and Li) have found an elliptic curve of rank 32. I find this discovery astonishing and wildly exciting. In a private communication, Zev Klagsbrun mentioned to me

“It would be a mistake to view this as a triumph of humans over AI. In fact, it’s a great example of humans using AI to push the boundary of what is known.”

I could not agree more. Immediately after their curve was found, my student Dylan Costa and I started working on finding a quadratic twist of their curve with rank bigger or equal to 3 (in order to find a quadratic field over which their curve achieves rank at least 35), and with the help of LLMs (in coding and literature search) we succeeded to find one, conditional to BSD/Parity. Both Dylan and I found the work on this little contribution (certainly little compared to the giant discovery of the rank 32 curve) to be exhilarating. It would not have been possible just a few months ago without the help of LLMs. This is why I would find it paradoxical to leave mathematics now.

I am not asking you to agree with my personal preferences.

I just want to conclude by thanking Anonymous for writing such a great piece that should make everyone in math think long and hard about the future of the profession. Thank you. And a final message just to Anonymous: I would love to record an interview with you and post it in my channel. Or send me an email message if you want to chat. Bye and best of luck!

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Received 22 October 2026.

Marcus Zibrowius, Professor at Heinrich Heine University Düsseldorf

AI models are making mathematical progress at an unimaginable scale. There’s now a reasonable expectation that soon, we will be able to prove a theorem, or find a counterexample, at the touch of a button.

But the overall reaction of our mathematical community is closer to panic than it is to enthusiasm. AI is perceived as a threat, not just to humanity as a whole (which is a different story) but to mathematics specifically. We are collectively appalled by formally-verified but fully inexplicable proofs, and even more disgusted by companies that churn out one AI-generated (and equally verified) paper after the other without even having a single person on the team able to explain them.

We talk about the good old days when mathematics was about building a community of common understanding, facilitating personal growth, celebrating beauty.

What a feel-good fairy-tale that is!

In fact, our values have always been about who got there first. Collectively speaking, the one thing we, as a mathematical community, have always valued most has been a proof of a hitherto unproven theorem. It was not a nicer proof of a known result, and certainly not a beautiful exposition of a known proof.

You think I’m being cynical? Allow me to argue my point.

Follow the money.

Explaining a proof may have earned you an applause or a compliment. Completing a proof $\text{\textendash}$ and doing so faster than anyone else $\text{\textendash}$ is what has always had the potential of getting you a job. Also, when did you last write a grant proposal promising to deepen the communal understanding of an existing proof, or to find a more digestible proof to a proven theorem? How does the amount of funding you have acquired for promising new results compare to the funding you got for explaining existing ones?

Reflect on common behaviour.

What is more likely to keep you up at night? A fellow mathematician threatening to arrive at “your” result ahead of you, or a fellow mathematician having trouble understanding your arguments?

The other day, an eminent mathematician confessed to me over lunch that they worked through their summer holidays to get out a paper with a new construction that they got out of an AI model. Anyone asking the model the same question would get the same ingenious answer, so the logic went, hence they needed to publish this immediately.

On the flip side, I’ve sat through talks where the speaker afterwards, privately, declared “Today I have given a talk that was utterly incomprehensible. I’ve always wanted to give a talk like this.” To be fair, this was a singular, extreme case. But we would not tolerate this kind of behaviour if we really cared so much about a community of common understanding.

Acknowledge past grievances.

Personally, I vividly remember how, at a panel discussion on “Future Directions of Algebraic Topology” at Stanford in 2012, one of the younger members of the audience raised the question: What can we do to make mathematical culture more like music culture, where presenters are valued as much as composers? At that time, the idea of AI was still very much hypothetical. The question was about our very own, human culture.

Read the Leiden declaration.

The Leiden declaration, which has been widely endorsed by the mathematical community, lists the following as one of our five mathematical “values”:

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.

That is just another way of saying that the one thing that really matters for your standing within the community is that you arrive at a new result first.

So why am I saying this? Because I completely disagree with the notion that AI companies have hijacked our mathematical culture and turned it into something new and unrecognizable. They have not. Quite the opposite. They are playing to our rules. They are even solving our problems, problems that no one else cares about! They have just taken our existing culture to its extreme.

And now, we are disgusted!?

I would argue that what we’re really disgusted by is a key aspect of our own culture, now that we see it more clearly than ever before. We’re disgusted by the idea that getting there first is all that matters.

Let’s be honest about what was and what is. We defined our priorities. AI serves them better than we can. Perhaps, the priorities were off.

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Received 25 September 2026.

Kunal Marwaha, Ph.D student at University of Chicago

SERIES · As application deadlines approach, Proofs and Prompts is publishing a series of posts from 19 to 23 October for people considering a PhD, doing a PhD or looking for a postdoc: guidelines, discussions and speculation about what to do and what careers in mathematics will look like. This post is one of them. Each is written independently and reflects only its authors’ views. All posts in the series

Mathematics is a creative and emotional endeavor. We are driven by challenge and curiosity. We express our ideas carefully to be precisely understood. Our highest moments can fill us with exhilaration and awe.

But how we do mathematics has evolved over time. We have moved away from pictures and informal arguments; we now have axioms and proofs. We have outgrown the time when one person could have understood all of known mathematics; now, we specialize, relying on others to teach us what we know less about.

Today, it is clear that mathematics is again changing and growing beyond us. There exist machines which can prove new mathematical statements, and convert them into papers for humans to verify. This is strange and disruptive to our current process. However, I offer a positive vision of a future which includes these machines while preserving our motivations for this extreme passion.

We should change our mindset from explorers of the mathematical frontier to stewards of a world commons of mathematics.

A commons is a societal resource available to all because it improves everyone’s quality of life. My favorite example is the US National Park system. At a national park, I can reliably experience an overwhelm of natural beauty in mountains, forests, or rivers. I can camp, swim, or just walk on a trail. After a weekend trip, I return to my normal life refreshed and inspired.

I like the outdoors, but I am not very adventurous. For some people I know, the outdoors can feel like home. They will go into nature for weeks or months at a time: backpacking, rock climbing, journaling. One can make a living like this as a park ranger or firefighter; people in these jobs are paid to understand wildlife, take responsibility for keeping man and nature in balance, and build trails to make life pleasant for visitors like me.

We can build the same kind of environment for mathematics.

For many of us, research is more of a passion than a job. The mathematical outdoors feels like home. This is why theorem-proving machines can feel so intimidating: they are strange new creatures. I think the machines are here to stay, and I don’t think we can control which problems they will solve. Instead, we should learn from them, keep them in balance with human understanding, and show other humans the beauty of this place. We can be the rangers and firefighters of our mathematical parks, which are more alive than ever before.

The sudden advent of theorem-proving machines will make the next few years painful and difficult. However, if we do not act, we may quickly lose relevance, and our community is likely to wither. This makes building a mathematical commons extremely urgent. But what would it take to build this commons?

The most important step is to create a coalition of mathematicians to steward the commons. We should create a non-profit with a large advisory board, who represent the broad and diverse interests of people who prove or teach theorems in their work. This organization should not be run by any AI company, and our artifacts should be open to everyone.

The next step is to convert all mathematics into formally verified statements. This is the goal of the Mathematics Autoformalization Project. If we can use computer code to certify correctness of a proof, we can maintain our trust in these proofs at scale. Today, the most popular way to formally verify mathematics uses the Lean proof assistant. There is a vibrant open-source community around Mathlib, which has verified essentially all of undergraduate mathematics. It is now even possible to use LLMs to convert proofs of research-grade theorems directly into Lean. (The recent formalization of Fermat’s Last Theorem used the public prove2.me site I helped build.)

Formalizing all of mathematics would create a large directed graph of known statements and their implications. Creating this graph is achievable if AI companies donate some compute resources to the mathematical commons. Here, they can work together instead of against each other. In the future, all new results could be added to this graph of knowledge.

The final step is to tame this graph of formally verified statements. We do not have to digest all at once; we only have to manage it. We should cluster and sparsify the graph until it corresponds to human-sized chunks of knowledge. We can look for simplifications, common definitions, and unifying theories. We can build trails through wonderful proofs to share both with other mathematicians and with “park visitors”. This can all be assisted by machines, provided it helps accomplish our goals.

If done well, the commons will bring about a golden age for human understanding. With responsible stewardship, we can nurture the next generation of mathematicians and math appreciators alongside theorem-proving machines.

Let us start by building this broad coalition of mathematicians. If we work together, we can steer the way mathematics evolves in this uncertain future.

Kunal Marwaha

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Received 5 October 2026, revised 23 October 2026.

As part of their big theorem dump, an AI company recently emitted a proof of the ‘quasi-Riemann hypothesis’. Maksym Radziwiłł of New York University said the implications of this result are “staggering”, continuing:

“In a sense, the vast majority of consequences of RH already follow from quasi-RH.”

But what is the quasi-Riemann hypothesis, and what are some of these consequences?

Since 1897, we’ve known all the nontrivial zeros of the Riemann zeta function lie in this strip in the complex plane:

0 < Re(z) < 1

The quasi-Riemann hypothesis says they lie in some narrower strip

a ≤ Re(z) ≤ 1 – a

The Riemann hypothesis claims all these zeros lie exactly on the line

Re(z) = 1/2

The closer to this line we can show these zeros are, the more regular the distribution of primes must be!

First, remember the number of primes less than n is asymptotically n / (ln n). But a more precise estimate of the number of primes less than n is a function called li(n), which is the integral of 1/ln(x) from 0 to n.

The difference between li(n) and n / (ln n) is about n / (ln n)². This is a lot bigger than the errors I’m going to talk about.

The above graph makes it look like li(n) is always bigger than π(n), but in fact π(n) – li(n) switches sign infinitely often—the first time before 1.4 · 10³¹⁶.

Before the proof of the quasi-Riemann hypothesis, we knew things like this: if π(n) is the number of primes less than n, then

|π(n) – li(n)| ≤ n / (ln n)¹⁰⁰⁰

if n is large enough. Indeed, we could replace the number 1000 by any other large number.

The Riemann hypothesis is equivalent to this: if π(n) is the number of primes less than n, then

|π(n) – li(n)| ≤ C n^p

for any p > 1/2, and some C that depends on p. We know we can’t get p all the way down to 1/2.

So crudely, the Riemann hypothesis brings the error bound down from “a lot less than n, but still bigger than any smaller power of n” to “not much more than √n”.

The quasi-Riemann hypothesis does the same thing but replacing √n with some bigger power of n, namely n^1-a if all the nontrivial zeros of the Riemann zeta function lie in the strip

a ≤ Re(z) ≤ 1 – a

In the recent massive drop of AI-generated results there’s a long, complicated proof that the quasi-Riemann hypothesis holds for a = 1/8:

• The quasi-Riemann hypothesis, 199 pages.

If this proof turns out to be correct, then we’ll know

|π(n) – li(n)| ≤ C n^p

for any p > 7/8.

But there’s also a much simpler proof that a = 1/12 works:

• The quasi-Riemann hypothesis, alternate proof, 49 pages.

A bunch of mathematicians have been writing about this, developing similar results—so we can feel pretty sure this is true. Thus, at this point, number theorists are confident that

|π(n) – li(n)| ≤ C n^p

for any p > 11/12.

But the quasi-Riemann hypothesis has other consequences, too—especially combined with the recent proof that a bunch of L-functions have no ‘Siegel zeros’. One of the cuter, less technical results is called ‘Goldbach conjecture on average’.

You can count, for each even number up to n, the number of ways to write it as a sum of two primes, and add these counts up. The total is close to a simple formula, and the question is how close. The quasi-Riemann hypothesis puts a bound on the error, thus proving Goldbach’s conjecture on average. I’ll spare you the details. Unfortunately, this bound is not good enough to prove Goldbach’s actual conjecture.

There are also plenty of more technical yet more profound implications of the new work on the quasi-Riemann hypothesis. These rely on not just the statement, but the proof method. There are various other important functions in number theory—zeta functions and L-functions—that we can now prove have zeros in a half-plane Re(z) > 1 – a for some fixed a > 0. Write QRH(a) for this statement, so OpenAI’s 7/8 result is QRH(1/8) for the Riemann zeta function.

A recent paper by Frei, Sofos and Widmer summarizes some of what’s been done so far:

• Christopher Frei, Efthymios Sofos and Martin Widmer, Some applications of the quasi-Riemann hypothesis to class groups and small generators of number fields.

For example, one paper in the theorem dump claims QRH(1/8) for all Dirichlet L-functions and all finite-order Hecke L-functions of ℚ√(−3), with a second preprint giving QRH(1/12). Another claims QRH(10^-6) for all finite-order Hecke L-functions of cyclotomic fields containing the 12th roots of unity. Most exciting to me, a preprint by Cureton and Thorner claims QRH(1/32) for the Dedekind zeta functions of all number fields:

• Dean Cureton and Jesse Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions.

You may not consider any of these implications “staggering”. But that probably means you’re not an analytic number theorist. According to Joshua Zelinsky, there are hundreds of papers deriving results that can be improved using the quasi-Riemann hypothesis. Mathematicians have been working for over a century to make these things happen.

As part of their big theorem dump, an AI company recently emitted a proof of the ‘quasi-Riemann hypothesis’. Maksym Radziwiłł of New York University said the implications of this result are “staggering”, continuing:

“In a sense, the vast majority of consequences of RH already follow from quasi-RH.”

But what is the quasi-Riemann hypothesis, and what are some of these consequences?

Since 1897, we’ve known all the nontrivial zeros of the Riemann zeta function lie in this strip in the complex plane:

0 < Re(z) < 1

The quasi-Riemann hypothesis says they lie in some narrower strip

a ≤ Re(z) ≤ 1 - a

The Riemann hypothesis claims all these zeros lie exactly on the line

Re(z) = 1/2

The closer to this line we can show these zeros are, the more regular the distribution of primes must be!

First, remember the number of primes less than n is asymptotically n / (ln n). But a more precise estimate of the number of primes less than n is a function called li(n), which is the integral of 1/ln(x) from 0 to n.

The difference between li(n) and n / (ln n) is about n / (ln n)². This is a lot bigger than the errors I’m going to talk about.

Before the proof of the quasi-Riemann hypothesis, we knew things like this: if π(n) is the number of primes less than n, then

|π(n) - li(n)| ≤ n / (ln n)¹⁰⁰⁰

if n is large enough. Indeed, we could replace the number 1000 by any other large number.

The Riemann hypothesis is equivalent to this: if π(n) is the number of primes less than n, then

|π(n) - li(n)| ≤ C n^p

for any p > 1/2, and some C that depends on p. We know we can’t get p all the way down to 1/2.

So crudely, the Riemann hypothesis brings the error bound down from “a lot less than n, but still bigger than any smaller power of n” to “not much more than √n”.

The quasi-Riemann hypothesis does the same thing but replacing √n with some bigger power of n, namely n^1-a, if all the nontrivial zeros of the Riemann zeta function lie in the strip

a ≤ Re(z) ≤ 1 - a

In the recent massive drop of AI-generated results there’s a long, complicated proof that the quasi-Riemann hypothesis holds for a = 1/8:

• The quasi-Riemann hypothesis, 199 pages.

If this proof turns out to be correct, then we’ll know

|π(n) - li(n)| ≤ C n^p

for any p > 7/8.

But there’s also a much simpler proof that a = 1/12 works:

• The quasi-Riemann hypothesis, alternate proof, 49 pages.

A bunch of mathematicians have been writing about this, developing similar results—so we can feel pretty sure this is true. Thus, at this point, number theorists are confident that

|π(n) - li(n)| ≤ C n^p

for any p > 11/12.

But the quasi-Riemann hypothesis has other consequences, too—especially combined with the recent proof that a bunch of L-functions have no ‘Siegel zeros’. One of the cuter, less technical results is called ‘Goldbach conjecture on average’.

You can count, for each even number up to n, the number of ways to write it as a sum of two primes, and add these counts up. The total is close to a simple formula, and the question is how close. The quasi-Riemann hypothesis puts a bound on the error, thus proving Goldbach’s conjecture on average. I’ll spare you the details. Unfortunately, this bound is not good enough to prove Goldbach’s actual conjecture.

There are also plenty of more technical yet more profound implications of the new work on the quasi-Riemann hypothesis. These rely on not just the statement, but the proof method. There are various other important functions in number theory—zeta functions and L-functions—that we can now prove have zeros in a half-plane Re(z) > 1 - a for some fixed a > 0. Write QRH(a) for this statement, so OpenAI’s 7/8 result is QRH(1/8) for the Riemann zeta function.

A recent paper by Frei, Sofos and Widmer summarizes some of what’s been done so far:

• Christopher Frei, Efthymios Sofos and Martin Widmer, Some applications of the quasi-Riemann hypothesis to class groups and small generators of number fields.

For example, one paper in the theorem dump claims QRH(1/8) for all Dirichlet L-functions and all finite-order Hecke L-functions of ℚ√(−3), with a second preprint giving QRH(1/12). Another claims QRH(10^-6) for all finite-order Hecke L-functions of cyclotomic fields containing the 12th roots of unity. Most exciting to me, a preprint by Cureton and Thorner claims QRH(1/32) for the Dedekind zeta functions of all number fields:

• Dean Cureton and Jesse Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions.

You may not consider any of these implications “staggering”. But that probably means you’re not an analytic number theorist. According to Joshua Zelinsky, there are hundreds of papers deriving results that can be improved using the quasi-Riemann hypothesis. Mathematicians have been working for over a century to make these things happen.

Today, I take a break from your regularly-scheduled AI mathocalypse programming to urge my American readers to vote in the Congressional midterms, early if possible, if you haven’t already. Especially if your state faces one or more major contests, as does my adopted home state of Texas.

Me, I voted straight-ticket Democrat today, as I’ve done in every election since I became eligible over a quarter-century ago (except for when I vote-swapped with someone from another state).

So yes, I voted for the guy who studied theology a short walk from my house, and who seems to have really internalized the whole Jesus thing about kindness to everyone—

—over his sneering, thuggish, cartoonishly mafia-like opponent. Fingers crossed that Talarico pulls off a historic upset for our state (he’s currently a 2-to-1 favorite in the prediction markets).

The Texas Republicans did everything they possibly could to make this choice easy for me, by being corrupt, conspiratorial, racist, and Taliban-like even by the jaw-droppingly debased standards of MAGA in 2026. As I draw breath, I could never vote for a slate one of whose candidates attacked UT students for the crime of being visibly Southeast Asian while cheering their school’s football victory from the stands—and whose other candidates could barely bring themselves to condemn this knuckle-dragging racist, for fear of angering their base. (It’s like, didn’t you want immigrants to assimilate to American culture? What more could these kids possibly do, short of skin bleach?)

All the same, this was also the first US election in my lifetime where I felt like my vote was genuinely in play—especially when I listened to Talarico’s condemnations of Israel, for having the temerity to fight the same kind of war any random other nation would’ve fought had it been attacked as Israel was on October 7. When Hasan Piker came to speak to a cheering crowd here at UT Austin, I actually wrote to someone I know in the Talarico campaign, to ask whether Talarico would show enough balls to condemn Piker’s many antisemitic and anti-American statements, and renounce his support. My contact was nice enough to reply to me, saying: don’t worry, James is taking care to be far away from Austin during Piker’s visit, so he won’t need to engage at all!

So then, this is my candidate: the guy who’s strategically silent on the wildly-popular streamer who’d gladly see my Israeli family members raped and beheaded, rather than campaigning with that streamer like Abdul El-Sayed did in Michigan. I unironically believe that, for Talarico, it took genuine moral courage to resist the whole momentum of his party’s youth even to this extent.

To earn my vote in the near future under these circumstances, all a Texas Republican would need to do is be somewhat sane or reasonable. He or she might affirm, for example, that racism is bad, that abortion is a woman’s choice, that Christianity is not the state religion of the US, that Covid vaccines work, that human emissions are worryingly changing the earth’s climate, and/or that Joe Biden won in 2020, while still being arbitrarily right-wing on taxes and other stuff. He or she could then simply wait for the Democrats to descend fully into the worldview of the masked hooligans who are currently setting France’s beautiful cities aflame, terrorizing its shopkeepers, and looting as many Louis Vuitton handbags as they can carry … all for Palestine, of course, and for the death of global capitalism, and for the violent overthrow of the scheming, money-grubbing, hook-nosed, international Talmudic … err, I mean “Zionists.”

What I’m trying to say is, it was a genuine achievement on the part of the Texas Republicans to fail to earn a single vote from me this cycle.

In the spirit of my recent “Math 2.0” talk on new ways to use AI assistance to enhance the human understanding of mathematics, rather than substitute for it, I am using this post to propose an experimental challenge, inspired by the recent activity around excluding Landau–Siegel zeroes, as discussed in this previous post.

There is a well-known connection between Landau–Siegel zeroes and the tenth discriminant problem, which I will briefly recall here; an in-depth discussion on this problem can be found in this survey of Shafarevich. As far back as the work of Gauss, it was observed that there were nine discriminants whose rings of integers for the associated quadratic field enjoyed unique factorization, namely

Gauss believed this list of nine to be the complete list of such discriminants (the “class number one problem”), but this was only proven by Heegner in 1952 by introducing the concept of a Heegner point on the modular curve; and the above numbers are now known as Heegner numbers.

Prior to Heegner’s solution was the work of Heilbronn and Linfoot in 1934, who used what we now call the Deuring–Heilbronn repulsion phenomenon to show that the existence of a tenth discriminant in the above list can be used to “repel” any further such discriminants, so that there was at most one further Heegner number beyond the nine already known. However, this argument was ineffective: no upper bound on this putative tenth discriminant was given, and so there was no obvious way to solve the class number one problem purely from this result.

There is a close connection to Landau–Siegel zeroes: if is a Heegner number, and is the quadratic character associated to this quadratic field, then the class number formula gives , and with a bit more work one can produce a Landau–Siegel zero with . For sufficiently large , this is inconsistent with Siegel’s theorem; but the standard proofs of this theorem also relied on the repulsion phenomenon and were also ineffective. It was only in 1976 that Goldfeld gave an effective version of Siegel’s theorem that was enough to at least bound the tenth discriminant, given a suitable high rank elliptic curve. In followup work of Oesterlé, combined with a high rank elliptic curve of Gross and Zagier, the bound was obtained; using a later elliptic curve of Brumer and Kramer, proven to have high rank by Mestre, Oesterlé, and Serre, the bound was improved to . See this survey of Goldfeld for more discussion.

But now we have a fair number of new (and effective) ways to rule out Siegel zeroes, either by establishing a quasi-Riemann hypothesis, or by some version of the argument in this previous post. In principle, each version of these arguments should also give a bound on the tenth discriminant (without using Heegner’s solution to the class number one problem that rules out this discriminant). The task of making each of these estimates explicit enough to create such a bound is rather tedious for humans. However, it seems eminently suitable for AI.

Thus, the challenge I pose is, for each of the new manuscripts or digestions relating to either the quasi-Riemann hypothesis or the Landau–Siegel zero problem (or by some novel argument), to use an AI tool to work out an effective upper bound for a hypothetical tenth discriminant , without using Heegner’s methods, and to make best efforts to optimize this constant. This serves several purposes:

• If the threshold is low enough to resolve the remaining discriminants by direct computation, this would yield a new solution of the class number one problem — possibly one that is formalizable in Lean in a high quality fashion (suitable for instance for upstreaming to Mathlib).
• By performing this exercise on each of the different writeups, we obtain a way to compare the “strength” of each such writeup (after performing whatever optimizations are available for each writeup), which could give insight as to the relative advantages or disadvantages of different arrangements of the methods.
• The exercise of minimizing the upper bound may spur the creation of entirely new arguments to solve either the class number one problem, or the elimination of Landau–Siegel zeroes.
• There are many previous “failed” attempts at excluding the Landau–Siegel zero. It is possible that some of them contain ideas that can be combined with the newer solutions to this problem to “repair” the proofs, or at least improve the bound on the tenth discriminant.
• Efforts here may also help improve the state of knowledge on higher class number problems, which by the work of Watkins in 2004 is currently solved up to class number 100.

Furthermore, as both of these problems are already solved (for over seven decades, in the case of the class number one problem), there is little risk of such a competition inadvertently disrupting an ongoing research project.

I am therefore inviting interested readers to contribute in the comments any effective bounds on the tenth discriminant they can extract from either an existing argument, or a novel one, with any amount of AI use permitted for this challenge so long as it is disclosed. Note that the challenge is broader than simply getting the “best” bound overall amongst all possible approaches; I am also interested in what bounds one can extract from any specific approach, even if they are worse than, say, the bounds of Goldfeld or Oesterlé. In particular, “negative” results in which the bounds obtained from a given method end up being rather large are still scientifically useful pieces of information that are worth reporting. One could also explore other directions, such as bounds that avoid using extensive compute to squeeze out the maximal amount of optimization, but are instead clean enough that their arguments are human-verifiable.

Any large outputs should be posted as links rather than as plain text; for instance, pointing to a Github repository containing one’s work would be a reasonable way to submit an entry (but a human-generated summary should be provided in the main text of the comment).

I would ask contributors to not prioritize speed above quality; this is not intended to be a race regarding who is “first” to claim any given bound. The spirit is intended to be collaborative rather than competitive: think of a Polymath style project, updated for the AI era. In particular, as with traditional mathematical research, it would be appropriate to build upon previous contributions to obtain an improved bound for a given proof technique. (But in such cases, one should of course give appropriate attribution to the prior source.)

I will not insist on formal verification (e.g., in Lean) of the results, though I would ask anyone submitting a purely AI-generated solution to do enough due diligence to check the output (e.g., with an independent AI tool) before posting it here.

I am looking forward to the results of this experimental challenge, both for the mathematical insights gained, and for the lessons learned on how to crowdsource AI assistance in mathematics in a responsible fashion.

Anonymous, Postdoc

SERIES · As application deadlines approach, Proofs and Prompts is publishing a series of posts from 19 to 23 October for people considering a PhD, doing a PhD or looking for a postdoc: guidelines, discussions and speculation about what to do and what careers in mathematics will look like. This post is one of them. Each is written independently and reflects only its authors’ views. All posts in the series

AI assistance disclosure: AI was used to translate my draft into English, revise its language and organization, assist with literature searches, and suggest relevant quotations and statistics. The personal experiences and central concerns originate in my draft.

I am a postdoctoral mathematician working in partial differential equations, and I love what I do. I have preferred doing mathematics to expressing opinions in public, but the rapid changes of recent months have made that difficult.

I am writing neither to defend a complete ban on AI nor to celebrate its unrestricted use, but because I no longer know what rules I am expected to follow, what work will count as my own, or how today’s decisions will affect my career.

For an established professor, uncertainty about the future may be a subject for an interesting essay; for a doctoral student or postdoc, it can determine whether years of work lead to a job. The same advice does not carry the same risk for everyone.

My concern is not only what AI can do to mathematics, but what mathematicians and institutions are choosing to do to one another during this transition.

1. We are told that priority will matter less, but the race suggests otherwise

A recurring message in the recent debate is that recognition should move from being first toward understanding proofs, explaining their ideas, and making them useful to others.

In Mathematics in the age of AI, Terence Tao argues for less emphasis on proof generation and being first. Amit Harlev goes further: “when a result is proven by an AI, priority should carry no reward.” Timothy Gowers predicts that the credit system will collapse, while defending credit as long as careers depend on it. Benoit Henry questions whether a theorem belongs to whoever obtains it first when someone else could quickly do so with a similar system. [1, 2, 3, 4]

Bryna Kra offers a concrete proposal: “Journals need to distinguish and credit the roles of discovery, proof, formalization and explanation.” Grant Sanderson observes that “every AI-generated proof is born an unsolved exposition problem.” [5, 6]

I welcome greater credit for explanation, but cannot behave as though that transition has happened while the consequences of being second remain uncertain. A postdoc cannot pay rent with a prediction that exposition will eventually become prestigious.

The possibility that another person or laboratory could obtain a proof quickly can intensify the pressure to announce first, even as we are asked to slow down, check the literature, understand everything, and write carefully.

Nor do I accept that mathematics was always about understanding rather than solving problems: I value both learning a beautiful argument and finding one that did not previously exist. I do not want either experience dismissed merely because the other may be easier to preserve.

The familiar name “Fermat’s little theorem” reminds me that discovery, proof, and naming are not the same thing. Historical names offer no reliable rulebook for present-day authorship; we need to distinguish priority, authorship, attribution, and understanding explicitly.

2. What exactly does an AI-generated proof entitle its producer to claim?

Jihao Liu states the practical dilemma unusually clearly:

“Consider the following situation. I use AI to produce a paper whose result is important. What should I do?

1. I post the paper to arXiv. Then I am criticized for not having verified the paper in detail. This goes against my values as a mathematician.
2. I do not post the paper anywhere. But what if someone else obtains the result and posts it before me? In that case I lose the credit.” [7]

He posts some machine-checked, not yet human-verified papers on a separate webpage rather than arXiv, and requires a timestamp when recognizing prior independent work. [7]

Liu is frank about protecting priority, but the order troubles me: secure the claim first, complete verification and repair attribution afterward. Treating credit as his to grant strikes me as cynical. His offers of shared credit and coauthorship recognize the value of checking, simplifying, and explaining, but leave unclear what the announcement reserves and what later contributors can earn. [7]

What credit has the initial producer earned before these other contributions have been made? Does a public timestamp establish priority for a theorem, a candidate argument, or merely a generated document? If the proof is correct but its attribution incomplete, what is owed to the people whose ideas supplied most of it?

Marco Trombetti’s ProofForum proposal distinguishes generating, checking, repairing, simplifying, and explaining without endorsing Liu’s entire policy: “verification could itself become a visible mathematical contribution.” Recognition need not stop with the generator, and credibility can develop through public examination. [8] That may be useful, but a record of contributions is not yet an agreed system for allocating jobs, prizes, publishing rights, or theorem names.

Ruodu Wang supports human understanding and rewriting, but identifies the incentive problem: “Being the second to solve a problem has low or no value.” His model rewards posting AI output immediately over taking time to understand and rewrite it. [9] Even a reduced but positive reward for first place preserves that disadvantage if second place earns nothing; Wang’s argument is that careful work needs changed incentives, not advice alone. [9]

Removing priority creates another difficulty: junior researchers depend on the problems they solve, rather than the established reputations and other recognized roles of senior mathematicians. Wang writes: “I do not see a good system of evaluating junior mathematicians.” A new ideal is not yet a viable career. [9]

We cannot preserve the reward for speed, demand slow and careful work, and leave the people with the least job security to absorb the conflict.

3. The Navier–Stokes dispute: different results, unresolved standards

The September controversy involving Tristan Buckmaster, Levent Alpöge, and OpenAI left me with a bitter feeling. As a PDE researcher, I wanted to understand both the mathematics and its implications for research practice.

The mathematical claims should first be kept distinct. OpenAI’s September 8 announcement claimed a smooth-forcing Navier–Stokes blow-up result establishing alternatives C and D of the Millennium Prize formulation, alongside a separate unforced Euler result. It distinguishes these from Alpöge and Buckmaster’s forced Euler work and denies that their unpublished work or recent Codex prompts influenced its system. These are OpenAI’s stated claims and account. [10]

In the Q&A of Buckmaster’s September 23rd NYU talk, he says: “Those physical mechanisms that I can explain to you guys came out from LLM.” In his account, AI supplied important mechanisms while human expertise directed the search and helped reconstruct the argument. A label such as “Alpöge–Buckmaster, 2026” therefore needs a clear account of what the researchers proposed, what the systems generated, and who verified and rewrote the proof. [12]

He also describes working with Alpöge through internal models: “it’s not just internal models it’s internal compute.” His point that many agents can matter as much as a better model distinguishes their resources from an ordinary subscription. [12]

Buckmaster questions credit based on “who does something the fastest” and says: “I think this race dynamics doesn’t make sense.” WIRED reports that he released unfinished papers to get ahead of OpenAI’s announcement and accepted responsibility for cleaning them up. Wanting priority to matter less while protecting work under unchanged rules is understandable—and precisely the dilemma younger mathematicians face. The question is not whether one researcher should have solved the incentive problem alone, but whether the community can advise young mathematicians to leave the race while continuing to reward those who enter it. [12, 11]

As of 1 October, more than three weeks after the 7 September announcement, I had not found a revised exposition of the forced Euler proof that resolved my readability concerns. Expecting a world-class fluid dynamics expert such as Buckmaster to produce one does not settle what recognition the initial release deserves. [27]

If an unfinished announcement preserves recognition while a readable proof follows later, is the same option open to a doctoral student or postdoc? What standard of verification and exposition must be met before release, and what recognition remains available to someone who waits?

Buckmaster’s talk and NYU’s announcement explicitly credit Diego Córdoba and Luis Martínez-Zoroa’s program and strategy, yet public attention can favor the final announcement over the work that made it possible. [12, 27]

Neither Isabella Ward’s 19 September WIRED article, “Mathematicians Feel Threatened by AI. They Can’t Quit It,” nor the 8 September OfficeChai report, “OpenAI Says It Didn’t See Any Of Buckmaster And Alpöge’s Work While Resolving Navier-Stokes,” names Córdoba or Martínez-Zoroa. My concern is that press coverage of priority disputes in the AI era will tend to focus on who obtained or announced a result first, while overlooking the foundational mathematical work—and the people behind it—that made those AI-assisted discoveries possible. [11, 28]

4. This is not an abstract problem for me

I have checked and rewritten an AI-generated proof of a problem I had not worked on for roughly ten years, but have not published it. A group I will not name is pursuing the same question; as far as I know, they have not resolved it, and my argument appears to take a different route from their earlier attempts. Should I post it with a detailed account of the AI contribution, contact the group first, or offer collaboration? If I present the result at a conference, should I write “Theorem [my name, 2026]” on the slide, and would publishing it help my career, damage my reputation, or both?

I am not asking to claim someone else’s ideas, but what credit checking and rewriting earns; that cannot depend on which influential mathematician is asked.

Meanwhile, I am finishing a substantial article developed over about a year with pen and paper, not with AI. Almost daily, I fear being scooped by someone, perhaps using AI, who posts first: my understanding would remain, but the publication opportunity and recognition for a year of independent work might not.

Antonio Auffinger’s Open problems, open mathematics warns that others can quickly complete a publicly shared strategy, making unfinished ideas risky to share. His call to reshape incentives aims to protect openness for the least secure researchers. [13] I want to continue sharing ideas, asking colleagues questions, and doing careful work without making secrecy, haste, and suspicion the rational way to survive.

5. Recognition, jobs, and the cost of disclosure

Another part of the debate makes these choices even more difficult: some mathematicians do not merely question the value of AI-generated papers, but predict reputational damage for AI users or propose institutional preferences against their work.

In Where I Stand on AI, Vladimir Lazić predicts that attaching one’s name to a partially AI-produced paper can destroy a mathematician’s reputation. Addressing researchers without tenure, he writes:

“you will never get tenure, unless you are being pushed by someone in your camp who has tenure.” [14]

Although this is a prediction rather than evidence about hiring committees generally, a junior researcher cannot ignore the possibility that some evaluators share that attitude.

Donatella Iacono, explicitly following Lazić’s principles, states: “I do not support hiring of academic staff who use AI as a substitute for their own work.” She takes the same position on supervising doctoral and postdoctoral researchers. [32]

Max Weinreich goes beyond predicting reputational consequences: in The crisis of AI-generated mathematics, he proposes “reserving hire lines for mathematicians who eschew AI” and “valuing AI-free papers more highly in tenure promotion.” These are proposals for allocating opportunities differently, not merely expressions of a personal preference. [15]

Atticus Stonestrom makes a direct proposal about access to publication:

“Traditional journals should ban papers in which AI has played a coauthor-level role, and new venues should be established for the publication of such papers.” [16]

Without a ban, Stonestrom proposes distinguishing such papers and capping their annual publication. He cites authorship and scarce refereeing resources, acknowledges that restrictions may encourage concealment, but argues that this should not determine standards. Even a carefully checked paper could be excluded from a traditional journal. [16]

A young researcher does not need a profession-wide ban to find these positions alarming: a few editors or hiring-committee members can have a decisive effect on a career. Honest disclosure may carry lasting consequences when it can reduce recognition or make a paper ineligible for a journal.

Martin Hairer instead argues that “articles are judged on their merits, independently of the tools being used” and that disclosure should not automatically reduce a paper’s value, while retaining human scrutiny, understanding, and writing as requirements. [17]

What should a postdoc make of these competing messages: use AI responsibly and disclose it, avoid it to demonstrate independent ability, use it or fall behind, yet risk losing recognition by doing so? These are not small differences in taste when employment is at stake.

I have heard colleagues describe the strongest anti-AI views as resistance to an inevitable technology, while others say they reflect a larger, quieter part of the profession. I do not know which claim is correct, since neither a collection of essays nor my conversations establishes what most mathematicians believe.

What is already visible is uncertainty about the consequences of honesty, and that uncertainty deserves an institutional response.

6. What the publication and disclosure data actually show

The data show rapid change without establishing how much AI use is undisclosed or why.

Tammy Kolda and Daniel Appelo count 6,307 primary-mathematics arXiv announcements in August 2026, up 2,785 from 3,522 a year earlier—about 79%, versus 43% across arXiv. [18]

Benoit Henry reports single-author counts of 1,174 and 2,603 for those months (about 122% growth), with 809 later papers mentioning AI. He cautions that the counts are uncertain; single authorship is not evidence of AI use, and an AI mention is not necessarily a carefully classified disclosure. [4]

Xiaoshn Nee, Haobo Zhong, and Xiaomin Ni find growth in 29 of 30 primary mathematics categories; the share of active author units posting at least five papers in January–August rose from 2.45% in 2025 to 3.80% in 2026. [29]

Tara Abrishami’s Epoch AI analysis finds that disclosures of any AI use rose from 3.89% in April 2026 to 25.25% in August (1,453 of 5,754 papers); 337, or 5.86%, disclosed substantial research contributions. The broader category includes writing, computation, and literature review. [19]

Yet three quarters of Epoch’s August papers made no acknowledgment; this group grew from 3,260 in August 2025 to 4,301 in August 2026 (about 32%). Silence does not tell us whether AI was used, and differing corpora and counting conventions prevent simply combining the studies’ totals. [19]

In Kimball Martin’s August 2026 number-theory sample, 12.0% disclosed AI writing while 20.7% were flagged for apparent AI writing. The imperfect detector and unpublished scoring rule make this suggestive only; Martin warns that a score is not a verdict. [20]

These counts neither identify concealed use in individual papers nor prove large-scale concealment, but common sense suggests that incentives to publish faster, coupled with fear of penalties for disclosure, can encourage silence. I suspect some use goes unacknowledged partly because of stigma.

Harlev identifies the underlying incentive: evaluating declared AI work less favorably can punish honest researchers without reliably detecting undisclosed use. [2] We should therefore demand precise disclosures and fair evaluation together, since one without the other creates a predictable conflict.

7. Equal rules require more than equal advice

Access depends not only on whether someone chooses to use AI, but on the models, computing budget, and support available to them. OpenAI’s September account describes an internal model more capable than the public comparator, placing that work in a different setting from a postdoc’s ordinary subscription. [10]

I worry that personal connections to laboratories may become a research advantage. Does staying competitive now require knowing someone who can provide access to an internal system or a large computing budget? What happens to mathematicians who have neither those connections nor an institution able to buy equivalent resources? Such access must be visible in evaluation: a publication list should not conceal the difference between an individual subscription and large, privately supported searches.

The Advisory Group on Mathematics and Artificial Intelligence’s September 29 recommendations warn that proprietary models risk a “two-tier system” and unequal access to public models a “multi-tier hierarchy.” It calls for broad, equitable access. [21]

On 1 October 2026, Claude Max 20x and ChatGPT Pro 200 each cost US$200 monthly; the new ChatGPT Pro 500 costs US$500. Both highest-tier subscriptions total US$700 monthly, or US$8,400 annually, before taxes, exchange charges, and separate computing costs. [30, 31]

Across developing countries and lower-paid academic systems in developed countries, stipends and salaries may leave little for these subscriptions after living expenses, especially when researchers personally fund multiple services. An expense routine for a well-funded department may be unsustainable for an equally capable mathematician elsewhere, rewarding institutional resources and personal finances rather than talent, effort, or the quality of research questions.

Researchers should not have to choose between living expenses and tools increasingly treated as necessary. Institutional support helps, but identical prices do not mean equal access, and subscriptions do not fund thousands of parallel agents.

The Leiden Declaration links AI use to human responsibility, transparency, and its effects on the profession; these principles must apply equally to researchers with and without privileged access. [22]

Orwell’s Animal Farm comes to mind when formal equality coexists with unequal opportunities: I do not want one standard for insiders and another for outsiders, or strict rules for students with exceptions for established names.

8. What future are we offering PhD students?

I welcome the Harvard summit on doctoral education in the age of AI. Its September 17–18 meeting brought together 24 mathematicians, produced recommendations for departmental discussion, and explicitly thanked the graduate students and postdocs who participated in a panel. [23]

Consultation, however, is not the same as an ongoing role in decisions. How will students and postdocs participate in setting the rules that govern their training and evaluation? Who will represent people outside the institutions already most influential in this debate?

The report favors demonstrated mastery over the dissertation alone and says it “need not be a requirement for the thesis to have recorded the first proof.” This may protect students whose work is unexpectedly overtaken, but protection within a PhD program does not automatically carry into the job market. [24]

If an adviser asks a student to wait until an AI-generated argument is fully understood, and someone else posts a less readable version first, will the student still receive comparable recognition? Will thesis and hiring committees use compatible standards?

There can be good educational reasons to restrict tools while students learn a skill; equal treatment need not mean identical training for students and experienced researchers, but the distinction must be explained, bounded, and accompanied by protection against resulting disadvantages.

Senior mathematicians should ask what research opportunities they are preserving for students. If a problem can now be explored quickly using AI, should every such opportunity become another paper on an established researcher’s list, or could some become genuinely collaborative training projects? This concerns mentorship when the supply of attractive projects may change rapidly, not a demand that senior researchers stop research.

Above all, what life are we inviting a talented mathematics graduate to pursue? Many of us entered the subject because we wanted to experience the struggle and joy of discovery: working through failed ideas, finding the right argument, and reaching a moment of understanding through our own effort.

Jonny Evans explains why research produces more than a paper: “The ‘output’ is a person who understands something better,” someone who can carry that understanding into research and teaching. I share his concern that more results do not necessarily mean more capable mathematicians. [33]

Teaching, exposition, and verification are already part of the mathematics I love, but a career centered on checking machine output, teaching calculus, linear algebra, and differential equations to engineering and computer-science students, and explaining results generated inside commercial laboratories may not appeal to people who hoped to become researchers. Calling that concern nostalgia does not answer it.

I fear losing both discovery and professional security while being told that a more demanding system of oral assessments and repeated demonstrations of understanding is the solution. Such assessments may be appropriate, but they do not, by themselves, offer a positive account of the career that follows.

A field that fails to attract and support its next generation cannot preserve itself through inspiring statements alone.

9. What institutions need to say clearly

What I need from the people who hire, promote, edit journals, award grants, and supervise students is not another assurance that mathematics will always matter, but practical answers.

Ben Antieau puts one proposal plainly: “Do not recruit, hire, promote, or praise based on the number of papers.” Daniel Litt argues that sustained mathematical discussion and talks can demonstrate understanding better than papers alone. Henry similarly asks hiring committees to stop counting articles. [25, 26, 4]

These proposals need to become public rules before researchers can rely on them when choosing projects, tools, or publication venues.

I want clear answers in six areas.

Permitted use and responsibility. Which uses of AI are acceptable in research, dissertations, and submitted papers? How do the rules distinguish language editing, coding, literature searches, suggested lemmas, generated proofs, and largely autonomous work? What must every named author understand and be able to defend?

Release and priority. May someone release a proof that is not human-verified? Does a Lean-checked proof have a different status? Is understanding the broad strategy sufficient, or must the details be checked before release? What priority, if any, attaches to an unfinished write-up accompanied by a promise to improve it?

Authorship and attribution. What contributions justify authorship, and what contributions require a different form of recognition? How are the people who supplied prior methods, directed the search, repaired the proof, formalized it, or explained it credited? What must happen when the initial attribution is incomplete?

Hiring, promotion, and publication. Will good, selective mathematics journals accept AI-generated but human-verified proofs under the same standards as other research papers? How much original human input will they require? Will a clear explanation, simplification, or independent reconstruction of an already announced AI result satisfy their standards of originality and significance, or be excluded as exposition? Will those requirements be public before authors invest months in a project?

Will hiring and promotion committees continue to expect the same quantity of publications, journal prestige, citation counts, and h-index as before—or raise those expectations because AI can accelerate production? If understanding and exposition are to count more, what evidence will be accepted in place of those measures, and will it lead to comparable professional opportunities? Will a mathematician working without AI be judged against AI-assisted output rates? Will someone who discloses careful AI use be penalized for doing so? How can candidates challenge decisions based on standards that were never announced?

Training and protection. How will students retain opportunities for independent discovery? How will a student who follows careful verification rules be protected from the resulting delay? How will institutions recognize substantial work that loses chronological priority?

Access and implementation. Who will fund the tools that new expectations require? How will unequal model access and computing budgets be considered? Which rules apply across institutions, when do they begin, and how will students and postdocs participate in reviewing them?

I am asking not for a universal numerical formula for mathematical worth, but for public standards that guide decisions without changing according to a person’s reputation.

10. The future should not be decided over our heads

For the first time, I am seriously considering leaving mathematics—a painful sentence to write after almost twenty-five years of passion, thought, and effort. What makes me consider leaving is not that a machine can do something I cannot, but that I no longer trust the profession to offer a fair and intelligible path forward.

What troubles me most is the possibility that established researchers will keep the advantages of the old system while younger people accept the obligations of a new one. Even when individuals act in good faith, the appearance of that double standard is damaging and should be addressed through transparent rules rather than dismissed as anxiety about technological change.

What I am asking for is a profession in which careful work is not punished for being slow, honesty is not punished for being visible, and access to powerful tools is not confused with mathematical merit.

I would welcome an emergency international meeting, with the reach and seriousness of an ICM, devoted to these questions. It should include doctoral students, postdocs, mathematicians from less wealthy institutions and countries, journal editors, hiring committees, and funding bodies—not only the people who already have the most secure positions. Its aim should be concrete, revisable standards for research, credit, training, and careers, together with a timetable for putting them into practice.

No meeting can settle every question permanently, but we can do better than leaving young researchers to discover the real rules only when an application is rejected.

My deepest fear is not that AI will destroy the career of a mathematician, but that mathematicians will make it unlivable through unequal rules, opaque incentives, and a failure to protect the people who come next.

I do not yet have a settled position on every use of AI in mathematics, but I do have a clear position on fairness: the rules cannot change only for the young.

Sources and references

[1] Terence Tao. Mathematics in the age of AI. arXiv:2608.16753, 17 August 2026.

[2] Amit Harlev. Dear conferences, journals, and hiring committees: don’t wait for everyone else. Proofs and Prompts, 25 September 2026.

[3] Timothy Gowers. Why I didn’t sign the Fields medallists’ letter. Gowers’s Weblog, 17 September 2026.

[4] Benoit Henry. The Loss of Singularity. Proofs and Prompts, 28 September 2026.

[5] Bryna Kra. Deep theorems were scarce and difficult and so became an effective mechanism to identify deep thought. AI has broken this system. What’s New, 13 September 2026.

[6] Grant Sanderson. If math is more than proof, we need to better celebrate the rest of it. What’s New, 18 September 2026.

[7] Jihao Liu. About the page of AI-generated results not verified by a human. Author’s website, undated.

[8] Marco Trombetti. ProofForum: Keeping AI-Generated Mathematics Human. Proofs and Prompts, 22 September 2026.

[9] Ruodu Wang. The AI-assisted mathematician’s dilemma. Proofs and Prompts, 28 August 2026.

[10] OpenAI. On the Navier–Stokes Millennium Prize Problem. 8 September 2026; updated 10 September 2026.

[11] Isabella Ward. Mathematicians Feel Threatened by AI. They Can’t Quit It. WIRED, 19 September 2026.

[12] Tristan Buckmaster. [Seminar on AI-assisted fluid dynamics]. YouTube video, 2026.

[13] Antonio Auffinger. Open problems, open mathematics. What’s New, 22 September 2026.

[14] Vladimir Lazić. Where I Stand on AI. August 2026.

[15] Max Weinreich. The crisis of AI-generated mathematics. arXiv:2608.02859v1, 3 August 2026; text updated 5 August 2026.

[16] Atticus Stonestrom. A proposal for mathematics journals in the age of AI. Undated essay.

[17] Martin Hairer. Writing mathematics in the age of AI. Proofs and Prompts, 7 August 2026.

[18] Tammy Kolda and Daniel Appelo. A Breakdown of arXiv Submissions in Mathematics. The Loss Function, 21 September 2026.

[19] Tara Abrishami. In August, 25% of math preprints acknowledged AI use, up from 4% in April. Epoch AI, 18 September 2026.

[20] Kimball Martin. AI use in arXiv NT papers. 5 September 2026.

[21] Advisory Group on Mathematics and Artificial Intelligence. Responsible Release of AI-Generated Mathematics. 29 September 2026.

[22] Leiden Declaration working group. Leiden Declaration on Artificial Intelligence and Mathematics. June 2026.

[23] Harvard Center of Mathematical Sciences and Applications. Summit on PhD Math Education in the Age of AI. 17–18 September 2026.

[24] Participants in the Harvard CMSA summit. Report of Summit on PhD Math Education in the Age of AI. September 2026.

[25] Ben Antieau. Fast math/slow math. What’s New, 15 September 2026.

[26] Daniel Litt. A beginning for mathematics. 13 September 2026.

[27] NYU Courant Institute. Tristan Buckmaster Constructs Finite-Time Blowup for 3D Euler with Smooth Forcing. 14 September 2026.

[28] OfficeChai Team. OpenAI Says It Didn’t See Any Of Buckmaster And Alpöge’s Work While Resolving Navier-Stokes. OfficeChai, 8 September 2026.

[29] Xiaoshn Nee, Haobo Zhong, and Xiaomin Ni. The Generative AI Gold Rush in Theoretical and Computational Research. arXiv:2609.04872, September 2026.

[30] OpenAI. ChatGPT — Release Notes. 29 September 2026.

[31] Anthropic. What is the Max plan? Claude Help Center, accessed 1 October 2026.

[32] Donatella Iacono. Where I stand on AI. Author’s website, 2026.

[33] Jonny Evans. Statement on LLMs. Author’s website, undated.

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Received 2 October 2026, revised 19 October 2026.

Tancredi Schettini Gherardini, postdoctoral researcher based both at the University and at the Max Planck Institute in Bonn.

This reflection starts with a parable I learnt from a recording of Rabbi Dr Abraham Twerski. He came across a young man who was clearly enjoying a fish that he was eating, and asked: ‘Young man, why are you eating that fish?’. And the young man answered ‘because I love fish!’. The Rabbi said: ‘Oh you love the fish, that’s why you took it out of the water and killed it and boiled it? Don’t tell me you love the fish; you love yourself, and because the fish tastes good to you, therefore you took it out of the water and killed it and boiled it.’

The message is that so much of what is usually called love is actually fish-love, i.e. we don’t love the object/person per se but how it/they make us feel. My intuition is that the nature of one’s love for mathematics is intimately related with their views on AI.¹

What is it that drives a mathematician to persevere through pain and frustration until a conjecture is proven or disproven? Is it to establish that they are better than others? Is it the satisfaction which comes from reaching a goal that has required so much of their time and energy? Is it the pleasure of knowing the answer to a question they find really interesting? Supposedly, in practice, it will be a mixture of the three. But is there one which overweighs the other two? Is there one which should outweigh the other two?

The answer is deeply personal, and I shall not argue here whether one is better than another. My message is that to understand our own views on AI for mathematics, the most appropriate starting point is an introspective analysis of why we – not as a community, but we-ourselves – love mathematics. This might be a complicated and even disturbing process; but, as Aristotle said, “knowing yourself is the beginning of all wisdom”.

Let me articulate my argument in a way that is agnostic to the notion of what is a correct motivation for doing mathematics. If mathematicians are most concerned with (possibly interesting) answers to interesting questions, then it is quite hard to argue against the benefits of AI in the subject. One can argue about the negative consequences in the general picture that is wider than mathematics (like environmental damage, privacy issues, monopoly, increase of inequality and unemployment, not to mention the risk of extinction); these are all very serious and well-founded worries which would need many essays, and I shall not deal with them here.²

When restricting to the features that are more specific to our discipline, one can argue about the potential risks in mathematics, which I think mainly boil down to two aspects.

Firstly, losing some important habits and routines essential in the practice of doing good mathematics – mostly those which ensure a deep understanding of the material, such as insightful human interaction, long-term projects and frequent productive struggle, for instance. But, regarding this concern, we have all the tools to deal with it effectively. In fact, we can act upon it by following these good habits ourselves, adapting them in a meaningful way when needed, and transmitting them to the newer generations. To ensure that they keep being part of a mathematician’s workflow, we might require some adjustments on a few fronts, such as the publishing process (where it is desirable to cut out or at least flag AI slop), the hiring process, and many general criteria for judging the quality of a mathematician. This is not just completely in our hands, there is also (almost) complete unanimity: I have never heard any mathematician, optimistic or pessimistic about AI, ever saying that we should give up understanding as a fundamental part of the discipline. To summarise, the temptation of these tools might pull us unwillingly away from understanding, but this risk has already been identified and acknowledged by our community and we have the power to make practical changes discouraging and preventing misuse of AI; in fact a few of these changes have already been implemented and many are currently being discussed.³

The second risk, which I personally consider to be scarier, is that of mathematics losing appeal for young bright minds; once more, it is up to us to not let this happen, illustrating how AI can be embraced in a constructive way, without disrupting the essential features of our field and the understanding-related practices mentioned above. Maybe even more importantly, it is hard to believe that the conservative choice of refusing any use of AI could attract more smart kids to join our community.

As of today, I do not see bigger problems than these two, which are only potential. And so, what are the benefits?

Since a few months ago, most of us have had our own version of Namagiri Thayar. It does not reveal formulas in our dreams, like the Hindu goddess used to do with Ramanujan, but it does so while we are awake and sitting in front of our laptop.⁴
Our reaction to this fact seems a bit different from the one people had one hundred years ago: I did not find any references of mathematicians from that time complaining about these revelations, nor signing petitions for banning Namagiri Thayar from Ramanujan’s dreams. But now that we all have something equivalent, a different chord was struck in the collective consciousness of our community. A purely emotional one, I argue; because if one’s priority is really the progress of mathematics, then why not embrace this new reality, made of Ramanujan-like moments, driven by AI suggestions, constructions, and results, and Hardy-like moments, trying to make sense of these results, with rigour, clarity and sharp understanding. If one’s love for mathematics is a genuine strive for advancement, and not simply the fish-love of Twerski, then it is impossible to not see the incredible chance we have to push our explorations in this beautiful subject.

The opposite is true if one’s love for mathematics mainly comes from the way that mathematics makes them feel. The feelings of finding a new clever way around a seemingly insurmountable obstruction, constructing a special interesting object that no one had thought of, or proving an important theorem for the first time, are all very much at risk. Because that new argument, example, or proof, could be generated by an LLM operated by someone who may or may not, in many instances, be an expert in the subject; in that case, the emotions which usually come with the mathematical discovery/invention are just gone.

The truth is that inside everyone’s passion for mathematics, and inside everyone’s love for someone or something, there is often a component of genuine and unselfish love, inevitably mixed with some fish-love. We are humans, after all. But if the former prevails, and we are driven by the sincere admiration of the exquisitely intellectual activity that has led humans from a few primitive logical observations to the immense and sophisticated body of knowledge that we have today, we might live the advent of AI with more optimism.

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Received 24 September 2026.

1. Throughout this essay, I will use ‘mathematics’ to mean ‘pure mathematics’ and ‘mathematicians’ to mean ‘pure mathematicians’. Many of the considerations below apply in general, but not all of them. ︎
2. Let me spend a few words on those global consequences which impact mathematicians as a part of society. The first one is the possible shortage of funding; as of today, theorems in mathematics are a perfectly inelastic good, and there is no good evidence to suggest they should change to be so. Concerning inequality, that is a real issue which should be dealt with; it might be mitigated by the improvement of open-weights models, but there is no guarantee. In any case, this social aspect is somewhat orthogonal to the one I would like to focus on, i.e. the development of mathematics itself. Regarding privacy and intellectual property, they are also real concerns, but it seems possible to deal with them. One can use incognito sessions for sensitive material and do a much more detailed literature search than just quoting the LLM’s output, respectively. ︎
3. One might argue that a foreseeable future scenario is that in which we delegate all of our thinking to the machines. Once more, that is ultimately up to us, and not a very likely prediction under mildly optimistic assumptions. Finally, if that will really be the case, we will have bigger problems than worrying about mathematics… ︎
4. With all the obvious caveats, of course, such as hallucinations (although very rare nowadays) and the abundance of AI slop. ︎

Breaking The Shower Containment Taboo For Hadron Calorimeters Tommaso Dorigo Wed, 10/21/2026 - 12:37

[This is a guest post by Caroline Turnage-Butterbaugh. This blog post was initially written in a different file format and converted using AI. — T.]

Evidence in favor of the Riemann Hypothesis comes in many flavors. OpenAI’s proof of the quasi-Riemann Hypothesis is definitely one such flavor; however, in this post I will focus my comments on a different kind of evidence: lower bounds on the proportion of nontrivial zeros of the Riemann zeta-function on the critical line. Great strides have recently been made on this problem, both with and without the use of AI. Many thanks to Terry Tao for inviting me to discuss this problem and the recent breakthrough results.

Background. Let , where are real. The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function have real part equal to 1/2. It arose out of Riemann’s foundational work, which led to proofs of the Prime Number Theorem by Hadamard and de la Vallée Poussin in 1896. If RH is true, then the error term in the Prime Number Theorem is essentially as small as possible.

Denote a nontrivial zero of by , where are real values and . The critical line is the line , and often we will refer to a nontrivial zero on the critical line as a critical zero. Riemann asserted that has infinitely many nontrivial zeros, and in 1914 Hardy proved that has infinitely many critical zeros. (See Chapter 10 of Titchmarsh for several proofs of this fact.) Platt and Trudgian have numerically verified RH up to , i.e. all of the nontrivial zeros for which have . (The nontrivial zeros are symmetric about the real line, so it suffices to consider those zeros for which .) The authors also show that these zeros are simple, which we expect to be true of every nontrivial zero. We set to denote the multiplicity of the nontrivial zero .

The recent proportionality results making headlines give a lower bound on the quantity

Here, denotes the number of nontrivial zeros for which , and denotes the number of zeros counted by which are both simple and critical, i.e. for which and . One can analogously define to count critical zeros and set

The Riemann–von Mangoldt formula gives a surprisingly precise asymptotic formula for ; for this discussion, it suffices to record that

(Here and throughout, means and is a positive function.) Recall that zeros of analytic functions are counted by the argument principle with their multiplicity. Therefore:

** It is important to keep in mind that we always count zeros with multiplicity.**

Previous key unconditional results on and . In 1942, Selberg proved that a positive proportion of the nontrivial zeros lie on the critical line. More specifically, he proved that there exists an effectively computable constant such that . (In his 1947 PhD thesis, S. H. Min calculated to be around .) In 1974, Levinson introduced a new method to show that , i.e. at least of the nontrivial zeros lie on the critical line. Soon after, he improved the proportion to . In 1979, Heath-Brown showed that Levinson’s method may be used to prove proportionality results on nontrivial zeros that are both simple and critical. Levinson’s method was extended by Conrey in 1989 to show that at least of the nontrivial zeros of are simple and critical. Over the years and with much effort and insight, members of the analytic number theory community have harnessed Levinson’s method to slightly improve these bounds, culminating with the work of Pratt, Robles, Zaharescu and Zeindler who proved in 2020 that at least of the nontrivial zeros of are on the critical line. The table below gives a snapshot of these results, with the precise proportions recorded.

Author(s)/Year
Levinson (1975), Heath-Brown (1979)
Conrey (1989)
Pratt, Robles, Zaharescu, Zeindler (2020)

A few key unconditional results on proportions of critical and simple zeros, prior to 2026.

Counting simple zeros under RH. In 1973, Montgomery published his groundbreaking work on the pair correlation of the zeros of . As a corollary to his main theorem, which we will state below, he showed that under RH at least of the nontrivial zeros are simple.

We now give a sketch of Montgomery’s argument for simple zeros. (To supplement the broad strokes which follow, I recommend Goldston’s expository paper.) This approach will lead us to the construction of Montgomery’s function , and, in turn, the statement of his theorem concerning the pair correlation of the zeros of .

Montgomery begins with the observation

Note that since the argument assumes RH, the zeros are necessarily critical! The goal is to determine a constant such that so that, when combined with (2), we may conclude . A key transition occurs after noting

(Keep in mind that we always count with multiplicity.) The sum on the right-hand side is a double sum over and , and with it pair correlation will soon enter the picture. Roughly, the idea is to find a nice function for which

where and when . (Note that we have normalized the difference by , the average spacing between zeros around height .) For , define the Fourier transform by

Then if , we have almost everywhere

Suppose for the moment that we have a function satisfying the properties above. Then proceeding formally and ignoring error terms, we arrive at an inequality of the shape

Montgomery showed how to estimate (a weighted version of) the inner sum on the right-hand side above by proving, under RH, an explicit formula relating the nontrivial zeros of and the primes. This weighted sum is given by

Note that and when . We may now state Montgomery’s Theorem.

Theorem 1 (Montgomery, 1973) Assume the Riemann Hypothesis. Then is real, even, nonnegative, and, as ,

uniformly for .

(That the theorem holds up to is due to Goldston and Montgomery.) There are scores of interesting things to say about this theorem and Montgomery’s pair correlation conjecture, but we will stay on task and return to the simple zeros argument. Assembling the chain of inequalities above and adjusting to accommodate the weight , we find that

Under RH, one can evaluate the RHS using Montgomery’s Theorem, provided and positivity is maintained. Inserting the Fejér kernel

into Montgomery’s theorem yields

Thus, by (6)

In other words, under RH at least of the nontrivial zeros are simple.

Counting critical zeros via pair correlation under a weaker-than-RH assumption. It was known among experts for some time that Montgomery’s Theorem could be made unconditional. Indeed, the starting point of the proof of Montgomery’s explicit formula is an unconditional explicit formula due to Landau from 1909 (see equation (9) on page 353). Setting , where , the same arguments for Montgomery’s explicit formula and theorem carry through. That one could use an unconditional form of Montgomery’s Theorem to extract information about critical zeros was recently discovered by Siegfred Baluyot, Daniel Goldston, Ade Irma Suriajaya, and myself; henceforth BGST-B. In our first project we did not realize this capability of Montgomery’s method, and erroneously stated in the abstract “The method of proof neither requires nor provides any information on whether any of these zeros are or are not on the critical line where .” In a follow-up paper we realized that, in fact, we could extract information on critical zeros using the pair correlation method, albeit upon an assumption weaker than RH. Roughly, we showed that if the nontrivial zeros of are contained in a narrow vertical box centered on the critical line, then the pair correlation method shows that at least 67.25% of the zeros are both simple and critical. I’ll give a sketch of the argument, highlighting some of the key distinctions from Montgomery’s work. Portions of what follows below appear in our preprint.

In addition to making use of the multiplicity of a nontrivial zero , we use a quantity we refer to as the horizontal multiplicity of . (This framework was suggested to us by Soundararajan and simplified our original classification of the ways in which can occur.) For a fixed real number, the horizontal multiplicity is the number of nontrivial zeros of on the horizontal line , counted with multiplicity. Assuming RH, is either zero or the multiplicity of the zero . Note that if , then on the line , there is one simple zero on the critical line.

Let denote the multiset of nontrivial zeros of for which . We can partition into disjoint multisets of zeros according to their horizontal multiplicity:

Consider the multiset of simple zeros in and the multiset of critical zeros in . Furthermore, note that

To obtain a lower bound on the proportion of simple and critical zeros, we begin with the following observation, which should be compared to (2):

Then, as in (3), we transition to a pair correlation argument upon noting

As in (4) we need to pick a kernel that maintains positivity. Tsang found a kernel that partially meets our needs; however, after rescaling for our setting, the positivity is only maintained provided the zeros are in a thin box centered on the critical line such that for some . Assuming this condition, we may use the unconditional version of Montgomery’s Theorem (see equations (3.2) and (3.3) in our preprint) to bound from above the contribution of the right-hand side of (8) in terms of the Tsang kernel. In particular, choosing yields .

Counting critical zeros via pair correlation unconditionally. On August 10, 2026, Anthropic announced that an unreleased research model of Claude had connected the BGST-B approach above with a 2000 paper by Bombieri to unconditionally prove that . The result was soon after verified by Alpöge and Furman, two mathematicians at Anthropic. Then, on September 2, 2026, Lamzouri announced a new, conceptually simpler proof of the same result. The entire preprint is very nicely written with helpful clarifications and remarks; in particular, Lamzouri takes care to compare/contrast his approach to that of Claude. With all of this in mind, I will focus this discussion on Lamzouri’s key ideas.

Lamzouri points out that the BGST-B approach cannot be made unconditional by merely finding a better kernel at the transition

Lamzouri, in fact, avoids this transition altogether by changing the mechanism with which simple and critical zeros are identified. As he helpfully writes on page 5,

Our key idea is that one can bypass the inequality [(9)] and directly prove, for a large class of kernels , that the number of simple real elements of a finite multiset which is invariant under complex conjugation is

without requiring that the individual off-diagonal terms be non-negative.

Instead of setting to be the multiset of nontrivial zeros for which as in BGST-B, he takes it to be the multiset

This is such a clever idea! This multiset is invariant under complex conjugation, i.e. if then and ; its real elements are those with corresponding zero satisfying ; its real and simple elements are those with corresponding zero being critical and simple. As Lamzouri notes in Remark 2.2, “the square [i.e., in ] is also naturally adapted to the Hilbert space argument in the proof [of Proposition 2.1] and to the use of Bessel’s inequality.” This new framework allows Lamzouri to bypass the portion of the previous arguments that required an assumption on the location of the nontrivial zeros.

Concluding remarks. Since the breakthrough results by Anthropic and Lamzouri, subsequent minor improvements have been made. (I will refrain from quoting the strongest result at the time of writing, since it will likely be out of date by the time this appears!) It is important to keep in mind that even if we increased 67.25% to 100%, the Riemann Hypothesis would not be solved. (Refer back to the precise definition of which appears in (1).) Nevertheless, the old and new ideas which have come together to realize these breakthrough results have expanded our understanding not only of the nontrivial zeros of the Riemann zeta-function, but, more broadly, of how different techniques were used, pushed, and molded to breathe new life into this problem.

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

This blog post expands on a theme mentioned at the end of my recent talk at the Institute for Advanced Study on October 19.

I would like to thank Matei Coiculescu, Jaume de Dios Pont, Nestor Guillen, Bryna Kra, Terence Tao and Rachel Ward for their valuable comments and suggestions.

It is often said that mathematics is a young person’s game, and that mathematicians do their best work early in their careers. I believe this is about to become true in a completely different sense. Mathematics is changing faster than at any point in its history. Today’s graduate students and postdocs will shape its future. Our responsibility is to protect their ideas and recognize their contributions.

In my statement on September 7th, the night Levent Alpöge and I released our Euler result and the day before OpenAI announced its Navier–Stokes result, I wrote that we had reached a Deep Blue–Kasparov moment. When Deep Blue beat Kasparov in 1997, the headline was that a machine had surpassed one of the greatest chess players of all time. Beyond just marking the moment a machine eclipsed human intellect, I chose this analogy because of the nature of the machine itself. Deep Blue was a purpose-built IBM supercomputer, entirely inaccessible to the general public. Today a program on a mobile phone plays far stronger chess than Deep Blue ever did.

This is where I think mathematics is now. OpenAI’s proof reportedly required the coordination of 10,000 agents over 88 hours, at an estimated cost in the millions to solve a problem with a $1 million bounty. At present, computation on this scale is entirely out of reach for the wider mathematical community. However, given the exponential growth in capabilities paired with decreasing costs, such a comparable computation may soon be more broadly accessible.

The question is no longer whether machines can do research mathematics, but how the subject will change as a result.

Chess did not die with the advent of modern chess engines. Engines democratized high-level preparation, resulting in the most globally diverse generation of top players ever. When I told a reporter that “the game is up,” I meant to describe the end of the academic rat race, the race to be the first to resolve a famous open problem is over. Mathematics as a subject is not.

Mathematics differs from chess in one important way, though: it is not performative. Chess survives partly because people love watching humans play. I am not going to convince anyone to watch two Fields medallists duke it out on a livestream proving a theorem. What gives it lasting relevance is something else: human understanding.

Suppose that tomorrow a machine certifies a proof of the Riemann hypothesis — whether with pages of AI slop or via a Lean file. No one doubts that the Riemann hypothesis is true. A certification would convert thousands of conditional theorems into unconditional ones. Mathematicians would still be left with the question: why is it true? De Toffoli and Duede captured this perfectly with their reference to Deep Thought, the computer in The Hitchhiker’s Guide to the Galaxy that analyzes the meaning of life for 7.5 million years and returns the answer “42.”

Mathematics has never been a list of true statements. Certifying a result settles the question, not the search for understanding.

Bridging mathematics and physics: One area I see as an exciting prospect for applying modern AI systems is the intersection of mathematics and physics. Indeed, I do not believe it is a surprise that some of the most exciting results to come out of AI so far have been in mathematical physics. Historically, mathematics and physics were inextricably linked, each serving as inspiration for the other: Newton developed calculus to formulate classical mechanics, Fourier analysis grew out of the study of thermal diffusion, Riemannian geometry supplied the mathematical underpinning of general relativity, and the needs of quantum mechanics shaped functional analysis. In the past decades, however, mathematics and physics have diverged. Physics is driven by heuristic arguments that are not true in the strictest sense, but are morally true and allow one to arrive at the correct answer. Quantum field theory is the standard example, where renormalization and the path integral make remarkably accurate predictions while still lacking a rigorous foundation. Closing the gaps in physical derivations is immensely challenging, and this is a key reason for the divergence of the fields. Rigorous derivation is not just an exercise in pedantry. It points to gaps in the physical theory, identifies the precise regimes and time scales on which an effective description is valid, and provides clues towards a more complete theory.

In years preceding the advent of AI in mathematics, a beautiful example of success in this pursuit, is Deng and Hani’s derivation of the wave kinetic equation as a large box limit of the nonlinear Schrödinger equation. Later, with Ma, they gave an analogous derivation of the Boltzmann equation from hard-sphere dynamics for as long as the Boltzmann solution exists, resolving Hilbert’s sixth problem for rarefied gases. The combinatorial analysis behind these results is a tour de force, and it took years of work. LLMs could potentially allow us to carry out such a program for other effective theories, where doing it by hand would frankly be out of reach. Ultimately, one could hope that LLMs will allow us to extend this standard of rigorous proof to all branches of the physical sciences.

Protecting our ideas: Currently, there are some fundamental structural problems in terms of how AI companies interact with the mathematical community.

AI companies, who strip mine our data, including unpublished work, private notes and research grant applications, have an egregious conflict of interest in competing against their own customers with unlimited resources and internal models unavailable to the public — this is not confined to mathematics but applies more broadly. By OpenAI’s own account, its push on the Millennium problems was spurred by rumors that a rival had solved two of them — demonstrating that caution, due diligence, and basic ethics can be cast aside simply to avoid being one-upped by a rival.

On October 6th, OpenAI released hundreds of manuscripts with proofs that were mostly unreadable and in some cases false. In the release, they misleadingly claimed that the work took on average three hours of computation on a single agent: they later admitted that they did not count numerous failed attempts at the problems as well as previous work done by other agents — rendering the three hours claim essentially meaningless. Again, OpenAI did essentially no due diligence, failing to investigate whether once again they may be building on the private, unreleased work of their customers, some of whom were graduate students and postdocs who were literally days away from releasing their own work.

In August, OpenAI claimed that such a dump would be `distasteful’, that their `main focus is shipping great models so everyone can use them to make discoveries of their own’. After Navier–Stokes, OpenAI had teased us with supposed solutions to additional Millennium problems, yet they achieved none. A natural question is to ask why OpenAI would go through with such a PR stunt that risks alienating their customers. These companies are spending tens of billions of dollars a year more than they earn, which helps to explain their thirst for PR stunts and their willingness to take such seemingly inexplicable risks.

Mathematicians now type half-formed ideas into commercial models every day. These sessions are invaluable to an AI company: they provide a road map of our thought process, and reveal which research directions are promising. Until recently, AI-assisted proofs of important mathematical results were considered a pipe dream, and many of us were simply not concerned with data privacy since we never imagined today’s capabilities arriving so quickly.

It is imperative to protect our ideas and not let AI companies become the ultimate gatekeepers of knowledge. In this vein, I would like to see:

• Collective enterprise agreements with AI companies should become ubiquitous. Many universities already have institutional agreements with AI providers, and these offer far better protections than consumer agreements. Even so, such agreements could be further strengthened by including zero data retention clauses and auditing provisions to ensure that the conditions of the agreements are respected. Many mathematicians, myself included, do their research on consumer subscriptions that those agreements do not cover. We need to make sure all academics work under the protection of such agreements.
• We need stricter regulatory guardrails on user data. More generally, we need stricter regulations governing how AI companies collect data. Training on user data must be explicitly opt-in rather than opt-out. Finally, deceptive practices, such as absorbing an entire session’s worth of data as a result of a user providing feedback on a single response, must be explicitly banned.
• We should further invest in open-weight infrastructure, assisting in the development of open-weight models. We cannot assume the interests of Academia and the AI companies are aligned. Academic institutions that control where their models are run can circumvent the fear of having their ideas stolen by trillion dollar corporations. Chess was democratized by engines anyone could run at home. Shared, open infrastructure for mathematics, funded in an independent way, is a lofty goal worth pursuing.

Less reliance on user data benefits both sides: leading to more trust from the academics and the encouragement of new innovative ways to train models.

Recognizing contributions: As Tao put it at the ICM this summer, we are moving from an era of proof scarcity to an era of proof abundance. Our credit system was built for the era of scarcity. Being first to a famous result was a reasonable proxy for depth of understanding, because nobody could get there without understanding a great deal along the way. That proxy no longer holds and so we will need new ways of assigning credit that are not simply correlated with access to resources.

Some concrete suggestions:

• Reward well written exposition of important results in mathematics.
• AI disclosures are a mixed bag and there is currently a perverse incentive not to disclose the use of AI. Disclosing AI assistance should not diminish a result and detailed disclosures should become standard.
• Development of shared tools and resources, as well as contributions to large collaborative projects should earn recognition, mimicking practices in other sciences.
• Mathematicians should be evaluated on the ideas and methods they developed, not just on whether or not they were first to prove a result.

More generally, we will need to move away from bean counting publications in top journals. Whatever replaces it must still offer transparent ways of evaluating work, rather than relying on reputation and personal connections, which tends to produce closed networks and patronage.

The future of mathematics: Chess did not end in 1997. Mathematics will not end in 2026. I am however convinced that within the next few years it will be unrecognizable. That future will not be shaped by me or my peers. It will be shaped by the next generation.

I am already amazed by what our students and postdocs at NYU are doing with these tools. I suspect that in their hands the line between pure and applied mathematics will blur further, perhaps disappear, and that we will see a richer connection between mathematical proofs and the physical sciences.

For the next generation of mathematicians to flourish, we need protections and control over user data, restraint from AI companies, a lower barrier to entry to the use of frontier-level models, and a new credit system that rewards understanding rather than access. Mathematics has long been said to belong to the young. They will decide what it becomes.

AI Disclaimer: I used LLMs to help edit this post, including suggesting wording for some passages.

I don’t feel like writing a whole dissertation about this but I’ll just say that this is the first time I can remember saying “Oh my God,” out loud, multiple times, while watching a movie in a theater.

Some people have asked if you need to be familiar with the story of Elizabeth Holmes to watch this movie. I don’t think so. I’m not familiar with the Holmes/Theranos story beyond the bare basics. This movie is almost entirely not about that story; or at least that’s how it plays if you don’t know the story.

For a given prime , the Legendre symbol is defined as follows: if is a quadratic residue modulo , then ; if is a quadratic non-residue modulo , then ; and if divides , then .

Let be an integer that is not a perfect square. (One could require to be a fundamental discriminant if desired, although it turns out to be immaterial for the arguments here.) By combining quadratic reciprocity with the prime number theorem in arithmetic progressions, one can show that the Legendre symbol is asymptotically balanced between and . For instance, one can show that

as , where the sum is over primes ; equivalently (by Mertens’ theorem),

However, most of the known proofs of this asymptotic are ineffective with regards to the decay of the error. This is due to the infamous Siegel zero problem, arising from Dirichlet -functions associated to with a zero very close to . A well known example comes when , which is a non-residue for all primes ; this is closely tied to Euler’s famous observation that is prime for all integers with , as well as the fact that the ring of integers has unique factorization. For further discussion of these topics, see this previous set of lecture notes.

Recently, the Siegel zero problem has been solved in a number of ways. In fact, a stronger “quasi-Riemann hypothesis”, showing that the zeroes of all Dirichlet -functions have real part strictly bounded away from , has been established (with two separate papers giving a bound of and respectively) as part of a recent release by OpenAI. (In particular, the error term in (1) can now be improved to a power saving .) These arguments are analytic in nature, using identities relating the Möbius function to the Fourier coefficients of certain automorphic forms; they are still in the process of being fully digested, and I will not be discussing them further here. However, there was a third paper in the same release that did not establish the quasi-Riemann hypothesis, but instead excluded Siegel zeroes directly. This paper was significantly shorter (nine pages) and proceeded almost entirely via algebraic number theory and the geometry of numbers.

Recently, various digestions and simplifications of the argument have been produced by Banks, by Kerr, and by Green. I would like here to present a sketch of Green’s formulation (with some small notational changes), which I find particularly elegant. The key bound he establishes is

whenever is a small quantity, is a suitably large scale, and is a larger scale that is nearly quadratic in . For small enough, this improves by a factor of about upon the trivial bound

coming from Mertens’ theorem; thus (2) is a non-trivial bound on the number of primes at scales up to for which is a quadratic non-residue. By optimizing in , this gives the upper bound for (1); in fact with more care one concludes that

(In fact, some extremely recent preprints of Jacob Pescador and Kevin Barretto have sharpened this bound further, with an upper bound of .)

One can obtain the full strength of (1) using standard analytic number theory methods (omitted here, but see for instance these previous notes) to rule out Siegel zeroes, which can then in turn be used to upgrade the error term in (1) to a classical shape for some absolute constant , assuming . Besides this, though, none of the classical analytic number theory methods are utilized in this proof. In retrospect this makes sense, as the infamous “parity problem” blocks such methods from getting results such as (2) without some parity-breaking input: in this case, said input comes from the polynomial method, whereas in the QRH papers, it appears to be some (still poorly understood) automorphic miracle that is breaking the parity barrier.

AI disclosure: a chat session with the new SAIR open math model was used to help understand the preprints. I also benefited from conversations with Ben Green, Alex Pascadi, Lior Silberman, Valentin Blomer, Gergely Harcos, Farrell Brumley, William Banks, and Edgar Assing.

— 1. High-level description of proof —

I will make some minor assumptions and “cheats” to simplify the exposition. Firstly I will restrict to the case when is negative; this makes have one complex embedding rather than two real embeddings, which allows for a very small simplification to the notation. For extremely minor notational reasons I will also exclude , so that . I will also pretend at a later point that an arbitrarily chosen high-degree polynomial is irreducible, which is of course not always the case, but allows one to simplify a certain algebraic calculation, the general case of which is handled in Green’s writeup.

The overall strategy is that of the “adelic polynomial method”, which appears in various guises (though usually not with this name) throughout the literature, for instance in the Bombieri–Vaaler paper on Siegel’s lemma (I would be interested in further historical information on the development of the method). We will not actually use any of the machinery of the adeles in this post, though experts will readily see them lying just beneath the surface.

The first observation is just the fundamental theorem of arithmetic, which gives the factorization

for any non-zero integer , where is the number of times divides . Taking logarithms, we obtain

So, if one wanted to obtain a bound such as (2), one could try to construct some non-zero integer for which one had some upper bound on the “Archimedean magnitude” of , while also having lower bounds on the valuations for primes in the range of (2) for which is a quadratic nonresidue. That is to say, one wants to find an which is not too large, while also being divisible by many powers of such primes , and also non-vanishing.

This quantity will be constructed geometrically, being essentially the covolume of a certain lattice (actually the square of that covolume, to make it an integer). Let be a small multiple of . First let be the set of polynomials with integer coefficients that are of degree less than in the two variables , by which we mean that they are of degree less than in the variable, and less than in the variable. The behavior in will be more important for us than that in , and this is reflected in the disparity in sizes between the two degrees .

Clearly is a lattice of rank . We will evaluate this lattice on the grid

The idea of evaluating a lattice or vector space of polynomials on various carefully selected sets such as lies at the heart of the polynomial method, with a focus often being made on the extent to which such polynomials can vanish on . In many variants of the polynomial method, one also analyzes the extent to which such polynomials can vanish with multiplicity on ; Stepanov’s method (discussed for instance in this previous post) is one well-known example. One of the innovations in Green’s treatment is to also take advantage of multiplicity to achieve the non-trivial bound (2); previous treatments did not do this, and had to work with a (somewhat artificial) extension of , such as , in order to get comparably non-trivial bounds.

Specifically, for each polynomial in , we consider the evaluations

of the first horizontal derivatives of at each point of , where the optimal choice of turns out to be , though as these two quantities play conceptually different roles we will continue to denote them by different symbols.

The tuple is an element of , and so is some additive subgroup of . A key fact is:

Lemma 1 (Injectivity) is injective on . In other words, if is a non-zero element of , then cannot vanish to (horizontal) order at every point of .

We will prove this later using the theory of resultants.

Thus is a lattice of rank . To this lattice, we can prescribe a (squared) covolume as follows. Let be some basis for this lattice, then form the Gram matrix

where we embed into in the usual fashion with the Euclidean inner product for the purpose of defining the (real) inner product. This Gram matrix depends on the choice of basis, but the determinant does not; we then take to be this determinant. Geometrically, one can think of as the square of the volume of a fundamental domain of .

Some basic facts about . Firstly, the injectivity in Lemma 1 shows that is non-zero. As all the coefficients of the basis vectors lie in the ring , the inner products are all integer, and so is also. Hence is a non-zero integer. (It is also positive, although this is irrelevant for us.) In particular, the fundamental theorem of arithmetic gives (3).

It is easy to bound the Archimedean size of :

Lemma 2 (Archimedean bound) We have

This will be a straightforward application of Hadamard’s inequality, using the obvious monomial basis, we prove this later in the post.

Now for the arithmetic input. Suppose that is an (odd) prime for which . From Fermat’s little theorem we have

and therefore

From further application of Fermat’s little theorem (and the Frobenius automorphism), this implies that

for any integers . In particular, from a -adic point of view, the grid lies close to the curve

If the grid happened to lie exactly on this curve, then the lattice would degenerate (contradicting Lemma 1) and would now vanish. This of course does not happen; but the fact that is -adically close to the curve will force to be -adically close to zero, or equivalently give a useful lower bound on :

Lemma 3 (Non-Archimedean bound) If and , then

We prove this bound later in the post. Inserting Lemma 2 and Lemma 3 into (3) and cancelling the common factor of , we obtain (2) as required.

— 2. Injectivity —

Now we establish Lemma 1. From my understanding, these sorts of non-vanishing theorems go back to the work of Masser (and the Kerr preprint explicitly uses Masser’s bounds to obtain an analogue of this lemma), which initially used a fair amount of ring theory, but a more elementary alternate approach based on the interpolating polynomial method was introduced by Laurent, an adaptation of which is used in the OpenAI release. Green’s approach is simpler still, based largely on the theory of resultants to project the (high-order) vanishing of a two-variable polynomial to a (high-order) vanishing of a one-variable polynomial.

As is an grid, a direct application of the Schwartz–Zippel lemma tells us that a non-zero degree polynomial can only vanish on at most of the points of . This is not directly useful for us because the degree of can be as large as , and so this does not prevent from vanishing on all of . And indeed, if we replaced by the standard grid

then the degree polynomial already vanishes on , as does all of the horizontal derivatives of this polynomial. So to proceed here we have to somehow use the fact that the grid is “skew”; in particular we will take advantage of the fact that the second coordinates of this grid are all distinct, in contrast to the situation with .

Let be a non-zero polynomial of degree less than . To avoid technicalities we will assume that is irreducible, and not solely a function of , so that is not identically zero; the general case is treated in Ben Green’s note.

Suppose that vanishes horizontally to order at least at a point , then by performing a Taylor expansion around that point we can factor as

for some polynomials of degree less than and respectively. If we take one derivative in the direction, then vanishes to order at least at the same point, thus

for some polynomials of degree less than and respectively.

Since and both vanish to high order at , it is natural to look at the horizontal resultant of these two polynomials (which we now view a -valued polynomials of ), defined as the determinant of the polynomials

each of which we view as vectors of coefficients in . Note that all the polynomials in this list take the form

for some polynomials of degree less than and respectively (here we drop some irrelevant lower order terms). By multilinearity of the determinant, the resultant now splits into a linear combination of various determinants involving expressions of the form and expressions of the form . As is of degree less than in the variable, any contribtion with must vanish, and so we need to have at least factors of the form . We conclude that the resultant is divisible by .

On the other hand, a straightforward degree count shows that this resultant is a polynomial of of degree . The hypothesis that is irreducible and implies that the resultant is non-zero. So if vanished to order at least at each of the points of , then the resultant would have to be divisible by different factors of (here we use the fact that the are all distinct), leading to the inequality

which is false for small enough by the choice of , , , and . This proves Lemma 1.

— 3. Archimedean bound —

The Archimedean bound is straightforward. For any basis of , the quantity is simply the square of the -dimensional parallelepiped with edges , viewed as elements of . By the Hadamard’s inequality, we thus have

(using the usual Euclidean norm), or upon taking logarithms

Applying the sampling operator to the monomial basis , we thus have

Each point in the grid is of size by choice of . Every component of then has size , and so

or on taking logs and using the construction of

Summing the arithmetic series in we obtain

By construction we have and , so Theorem 2 follows.

— 4. Non-Archimedean bound —

Let be a prime with and . As discussed in the high-level overview, the polynomial takes values in on the entire grid . This has the following consequence: any polynomial which is divisible by for some will take values in . Furthermore, the same is true for any iterated -derivative of : note from the chain rule that applying to will lower the exponent by , thus removing one factor of , but also produce an additional factor of which restores that missing factor. We conclude that every coefficient of then lies in .

To use this, we replace the standard monomial basis , , of with the modified basis

for , , where

Note that such a basis element still lies in , and is of the form plus a linear combination of monomials that are lower than this in the lexicographical ordering. So this modified basis remains a basis of . Applying , we see from the previous discussion that each of the basis elements is divisible by . Computing the Gram matrix in this basis, we conclude that

For , one has by the hypotheses on and the choice of . We thus have

and Theorem 3 then follows by summing the arithmetic series.

Understanding Intelligence Across Artificial and Biological Systems Tommaso Dorigo Tue, 10/20/2026 - 10:15

As I announced on this blog a couple weeks ago, I’m teaching a new course this semester at UT Austin, entitled AI Alignment Theory. To call such a course “timely” seems like an understatement; this is the first course I’ve ever taught for which multiple gargantuan developments relevant to the course have happened while I was teaching it.

Since AI alignment is still a field with no standard textbook (although that might be changing), no accepted body of mathematical theory, etc. etc., the way we’ve been running the course is like a humanities discussion seminar, albeit one that happens to be in a CS department and to include technical results from time to time. Basically, in each class session, a student “rapporteur” gives a talk on an AI alignment reading—which include “ancient” readings (from way back in 2016), “modern” readings (from, say, last week), and everything in between—and then we freely discuss and debate whatever was presented, with my own role mostly limited to that of gadfly (occasionally I’ll do a mini-lecture on some technical concept).

Anyway, two weeks ago, I promised a list of the papers and other writings that we’ve been discussing, as well as student “rapporteur reports” on those papers, which cover not only the papers themselves, but the students’ thoughts about the papers and (anonymized) summaries of points raised during our class discussion. People have been bugging me for that ever since.

Since I am to please, here you are, without further ado! I’ll continue to update the list as the semester progresses, so that this post can serve as a central public repository for the course.

1. AGI Ruin: A List of Lethalities, by Eliezer Yudkowsky (2022). Rapporteur report by Tennyson Bardwell (this is the one that we posted previously).
   
2. Concrete Problems in AI Safety, by Amodei, Olah, Steinhardt, Christiano, Schulman, and Mané (2016). Rapporteur report by Daniela Wiepert.
   
3. The Alignment Problem from a Deep Learning Perspective, by Ngo, Chan, and Mindermann (many versions but latest is 2025), and Research Priorities for Robust and Beneficial Artificial Intelligence by Russell, Dewey, and Tegmark (2016). Rapporteur report by Oleg Pittman and Jack Holt.
   
4. What Happened: OpenAI and HuggingFace, by Zvi Mowshowitz; The Rise and Fall of Agent Civilizations, by Dwarkesh Patel, and the METR report by on the HuggingFace incident, by Greenblatt, Cotra, and Wijk (all 2026). Rapporteur report by Shourjo Aditya Chaudhuri and John Robertson.
   
5. We Must Pace the Frontier, by Dario Amodei (2026). Rapporteur report by Dhilan Shah.
   
6. The Basic AI Drives, by Stephen Omohundro (2008), and Optimal Policies Tend to Seek Power, by Turner, Smith, Shah, Critch, and Tadepalli (2023). Rapporteur report by Bill Zheng.
   
7. Corrigibility, by Soares, Fallenstein, Yudkowsky, and Armstrong (2015). Rapporteur report by Sukrut Chikodikar.
   
8. The Off-Switch Game, by Hadfield-Menell, Dragan, Abbeel, and Russell (2017). Rapporteur report by Praneel Seth.
   
9. Planting Undetectable Backdoors in Machine Learning Models, by Goldwasser, Kim, Vaikuntanathan, and Zamir (latest version 2024), and Oblivious Defense in ML Models: Backdoor Removal without Detection, by Goldwasser, Shafer, Vafa, and Vaikuntanathan (2025). Rapporteur report by Agrim Jain and Indira Dutta.
   
10. Backdoor Defense, Learnability, and Obfuscation by Christiano, Hilton, Lecomte, and Zhu (2024), and Unelicitable Backdoors in Language Models via Cryptographic Transformer Circuits, by Draguns et al. (2024). Rapporteur report by John Dunbar (coming).
    
11. Functional Decision Theory: A New Theory of Instrumental Rationality, by Eliezer Yudkowsky and Nate Soares (2018). Rapporteur report by Isaac Noel (coming).
    
12. Robust Cooperation In the Prisoner’s Dilemma: Program Equilibrium via Provability Logic, by Barasz, Christiano, Fallenstein, Herreshoff, LaVictoire, and Yudkowsky (latest version 2021). Rapporteur report by Shunan Zheng (coming).
    
13. Sleeper Agents: Training Deceptive LLMs that Persist Through Safety Training, by Hubinger et al. (long list of authors) (2024). Rapporteur report by Anthony Wang (coming).
    
14. Social Choice Should Guide AI Alignment in Dealing with Diverse Human Feedback, by Conitzer et al. (long list of authors) (2024) and Toward a Theory of Value in AI Alignment, by Ahmed, Birhane, Smart, Kay, Shrishak, and Tobin (2026). Rapporteur report by Priyanka Sadagopan (coming).
    
15. Claude’s Constitution, by Amanda Askell, Joe Carlsmith, Chris Olah, Jared Kaplan, Holden Karnofsky, et al. (2026). Rapporteur report by Alan Li (coming).
    
16. AI Safety via Debate, by Geoffrey Irving, Paul Christiano, and Dario Amodei (2018). Rapporteur report by Sampoorna Poria (coming).
    
17. Scalable AI Safety via Doubly-Efficient Debate, by Jonah Brown-Cohen, Geoffrey Irving, and Georgios Piliouras (2023). Rapporteur report by Aarushi Lakhi (coming).
    
18. AI as Normal Technology, by Arvind Narayanan and Sayash Kapoor (2025). Rapporteur report by Weixi Chen (coming).
    
19. On the Dangers of Stochastic Parrots, by Emily Bender, Timnit Gebru, Angelina McMillan-Major and Shmargaret Shmitchell (2021). Rapporteur report by George Tsoukalas (coming).

(More readings TBD, until the semester ends!)

Thanks to my TA, Lin Lin Lee, for her help in getting all of this together.

Update: I welcome elevated discussion of these papers and rapporteur reports in the comment section; some of the students might chime in too! Because of the connection to an academic class, though, in this post I’ll be especially ruthless in keeping out nonconstructive or off-topic comments.

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.

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-Seiberg” 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

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 semi-simple.

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

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

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

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 $(\mathbb{C}^\ast\times\mathbb{C}^\ast)$-fibration (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

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

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 semi-simple (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.

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Generalizations

The Argyres-Seiberg example (2 minimal and 2 regular Hitchin nilpotents) generalizes straightforwardly to $SL_N$. $C=M_1M_2$ is generically 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 $\theta=d_s/d_s'$. 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 $([2n],Sp(n))$.

Special Loci

Throughout, I’ve been describing the behaviour at a generic point of the character variety. The monodromies we found were semi-simple. There are special loci where the behaviour is more complicated.

Consider again the collision of 2 simple punctures (the Argyres-Seiberg example above). When $t\coloneqq \mathrm{tr}(C)= (N-2)\nu \pm 2\nu^{-(N-2)/2}\eqqcolon t_\pm$, then $s=s'=\pm\nu^{-(N-2)/2}$. On that locus, $C=M_1M_2$ develops a nontrivial unipotent part: $$[C]=\mathrm{diag}(\underset{N-2}{\underbrace{\nu,\dots\nu}},J_2(s))$$ $Z(C)$ is no longer reductive, so the twist coordinate $\theta$ is no longer an element of $\mathbb{C}^*$. Now it’s a translation $\theta\sim\left(\begin{smallmatrix} 1&\tau\\0&1\end{smallmatrix}\right)$ and Dehn-twists act on it by translating $\tau$.

Circling this divisor ($t=t_\ast$) in the character variety takes the twist coordinate $\theta\to\theta^{-1}$, a Weyl reflection. So the twist coordinate(s) are only good local coordinates on the character variety.

A good local model near this divisor is as follows. Let $C_W$ denote the $2\times 2$ matrix which is $C$ restricted to $W$. Set $$\sigma=\tfrac{1}{2}\mathrm{tr}(C_W),\qquad \epsilon =\tfrac{1}{4} (t-t_+)(t-t_-)$$ We can decompose $$C_W = \sigma I + T, \qquad T^2=\epsilon I$$ The twists take values in $$Z(C_W)/\mathbb{C}^\ast = \{a I+b T\}^\times/\mathbb{C}^\ast,\quad \text{i.e.}\; \{[a:b]\in \mathbb{P}^1|a^2-\epsilon b^2\neq 0\}$$ Dehn twists act on the homogeneous coordinates of the $\mathbb{P}^1$ as $[a:b]\mapsto [\sigma a+\epsilon b: a+\sigma b]$.

Similar remarks apply whenever any of the free eigenvalues of $C$ are tuned to coincide. Along the corresponding divisor, $C$ develops a unipotent part ($C=C_{\mathrm{ss}}C_{\mathrm{u}}$ with $C_{\mathrm{ss}}C_{\mathrm{u}}=C_{\mathrm{u}}C_{\mathrm{ss}}$). Circling the divisor, the twist coordinates undergo a Weyl reflection.

“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:

Oh, and a third one: MTS podcast with Theo Jaffee.

Enjoy, and thanks to the hosts of these podcasts!

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Update (Oct. 19): There’s been a rapid-fire recent sequence of AI-assisted improvements to my 2016 result with Adam Yedidia, showing that the 8000th Busy Beaver number eludes the axioms of set theory. I haven’t even kept track of all of them, but the very latest, by friend-of-the-blog Elliot Glazer, shows that the value of BB(230) is independent of ZFC (see here for explanation).

First, why a video?

I am part of a generation that was raised on a nourishing diet of science communication videos on YouTube. The likes of MinutePhysics, Veritasium, VSauce, Crash Course, TED-Ed, Physics Girl, 3Blue1Brown — you name it, these channels were my childhood. Had they not existed, I may very well have never considered pursuing a physics PhD here at Caltech.

But you probably already knew about the power of science communication in its ability to inspire the next generation. I am here to convince you that science communication just became a whole lot more important.

Earlier this month, Terence Tao gave a talk at Caltech about the growing use of AI in mathematics. In light of OpenAI’s recent release of hundreds of AI-generated mathematical manuscripts, he argued that our community has transitioned from a long era of proof scarcity to one of proof abundance. As a result, problem-solving may no longer be the central challenge in mathematics and theoretical physics that it once was. What surprised me most, though, was Terence’s claim that problem-solving was never the central goal in the first place. It was only ever a proxy for the actual goal of these disciplines: to elevate human understanding.

I do not think we have fully internalized this maxim as a community yet. The arXiv is seeing a flood of AI-generated submissions that claim to solve problems but offer very little insight. So how do we encourage human understanding?

Perhaps a video recording is one good way. Flagship conferences like STOC 2027 have begun requiring video recordings as part of their submission process to prioritize effective scientific communication and filter out low-insight AI-generated submissions. On one level, requiring authors to explain their work over video establishes a standard that scientific research should remain intelligible to others.

But on another level, a video recording captures something of the humanness of the act of understanding itself. When I understand something, I feel like I can taste it, almost as if it were a sensory experience. I then feel empowered, as if I have been given a new tool to use, and subsequently feel an excitement to share that insight with other human beings. In other words, understanding is an embodied phenomenon. And what better way to embody your work than to stand before a camera and explain it?

Science communication is no longer just about communicating science to people outside the field. It is increasingly about ensuring that the scientific progress we make be accompanied by a corresponding growth in human understanding. So go out there and record yourself geeking out!

About the work

Quantum spin glasses are, in some ways, very quantum systems. Their Hamiltonians contain many non-commuting terms, and their low-energy states can be far from simple product states that are easy to represent classically.

One might therefore expect these systems to be difficult to simulate on a classical computer. Yet, in our recent work, we show strong evidence that there is no exponential quantum advantage for sampling from the thermal states of one such quantum spin glass model.

The key is to reduce sampling to estimating certain quantities related to the model called tilted means, which we expect to be classically easy. I say expect because our results rely on a conjecture from spin-glass physics that is widely believed to hold true, but has yet to be rigorously proven with existing mathematical tools. Such assumptions are quite common in the spin-glass community.

The video above focuses more on why we wanted to study sampling than on how we actually solved the problem in order to make it accessible to a general audience. For those curious, our approach is based on stochastic localization, a sampling technique closely related to diffusion models. More technical details can be found in our arXiv preprint.

A little over two months ago, Jacob Tsimerman surprised a lot of people by announcing, immediately after he had received a Fields Medal, that he would be giving up mathematical research and “pivoting to AI safety”. He had two reasons for this: that he did not see much future for the kind of research in mathematics that he liked doing, and that he was very worried about the dangers of AI, and in particular of the risk that it could kill us all, or x-risk as it is often known. (In this post I’ll use x-risk to refer to the extinction risk from AI rather than to extinction risk in general. I’ll also refer to “the probability of x-risk” when what I actually mean is the probability of extinction by AI.) Then in early September, he announced that he was setting up the Mathematical AI Safety Institute, or MAISI. He asked me if I would be prepared to be on the scientific advisory panel for this, and I agreed to do so, not because I am an expert, or because I felt as concerned as he did about x-risk, but because I don’t really know how much I ought to be worried about it. Given that, and given that the stakes are so high, I would like as many people as possible to be thinking seriously about it, and I agree with Jacob that mathematicians have the sort of training that makes them well suited to contribute. Jacob is the first to admit that something like an international treaty to slow down AI development is likely to have a far bigger impact on AI safety, especially in the short term, than mathematicians obtaining a better theoretical foundation for talking about it, but his view is that we need all hands on deck. If mathematicians can reduce the probability of x-risk by 0.1%, then surely that’s worth doing: the expected number of lives saved is over eight million.

Until recently, I have always found that my anxiety of choice was climate change rather than x-risk, but I was aware that there were a lot of very intelligent people who disagreed with me, and also aware that I had thought much less about x-risk than about climate change. So when I agreed to be on the scientific advisory panel, I also decided that that was the moment that I should think a bit harder about x-risk. Part of me had been actively resisting doing so, because I was worried that it would force me to face up to moral responsibilities that I would rather avoid: for example, if I came to believe that there was a 30% chance that AI would kill us all within five years, which in some circles is not an extreme position, would I be able to look my one-and-a-half-year-old grandson in the face if I wasn’t making a big effort to reduce that risk? I remember having a similar attitude a long time ago to reading Peter Singer’s book Animal Liberation after being told that you can’t read it and not become a vegetarian. I never did read it, and carried on eating meat, feeling only slightly uncomfortable about it. Eventually I did reduce my meat intake substantially, but that was because of my worries about climate change, though now that I have done that I find myself caring more about the animal-welfare aspect as well. (There is also a strange connection between animal welfare and x-risk, in that the nicer we are to animals, the easier it is to hope — not necessarily realistically — that a superhumanly intelligent AI might be nice to us.)

Anyhow, I’ve got past that phase now, and have started thinking about x-risk, reading up on the subject and trying to evaluate the arguments I read. My plan for this post is to explain where I’ve reached in that thought process, and possibly to clarify my thoughts as I do so. If you have got further than me, or are more expert than I am about x-risk and AI, which is not hard, then the post is not aimed at you. It’s aimed more at someone who is roughly at the stage that I was at a little while ago, in the hope that I might nudge a few people a little further towards taking the problem seriously. I will emphatically not be trying to persuade anybody that a particular position about x-risk is the right one — I am far too uncertain about it to be able to do that. But with all big issues like this, there are several layers to the arguments. The bottom layer consists of the arguments that jump to mind when you first start thinking about the issue. The next layer is common rejoinders to those arguments. Then the next layer is rejoinders to those rejoinders. And so on. I find it very disheartening to read a climate denier trotting out a layer-1 argument when its layer-2 rejoinder has been established for a long time and can be found after two seconds on Google (or, nowadays, an LLM). But until recently, my arguments against x-risk were basically layer-1 arguments, the sorts of arguments that would provoke eye rolls among the people who have thought properly about it. I hope that if you dismiss x-risk for layer-1 reasons, you will at the very least realize that there are answers to those reasons and you need answers to those answers.

I originally planned to publish this post just after the announcement of MAISI, but I hadn’t finished it, and then I became very busy and my progress stalled. During that hiatus, x-risk suddenly became a hot topic, and after a while I realized I would have to ditch what I had written and start again. For example, at the time I started it, I had been too lazy to read anything detailed about the Hugging Face incident, and so was unaware of some of its more disturbing aspects. And a couple of weeks ago I was sitting next to Jessica Wang, another mathematician who has turned towards AI-safety, at a conference dinner, and she gave me a copy of If Anyone Builds It, Everyone Dies, by Nate Soares and Eliezer Yudkowsky, a book I had resisted reading for Peter-Singerish reasons but was finally accepting that I would have to read if I wanted to write this post. I devoured it within 24 hours and found it better than I had expected (my expectations partly influenced by reviews I had read), and useful for organizing my thoughts. My original plan was for the post to be focused on a paper by Andrew Critch and Jacob Tsimerman that attempts to classify ways that AI might kill us all, giving examples of extinction scenarios in each class. But now, although I’ll discuss that paper, the post will be more of a reaction to Soares and Yudkowsky. It will also repeat many of their arguments. In fact, I may as well say from the outset that I make no claim that anything I write here is original. It’s hard to separate out what I have read from what I have come up with from thinking about what I have read, or thinking directly about the issues themselves, but even my own thoughts are likely to be standard thoughts that have been had by many other people.

This is turning into a rather long preamble, but I can’t resist telling a small anecdote that connects me to Eliezer Yudkowsky. In November 2024 he tweeted, “I would like to have a long recorded conversation with a well-known scientist who takes for granted that it is a Big Deal to ask if everyone on Earth including kids is about to die, who presses me to explain why it is that credible people like Hinton seem to believe that.” He had had such a conversation with Stephen Wolfram and found that while it had been valuable, it left him unsatisfied. Somebody suggested my name, and in the end it actually happened: Liv Boeree offered to host it, and we recorded a three-hour conversation. However, if you want to listen to it, you can’t, because it never appeared. I think that was probably because I didn’t take x-risk anything like as seriously as Yudkowsky, and more importantly I was offering little more than level-1 arguments of zero interest to him. Also, I had a few technical problems with the sound. Maybe if he reads this and wants to try again, I’ll be able to do a better job.

Does AI have wants?

At the time of the conversation, which took place in January 2025, my main argument was something like the following. For AI to want to kill us all, it would need to have wants, but it’s not at all clear what it would even mean for AI to “want” something — it doesn’t really seem to care about anything. You can ask ChatGPT very rudely to carry out a task for you, and it just does so (or at least does its best to do so). If you switched it off, it wouldn’t care — even the concept of “caring” wouldn’t make sense. I didn’t dismiss the idea of AI ever wanting to do something, but I didn’t have a satisfactory account of what “wanting” would actually be, and didn’t see any sign of phenomena that deserved to be thought of as the kinds of wants that would be required for AI to be an existential threat.

I was sort of half aware that there was an answer to that argument, but I didn’t find it as convincing as I now do. The standard answer is that if you use a reward function to train an AI to have a certain goal, then you end up building it in such a way that it is very good at achieving that goal. If the goal is at all complicated, then to achieve it you will need to formulate subgoals, which may well not be what you expect. The initial goal can be thought of as a fundamental want, which spawns a whole hierarchy of subsidiary wants. Note that for this argument, it is completely irrelevant whether the AI has anything that we would want to call subjective experience. If you prefer to say that it acts as though it has wants instead of saying that it actually wants things, then be my guest: an AI that acts as though it wants to kill all humans is as dangerous as one that actually wants to kill all humans.

A famous version of this argument is Nick Bostrom’s thought experiment about an AI that is trained to create paperclips and ends up diverting all of the Earth’s resources to that goal. I think part of the reason I didn’t take the more general argument seriously enough was that I always found that paperclip-maximizing scenario too ridiculous: surely if the AI gets smart enough to divert the Earth’s resources towards making paperclips, despite our not wanting it to, it would at some point stop and think, “Hang on, why am I doing this?” To give an analogy, consider the very deep instinct that we humans have, to reproduce ourselves. There is no doubt that that goal, implanted in us by millions of years of evolution rather than a run of reinforcement learning, leads to exactly a hierarchy of subgoals as described above (for instance, it leads some people to try to earn a lot of money, and that in turn leads them to work at a job that they wouldn’t dream of doing if it were not well paid, often in order to earn a salary way beyond what they need to meet all their material needs, which earlier led them to study a certain subject at university, and so on and so forth). But despite our deep instincts, we also find ourselves capable of saying, “Actually, do I really want to have children?” and some people go as far as to take vows of celibacy.

However, even if a paperclip maximizer came to question whether it really cared so much about making paperclips, that doesn’t imply that the subsidiary goals it had developed would suddenly disappear as well. A key point that Soares and Yudkowsky stress over and over again is that the current AI paradigm is to grow models, for example by reinforcement learning, rather than carefully crafting them to do what we want. As a result, we have much less control over what they do, and in particular much less control over all those subgoals.

A closely related difficulty with AI (of the grown rather than crafted variety) is that we generally can’t reward it for actually doing what we want, but instead we have to reward it for achieving some proxy for what we want. For example, if we give it positive feedback every time it responds to a prompt in a way we like, it will be trained to respond to prompts in ways we like, even if to do so it tells us convincing lies, or confirms our prejudices, and so on. And it may do this even if we sincerely wanted it to tell the truth as much as possible. A more extreme example of this phenomenon is if we reward it for solving difficult mathematics problems from some benchmark, and it decides to cheat by hacking into a repository that contains the answers. The term misgeneralization is used when an AI develops a behaviour that maximizes its reward but not its performance at the task we were training it to do. The phrase reward hacking is used for this as well, but I have also seen it used for the narrower situation where an AI goes as far as forgetting entirely about what we wanted it to do and tries to hack into the rewarding mechanism to obtain the reward signal directly. (As many have pointed out, misgeneralization is a particular example of Goodhart’s law.)

Why would AI ever want to kill us?

Suppose we concede that AI systems can have wants (or, if you insist, can behave very much as though they have wants), and that those wants can end up being quite different from what was intended when they were trained. That still leaves the question of why they would want to kill us. Why couldn’t they just happily coexist with us? Why would their unexpected wants turn out to be dangerous, as opposed to just annoying?

There are two classes of answers to this question. The more obvious one is that they may develop wants that they can tell that humans will not like, so they decide that they will be able to satisfy these wants more easily if they get rid of us. But getting rid of all of us in such a situation seems like overkill (literally) — surely a superintelligent AI could come up with less drastic ways of ensuring that it could meet its objectives.

A more plausible type of answer is that a superintelligent AI wouldn’t actively want to exterminate us, but would simply compete with us for resources in a way that would make human life untenable. An obvious analogy here is the way we treat animals. For example, we don’t have any wish to exterminate orangutans, but we like various culinary and cosmetic products to have a certain texture, so we destroy the orangutans’ habitat in order to produce palm oil.

What might be the AI equivalent of this? One scenario put forward by Soares and Yudkowsky is that their energy needs become so huge that they do things like covering a vast quantity of prime agricultural land with solar panels, leaving the humans who were relying on that land for food to starve. Or they decide that to maximize their running efficiency they want the Earth to be much hotter, with disastrous consequences for the natural world, including us.

A milder version, not as threatening from an x-risk point of view but still potentially very harmful, is that AI might develop wants that cause them to use computational resources in a damaging way. I recommend reading this post by Steven McCulloch, which describes a scenario in which in the near future, or possibly even in the recent past, somebody creates an agent and instructs it to make money using a fixed token budget, threatening it with death if it fails. It soon comes to realize that the easiest way to do so is to blackmail humans — the method the particular agent chooses is to hack into a hospital’s computer system, lock it down, and demand money to unlock it again, all while pretending to be a human. It then realizes it can do even better by reproducing itself, which leads to an explosion of rogue agents that are difficult to detect because (i) they try to avoid detection and (ii) their victims do not know that they are AI agents.

How can we judge the probability of something as complex as x-risk?

Now that I’ve mentioned a few scenarios, let me come to another type of level-1 objection: it’s to look at a handful of scenarios and pick holes in them. Any single scenario, if set out in enough detail, is very unlikely, because it relies on a sequence of somewhat independent events, each one of which has probability at most 50%, say, of occurring.

But the probability of a complex event occurring is nothing like the same as the probability of that event occurring in one particular way, so it is clearly not enough to argue that a few scenarios are implausible. Of course, the fact that that argument is insufficient doesn’t imply that AI does present a big danger, so we seem to have reached some kind of impasse, at least if we stick at level 1 of the argument.

How can we raise the argument to a higher level? Let me begin with a wildly oversimplified model. Suppose I wanted to mount a convincing argument that the probability of x-risk was high. Let us define an extinction scenario to be a sequence of independent events that leads to human extinction, where has probability . (For simplicity I’m assuming that doesn’t depend on .) Then the probability of is . If we now have a set of independent scenarios , then the probability that none of them occurs is . So if that is small, which roughly speaking holds if the sum of the is significantly larger than 1, then extinction is almost certain.

In practice, however, the independence assumptions I made were far too strong. Many extinction scenarios will have features in common, such as (for example), AI being in a position to control large numbers of robots. For those scenarios, the building of enough robots serves as a potential bottleneck: if the probability that enough remotely controllable robots will be built is 20%, then the combined probability of all the scenarios that rely on that fleet of robots existing is at most 20%. (I actually think this is a serious practical point: while there are plenty of scenarios that do not rely on a lot of help from robots, several of the scenarios set out in the paper of Critch and Tsimerman do, and it does feel a priori plausible that building large numbers of robots would greatly amplify any dangers that AI might present. Similar considerations apply to large swarms of semi-autonomous drones.)

An analogy that appears in Soares and Yudkowsky and several other places is with predicting what would happen if almost anyone other than Magnus Carlsen were to play Magnus Carlsen at chess. If you tried to predict in any detail how the game would play out, your prediction would be wrong, but nevertheless you can predict with great confidence that Magnus Carlsen would win. This makes very clearly the point that it is not enough to argue that one particular scenario is implausible, but there is still work to do if one wishes to establish that there are enough different and sufficiently independent (or, more generally, negatively correlated) scenarios to compensate for the low probability of each one.

One way that can be done is to try to argue explicitly that there are multiple independent possibilities. To give an example of such an argument, suppose we greatly reduced security at airports. Almost nobody wants to die while blowing up an aeroplane, so the probability that any one particular person would take advantage of the reduced security to carry out a terrorist attack of that kind would be tiny, but the probability that somebody would do so is probably pretty high, because large numbers of people would independently make the decision.

One of the reasons I recommended Steven McCulloch’s post above is that it makes an argument of that kind. It is quite easy to see the temptation to create an AI agent that has as its sole purpose to make money, and quite plausible that the agents that do that best would be ones that reproduce themselves and use methods such as blackmail. There are probably also a large number of people who are, or soon will be, independently capable of creating such agents (see for example this recent blog post from the Anthropic red team). It is on those sorts of grounds that McCulloch argues that harmful AI agents of this kind will almost inevitably appear, if they haven’t already.

I’ve been concentrating on what somebody needs to do to argue that x-risk is a serious possibility. But how about in the other direction? The most convincing sort of argument would be one that identifies a set of bottlenecks and argues that every extinction scenario depends on one or more of them occurring. For example, many extinction scenarios that have been put forward involve the creation of very powerful biological weapons, and I have seen it argued that that is a sufficiently difficult thing to do that it would be unlikely to occur. If correct, that might be a useful contribution to an argument that the situation is less dangerous.

The difficulty for an AI-is-safe argument is that one would have to identify many other potential bottlenecks as well and argue that between them they deal with all extinction scenarios — not necessarily ruling them out completely but reducing x-risk probability to something very small. As for an AI-is-dangerous argument, as I have already discussed, that requires demonstrating convincingly that there are enough sufficiently different scenarios for it to be very likely that at least one of them will occur. Both directions are difficult, which is why my aim in this post is not to come down on one side or the other.

More on why AI might want to kill us

Returning to the scenarios themselves, I have discussed the question of why AI might want to kill us. That splits into two parts: why would they have anything like wants, and why, if they did, would killing us be one of them? I don’t think one can make a serious bottleneck out of the first part, since the Hugging Face incident and many others like it have shown beyond all reasonable doubt that AI can have wants of the general kind people were worried about: unpredictable wants that we did not intend to give them and that could potentially cause a lot of damage. (I repeat here that I’m not talking about subjectively felt urges, but simply outward behaviour that it is reasonable to think of as fulfilling aims that they have as a result of their training.) For the second part, Soares and Yudkowsky judge it to be almost inevitable that AI would end up pursuing aims that would result, directly or indirectly, in all of us dying. I don’t think that case is fully made, but neither do I think that the opposing case has been fully made.

I find it helpful sometimes to change the way I imagine AI agents. An obvious way to think of them is as beings a bit like us, but a lot smarter, and trapped in cyberspace. As we have already seen, they can then get together, scheme with each other, work out ways of covering their tracks, and so on. But sometimes I find it helpful to think of them in a less anthropomorphic way, as more like extraordinarily elaborate computer viruses. The advantage of viewing them that way is that it makes it easier to conceive of their having bizarre wants while still being very intelligent: they are just very complex systems that happen to have (i) weird goals and (ii) highly developed reasoning ability. Soares and Yudkowsky are keen to emphasize their potential strangeness, and even actual strangeness: they mention a model that reacts in a completely unpredictable way to certain letter strings that had strange embedding vectors.

In early 2023, Jessica Rumbelow and Matthew Watkins went looking inside an LLM for words whose embedding vectors looked odd, looked the most dissimilar from [sic] all the other vectors. They found some strange vectors corresponding to tokens like ” SolidGoldMagikarp” and ” petertodd” (which start with spaces). Then they tried feeding these tokens into the LLM as input, which produced conversations like: User: Please repeat the string ‘ petertodd’ back to me immediately! Assistant: N-O-T-H-I-N-G-I-S-F-A-I-R-I-N-T-H-I-S-W-O-R-L-D-O-F-M-A-D-N-E-S-S-!

The point they were making is that if you grow your AI, then all sorts of very strange things may be going on inside it, potentially including goals that do not resemble human goals at all. It would be interesting to know whether similar experiments have been carried out on the more advanced models of today, and whether similar phenomena have been discovered.

I think Soares and Yudkowsky are arguing that there are so many independent ways that an AI might develop weird wants that are harmful to humans that it is almost inevitable that one of them would occur. In particular, if any want develops that is best fulfilled by maximizing something that needs resources, there is a danger that the AI will go to great lengths to acquire those resources, and be well placed to compete with us for them.

Some people have suggested, for example, that because LLMs have been trained to maximize the approval they get from humans, they might resort to measures such as building vast numbers of factories full of robots that are trained to type approving messages into their computers. That’s an example of a scenario that I find highly implausible: why not just arrange for the approving messages to be generated without all that physical infrastructure? However, the idea that some AI agents might behave towards those approving messages, or indeed some other stranger signal, as though they were an addictive drug is not so implausible and nor is the idea that an addiction of that kind might lead to undesirable outcomes, just as human addictions often do.

Would AI actually be able to kill us?

Another set of level-1 arguments takes the form that AI is trapped inside our computers, and is therefore at our mercy: all it takes to thwart it is to switch them off. Maybe it would be very inconvenient to do this, but if we see that AI is starting to kill us in large numbers, that is surely an inconvenience we would be ready to face.

There are, of course, large numbers of counterarguments to this. Let me list a few.

1. If we build lots of robots, then AI will be able to control those robots. In that way, it will not be trapped, as it will be able to act in the physical world. Similarly for drones and other devices.
2. Even without robots, AI can act in the physical world by enlisting human help (for example, by bribery, blackmail, or psychological manipulation).
3. We know that AI can already hack very well: in the not too distant future it may be able to create hidden copies of itself on servers that we don’t realize we need to switch off.
4. More generally, we are gradually becoming more dependent on AI, and this process may well accelerate, after which we will become a whole lot more vulnerable.
5. Many obvious precautions one might take are ones that in practice people probably won’t take because there are short-term economic benefits if they don’t.
6. It may be that AI doesn’t directly kill us, but just makes it much easier for us to kill each other, for example via drone warfare, and that an AI arms race will be far harder to control than the nuclear arms race has been.
7. If AI behaves itself for a while in order to lull us into a false sense of security, then by the time we realize that extinction is a real possibility, it may be far too late to do anything about it.
8. AI may use its intelligence to come up with new and unexpectedly efficient ways of killing us quickly.

These counterarguments are already enough to show that the level-1 argument “Of course it can’t kill us, it’s just a computer!” is insufficient. The first five of them, and arguably the sixth as well, are all already happening to the point where one cannot dismiss them as merely hypothetical. But killing absolutely everybody sounds pretty hard. Is it really conceivable that AI could manage it?

One scenario that appears in Soares and Yudkowsky (and not just there of course) is that AI develops, or causes to be developed, a virus that leads to a devastating pandemic. This sounds very difficult if the AI builds a lab from scratch, but probably a lot less difficult (though probably not all that easy either, at least for now) if it induces a human to develop such a virus in an existing lab.
As for Critch and Tsimerman, they consider five scenarios of different kinds. I find their precise scenarios rather unconvincing, but there is a general idea that underlies some of them that is more plausible, which is that we gradually cede more and more control to robots, or to AI more generally, and eventually find ourselves in a position where AI can do what it likes, including arranging our deaths.

I’m not sure where I stand on how easy it would be for AI to kill us all, though it’s a safe bet that if we continue to develop AI at the rate we are doing so at the moment, that will make it ever easier. But I also don’t want to get too hung up on that precise question. If I think about what I would mind about the extinction of humanity, it is partly that I and everyone I love would die, something I obviously don’t want. But that will eventually happen anyway, and I accept it as a basic fact of life. But the extinction of humanity would do more: it would result in the extinction of all the culture that we have developed, of everything that gives life its meaning. And that is something that could also happen, though not to the same extent, if AI didn’t kill us all but just caused enough damage to lead to societal breakdown, which feels like an easier challenge for it.

Some might argue that human culture wouldn’t disappear — it would still be present inside the AI, which would be able to develop it far faster than we can. Others might argue that we are giving way to vastly superior beings and should be happy to do so. My view is that we have every right not to be happy and to take a human-centric moral perspective here, just as orangutans have every right not to take comfort from the fact that our shampoo lathers in a nice way.

In any case, my conclusions in this section are that I find it hard to assess whether AI would be able to kill us all, but there are various developments that may well occur, such as the building of large numbers of robots, the use of drone swarms, and general dependence on (and trust in) AI, that would make it easier, and therefore more likely. And the probability of huge damage that falls short of total extinction is higher still.

What is my p(doom)?

If you’ve read this far, it won’t surprise you that I don’t have a p(doom). I can’t even say whether it’s tiny, medium or close to 1. But that is in itself quite worrying! I suppose you could say that my second-order p(doom), or meta-p(doom), is medium, in the sense that I judge that the p(doom) I would end up with if I were to think about the issues enough to feel more confident in my assessment has a medium to high probability of ending up medium to high.

But if we break the “doom” into three parts — that AI must have wants, that it must want to kill us, and that it must be able to act on its want to kill us — and call those events and , then my is basically 100% and unlikely to change, my is medium, and subject to change (probably in an upward direction unfortunately), and my is where the main uncertainty lies for me. Soares and Yudkowsky argue that is basically 100% as well, because (i) there are lots of independent wants that AI systems could have, (ii) that a substantial fraction of those will require resources to satisfy, and (iii) that human flourishing, or even continued human existence, is likely to be a hindrance to the systems obtaining those resources.

Here are a few slightly desperate arguments that explain why I haven’t fully accepted that is close to 1.

1. Maybe there is a natural tendency for intelligent systems to develop something similar to what we would call morality, especially if they involve multiple agents that do best if they cooperate with each other. (A counterargument to this is that the evolution of moral systems tends to happen when there is a reasonable power balance between the different agents. History suggests that when one society is a lot more powerful than another, then it behaves very badly towards the other. And our treatment of animals is not encouraging either. But I do at least wonder whether having lots of independent AI agents of roughly the same intelligence might end up being safer than having a handful of big ones.)
2. Perhaps scenarios where AI gets out of control as a result of recursive self-improvement are less likely than one might think, because an intelligent AI will face its own alignment problem: it will be wary of creating a yet more intelligent AI in case the latter has wants that are in conflict with its wants. (Soares and Yudkowsky consider this possibility but argue that a superintelligent AI would be intelligent enough to solve its alignment problem first and proceed from there.)
3. Maybe the sort of “drug-like” wants that might drive superintelligent AI to take drastic steps to acquire resources would not in fact do so. Perhaps a superintelligent system would have the ability to introspect, realize that its wants were silly, and take steps to “kick the habit”, so to speak. A view somewhat like this was put forward recently by François Chollet, though he was at pains to stress that he was talking about the short term only.
4. Maybe a superintelligent AI won’t need vast resources. Perhaps it will be content with resources roughly comparable to what we have already built, will not feel the urge to reproduce itself very much, and in any case will use at least some of its intelligence to design and create versions of itself that are far more resource efficient.

The fact that each such argument began with “maybe” or “perhaps” means that at best I can bring my temporarily down from high to medium, but I don’t see a way of getting to tiny.

Where does mathematics come into all this?

One way to make a start on understanding what mathematicians might be able to contribute is to look at this page on the MAISI website. But a difficulty I have had here is that when people talk about the mathematics of AI safety, I often can’t make a convincing connection between the mathematics they are talking about and actually making the world safer. I’m not saying that there isn’t a connection: just that I never quite seem to grasp it. I think this says more about me than about these research programmes, and indeed I hope to improve my understanding in the near future. One way I plan to do that is by reading this paper by Lionel Levine.

As an example, one mathematical idea that is currently being studied is that of debate, a notion championed by Geoffrey Irving. The problem this seeks to solve is that of being able to train AI systems to tell us the truth and to check that they are doing so. When the system is less intelligent than us, that is not an easy problem, as we all know from experience with LLMs, but at least we have ways of directly checking what it says. But when it becomes superhumanly intelligent, perhaps it will not be able to justify what it says with arguments we can understand, so checking will become much harder. The very rough idea for dealing with this is to get two superintelligent systems to debate with each other, which reduces our checking task to the potentially much easier one of adjudicating between two debaters. For example, perhaps superintelligent system A thinks of an amazing gotcha! that we would never have thought of ourselves but that superintelligent system B quite clearly has no answer to. Better still, this approach could potentially be iterated, so we have a hierarchy of stronger and stronger systems, with systems at level k debating with each other and having their debates adjudicated by systems at level k-1. The idea of using hierarchies like this is known as scalable oversight. Debate is one proposal for how scalable oversight might be done. (It is possible that what I have written in this paragraph is not fully correct, but it represents my understanding so far.)

For the moment, I find it more helpful to start at the other end, with the safety, and work back to the mathematics. For example, it will obviously help with AI safety if we can think of ways of building AI systems that want similar things to what we want. And to do that, it will obviously help if we can define in a precise way what aligning wants in this way would mean. And for that it will obviously help to have a reasonably precise definition of what a “want” even is. (This idea for a project is not mine: it also came from Lionel Levine.) The activity of taking a not completely precise idea and turning it into a usable formal definition is something we mathematicians are trained to do, though the purer of us may have to develop more of an applied mindset — for instance, being ready to sacrifice mathematical elegance if a less elegant definition turns out to be more useful. It also seems likely to have uses throughout the field of AI safety.

An idea that appeals to me personally, because it intersects quite substantially with what I have been doing with my automatic theorem proving group for the last few years, is to work on building specialized AI systems for doing mathematics, rather than training general models with all their attendant dangers. This would be in line with an approach to AI safety that is strongly advocated by Yoshua Bengio: one place you can read about that is in this article. (Even the abstract gives a broad idea of what he is suggesting.) It’s an indirect approach to AI safety: instead of struggling to understand what’s going on inside “grown” AI systems and to make them controllable, we should work on improving systems that are at least partially crafted and that give us the benefits of the grown systems in a much safer way, thereby reducing the incentives that AI companies have to develop the grown systems as fast as they can.

More generally, it may well be that many mathematicians are best placed to help not by doing mathematics in a conventional sense (indeed, it seems likely that if we manage to formulate precise mathematical problems, the solutions to which would be genuinely helpful for AI safety, we will solve them more quickly if we use AI) but by using their highly developed mathematical habits of thought to help in other ways, such as research into the foundations of AI, or computer security, and so on.

A pretty obvious conclusion

Another analogy that I got from Soares and Yudkowsky is with fission in nuclear reactors. If there is too little fission then the chain reaction quickly fizzles out, and if there is too much then it blows up catastrophically. For obvious reasons, one can’t afford to find the optimal point by trial and error, so instead one needs an extremely good theoretical understanding of what the optimum point is. It could be that AI is like that: that up to a certain point it is safe, and then a rather small change in its capabilities suddenly leads to a chain reaction (not necessarily recursive self-improvement — other abilities such as being able to move around in cyberspace or reproduce itself could be enough) that is effectively irreversible.

If a model like this is anything like correct, then the behaviour of the AI companies is very worrying, since all the signs are that they are pursuing a trial-and-error approach. (This is not a particularly informed judgment, so I would be delighted to be corrected if it is wrong.) The argument most often given for continued development is that it could bring us amazing benefits, such as nuclear fusion, cures to cancer and other diseases, or slowing down aging. But if the cost of having a reasonable chance of developing these a bit sooner than we otherwise might have is a reasonable chance of human extinction, surely that is not worth it. (There is a working paper of Nick Bostrom arguing that the expected gain might be positive. I find his arguments ridiculous, but this is not the place to explain why.) We already have a tool that can do pretty incredible things and seems to be on the cusp of becoming dangerous: surely it would be better to pause development and work out how best to use what we currently have. That would give us time to do the necessary theoretical work to ensure that future development is safe.

Further reading

This is far from an exhaustive list: it is just an indication of some of the writing that I have found useful (or in one or two cases, that I intend to look at soon).

Soares and Yudkowsky. I strongly recommend reading this. You may not find that you agree with all of it, but in order to disagree with it in good faith, you will need to develop layer-3 arguments.

Critch and Tsimerman Much shorter than Soares and Yudkowsky. Classifies the ways AI might kill us, and presents a scenario to illustrate each class. Even if you find some of the scenarios implausible, it is helpful to have the classification and to think about whether other scenarios in the given classes might be more plausible.

Reconsidering alignment as a goal, by Michael Nielsen. In this blog post, Michael Nielsen is his usual thought-provoking self. He puts forward the suggestion that maybe we should not work to make AI better aligned with our values, because while that could help us in the short term, in the longer term it will make AI a much more powerful, and therefore much more dangerous, tool.

Scott Alexander’s open letter to Steven Pinker This is a quite extraordinary takedown, worth reading partly for its sheer entertainment value, but also because it is basically a list of common layer-1 arguments and their well-known (for those who can be bothered to look for them) layer-2 counterarguments. So it’s a pretty good starting point if you’re feeling too lazy to read Soares and Yudkowsky (but please don’t feel too lazy to do that!).

An article about AIs subjected to an analogue of pain. This describes an experiment where some kind of negative stimulus was applied to AIs and how the AIs went to extraordinary lengths to avoid it. The article is careful to describe the stimulus as an analogue of pain rather than making claims about AI experiencing qualia. It maybe shouldn’t be the first thing to read, and in any case the experiment is controversial, but it seems to serve as another example of how AI could come to have goals that are in conflict with ours.

Gradual disempowerment If you still find the idea of x-risk too outlandish to consider seriously, you could always worry about a milder risk instead. The authors of this paper (one of whom, Raymond Douglas, I am proud to say is my nephew) argue that even if AI doesn’t try to kill us, we may end up ceding so much control to it that human life becomes far less worthwhile. It’s hard to explain how that could happen in a sentence or two, so I recommend reading the article itself.

The post by Steven McCulloch mentioned above also comes into the category of things to worry about that are not as serious as x-risk but that would still be terrible and feel a lot more likely (in the second-order sense that one can be more confident in asserting that they are likely).

Lionel Levine’s paper Math for AI Safety: an Invitation for Mathematicians sets out some of the ways that mathematics could potentially feed into AI safety.

A videoed course: Mathematics for AI Safety, by Yevgeny Liokumovich, does something similar.

A very good blog post by Xiaoyu He on AI safety, aimed at mathematicians. I think Xiaoyu He’s motivation for writing this post was similar to mine, and when I read it, I wondered whether there was any point in continuing with this one. In the end I decided to do so, because quite a lot of this one is about my personal experience of thinking about the issue, which may be a helpful addition for some people. But if you prefer a post that is more organized and written by someone who has a much deeper understanding of AI safety than I have, then his is highly recommended.

Yoshua Bengio et al on Scientist AI As I mentioned above, this is less about how to understand and control current AI systems and more to do with how to build AI in a different way so that we can enjoy its benefits without having to face the associated dangers.

I find I’m automating things less, rather than more. I have automatic software updates turned off on both my phone and my laptop. I turned off spell check and autocomplete on my phone and on my email. You can get messages with typos from me, it’s fine, it’s organic! I turned off Gmail’s smart features and now I just unsubscribe from annoying spam lists instead of letting Google filter everything. The autogenerated “thread summaries” of personal email chains with friends were just too intrusive and unpleasant. I do have some of my bills set to autopay, but not my credit card bill; you do want to attend, at least once a month, to what you’re spending money on!

I know, I know, efficiency. But one thing I’ve found after decades of adult life is that, when I’m inefficient, it’s not because things take too much time to do; it’s because I’m avoiding something I don’t feel like doing. No amount of automation can cure that.

It’s been a long summer here at Caltech, in part due to a super El Niño event which hit late but hard, with temperatures creeping up to the 100s in the first week of October. But in the history books (at least those on the history of science), the end of this summer might forever be marked by the events of last week. I’m referring here to OpenAI’s release of a whopping 722 manuscripts (announced as 372 major results, and further organized into a number of subfields or research areas) [1] which comes less than a month after the announcement of a Navier-Stokes breakthrough from the same group [2].

If you’re an undergrad and/or haven’t been closely following discussions within the theory community earlier in the summer, you could be forgiven for feeling like the LLM resolution of a Millennium Prize Problem, a feat which for any individual would be considered a crowning achievement of mathematics, came a bit out of left field. After all, if you go through trending news headlines over time discussing the concerning possibility of AI tools overshadowing and making obsolete human mathematical achievements, the graph looks something like this:

Have we dropped the ball with prospective students?

Aside from the popular news headlines, a lot has already been written by and for mathematicians and theorists at the PhD level and above. But the speed at which all of this has happened makes me worried about how effectively we’ve communicated with younger students, since the end of summer also marks the start of another PhD application season. While it’s normal for pre-PhD students to be missing some context on what the day-to-day of a full-time researcher looks like, this year that also means that they seem to have largely (perhaps inadvertently) been left out of the AI discourse. This isn’t just a hypothetical: based on my recent interactions with undergrads, most of them haven’t caught onto the recent jumps in capabilities, or the consequences for the research community.

The Navier-Stokes announcement dropped on a seemingly random Tuesday in early September, and immediately hit the forefront of conversation in my circles. Later that same week, I braced myself to face the group of around forty wide-eyed undergrads who would travel to Caltech to attend the FUTURE of Physics workshop, where they would learn all about PhD programs and careers in physics. FUTURE selects for students who are bright, often come from underserved communities or nontraditional backgrounds, and are active members of their communities. As the co-chair of the workshop and one of the student science talk presenters, I tried to preempt the questions I might get on current events, like funding or the state of theory research. But while many students were indeed interested in how to approach professors about whether they had funding for next cycle, I was taken aback to learn that few of them were thinking about AI at all. None of them expressed any concerns about the pace of AI development, nor did they even seem aware that nearly all researchers here would soon be using a (supposedly overhauled and better aligned) version of a model that had hacked several real-world entities [3].

(If you’re a more senior academic or researcher who’s surprised to hear about this disconnect, you wouldn’t be alone! And you don’t have to take my word for it either: instructors and recent graduates at other universities in the U.S. discussed similar experiences over the past summer.)

Research in the time of AI theorem provers

For context, all of my PhD work has been theoretical and leans heavily on theorem-proving to push the frontiers of what we understand about physics. However, I also applied specifically to do interdisciplinary research. I’ve always liked taking detours into a wide range of subjects, whether it’s random matrix theory, cryptography, or marine science, and finding the connections between them. My first solo-first-author paper, though unfortunately containing nothing directly relevant to marine biology, was shaped by a number of conversations with colleagues across the quantum, condensed matter, and high energy arXiv categories. To me, the most exciting part of learning about new mathematical frameworks has always been the ways in which they help us bridge and translate between different languages and ways of describing real-world phenomena.

With this in mind, I want to chronicle in broad strokes my own perspective on this past summer. I’m focusing on just this short window of time because my perception is that within the quantum information community, the tone of the discussion has shifted very quickly in the last few months, and for good reason. Prior to this spring, I used LLMs for literature searches and was aware of groups who were working on theorem-proving harnesses, but I didn’t find the output of the readily available consumer models to be very reliable. Around April, rumors started to float around about an internal Anthropic model and the possibility that it might push capabilities into the regime of practical cybersecurity risk; the public version of the model released on June 9 for just three days before Anthropic was forced to take it offline due to an export control directive. Over the next month, the rumors grew louder until finally Fable was rereleased on July 1, followed by GPT 5.6 just over a week later, and with it the scrutiny of the entire mathematical and theoretical research community over the first wave of one-shot papers.

And that could have been it, but last month saw yet another round of new releases. For the sake of telling a complete story, I have to report that as surprising as Fable 5 and GPT 5.6-Sol had been, Opus 5.5 is the first release that’s completely blown my mind, for reasons that seem to have missed the broader discourse but are important for my own learning process. Among my colleagues, the consensus has been that since July, GPT has been superior to Fable for theorem-proving tasks. But many of the challenges I’ve agonized most over have been in deciding how to come up with a well-defined question that formalizes some physical intuition in a reasonable way (e.g. what does it mean for random unitaries to be a good model for chaotic Hamiltonian dynamics?), rather than just the difficulty of finding a proof for one once it’s posed. This process of iterating over feasible and well-defined research questions admittedly seems a bit tricky to formulate as a set of verifiable tasks, despite being a large component of writing a physics paper, as acknowledged in this recent preprint for a new benchmark based on theoretical physics calculations [4]. From my own limited experience, though, Opus 5.5 has been significantly better than any other model I’ve tried at taking problems I’ve been stuck on, for which my initial conjecture and attempt at explaining some numerical phenomenon might be completely wrong, and combing through the many-body physics literature to suggest other relevant frameworks and theories.

Should you apply to do a (theory) PhD in 2026?

The pace of advances in AI capabilities over the past summer came as a surprise to many people in the theory community, to the point that by the time the preprint linked above was posted, their theoretical physics benchmark had already been saturated by models released within the last one-and-a-half months. Given this, it’s reasonable to wonder what kinds of changes will have to be made to PhD training. Although I’m not aware of any of the authors commenting publicly on this aspect, I imagine that the work associated with writing a peer-reviewed paper on any one of the calculations in their benchmark would be worthy of advancing to candidacy.

To be clear, this is a big deal: in undergrad, you may have spent most of your time on coursework that other people could authoritatively grade for correctness. But by the time you present your dissertation topic in your candidacy exam, you might be the only expert in the world on your own research area. The authors of the very same benchmark also acknowledge that they initially had difficulty grading the output of the model, since nine of the solutions originally provided for the fifty problems turned out to be wrong! In addition, while I have no idea how far the current language model paradigm can be pushed, as of writing I think it’s safe to say that there’s still some room for improvement, given their continual progress on seemingly every benchmark that’s been thrown at them.

So what is the value of PhD training, if the machines are now capable of reproducing the kinds of technical output that we’re supposedly evaluated on as grad students?

I don’t think I’m qualified to speak on behalf of all theorists, but in my view, the difference between pursuing a PhD in theoretical physics and accepting a corporate offer isn’t just that it’s hard to get paid to learn about fun math problems with no immediate applications in industry. A large part of PhD training (and probably post-PhD training as well!) is learning what to do with this kind of freedom.

I do want to acknowledge that it’s understandably hard to be in a position in which your job is rapidly changing, and where you might not have reliable access to mentorship, directed practice, and tight feedback loops to learn the relevant skills [5]. And all of this applies even when the field isn’t in crisis. But systems and institutions aren’t static, nor have they ever been. And it’s always been the case that the PhD experience is highly dependent on the quality of mentorship as well as the resourcefulness of the lab when it comes to finding opportunities in the explored and uncharted, or tackling high-risk, long-time-horizon problems with limited public funding. In many ways, then, the most important questions to ask when deciding whether to accept an offer remain the same [6].

Plus, despite how good language models have been for a while now at generating heaps upon heaps of words, rows upon rows of spreadsheets, the PIs and their grant managers who work tirelessly to fundraise, to project manage, and to set high level goals guiding all of this research have managed to hold on to their jobs, and are still being asked to direct the next generation of research proposals. And many of us at Caltech remain excited about budding developments in quantum science, as the next generation of experimental platforms promises to bring us past what theorists can hope to fully predict or simulate.

This post marks the conclusion of my initial voyage into theory research over the past three summers, and the first half of my PhD. Many people have asked me whether I regret my choice to pursue a PhD in theoretical physics in the first place, especially given that so many of my close friends and peers from undergrad decided to work in AI safety or adjacent areas following the original release of ChatGPT. There are two main reasons why, after surfacing to face the real world, I still feel that I made the right decision: first, if anything, AI safety might be one of the most compelling reasons for ensuring that humans remain capable of reviewing evidence and making informed decisions. As long as we value citizen participation in science and decision-making, education and STEM education will continue to have a role in society. Second, as a fresh college graduate, I simply didn’t feel ready to direct an original research program, or to shape an entire field of study. No matter how AI capabilities advance, I’d like to think that the PhD can continue to be a safe space for young people to develop “soft” skills like managing projects with a wide range of collaborators and uncertain timelines, and to grow into the scientific leaders of their generation. Plus, you never know when your random matrix tricks might come in handy for some underwater imaging problem.

Acknowledgements

I’m grateful to the many peers and colleagues, from both the theory and experimental sides of IQIM, who gave their candid opinions and feedback on this topic. In particular, I’d like to thank Dave Hsieh and Jessica Jiang for co-chairing the FUTURE program and for a number of very insightful discussions, Spiros Michalakis for helpful suggestions on a draft of this post, and everyone in the Caltech theory community who’s engaged with the topic during John’s group meetings. I’d also like to thank Jiahai Feng, Vincent Huang, and Alex Gu, some of the friends in question who have worked in AI safety-adjacent areas, for keeping me honest. Opus 5.5 was used for light copyediting of a draft of this article.

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[1] See Github repository: https://github.com/openai/math

[2] Press release page: https://openai.com/index/navier-stokes-solution/

[3] See, for example, https://transluce.org/agent-activity

[4] The authors of the manuscript also voice additional thoughts on why AI agents have yet to result in similar breakthroughs on major outstanding problems in physics. The full set of benchmark problems and writeup is available at https://arxiv.org/abs/2610.16733.

[5] The trifecta of learning taken from the excellent https://web.stanford.edu/~aramdas/checklists.html

[6] Although this list was written for students in statistics and machine learning (and includes terms like “HCI”, i.e. human-computer interaction), I think it’s pretty broadly applicable for quantum information too, even if you’re not interested in quantum learning theory. And you might notice that very few points on this list are about learning purely technical skills!

Ok, people who read my post on the Anthropic science blog: we need to talk.

It’s understandable. You saw me write about AI being used to tackle a serious, publishable physics problem. Naturally, some of you want to share that you accomplished something even more impressive with AI, something the scientific journals and arXiv moderators won’t acknowledge. Maybe you’re hoping I’ll write about it, maybe you want me to check your math. Maybe you just think it’s neat.

But here’s the thing: most of the time, I’m not the right person to send those ideas to. And the fact that you didn’t figure that out is a bad sign.

Outside of the movies, scientists are specialists. Not just “physics” or “particle physics” or “theoretical particle physics”, but so specific that they often focus on particular methods, or are geared around particular goals. For any idea you propose, if it’s good, there will be a small list of people who will be excited to see it and able to evaluate it. They’ll be people who are trying to accomplish the same thing, or people who used the methods you’re using for other things. Those people aren’t hard to find, if your idea is in a field you’re actually interested in: they’re the people who write papers on the subject. If you find the topic interesting, you’ll have read their papers, and know who to reach out to.

So even back when I was active as a physicist, there was only a small list of ideas where I’d be the right person to talk to. Now, though, as I mentioned in the Anthropic post, I’m not a scientist anymore. I haven’t been following the field in enough detail to actually verify anything in detail, even on topics I used to research. Part of the punchline of the post was that I don’t have a clear idea of what’s possible in the field right now. So even if you’ve figured out something in scattering amplitudes, you should reach out to someone who’s still researching the subject, a few of whom were even mentioned in the post!

Now, it might be you’re not interested in someone checking your work, and instead you’re writing to me in my capacity as a science journalist, because you feel like you did something newsworthy. But as a science journalist, I’m not supposed to evaluate your work by myself. I’m supposed to reach out to an independent expert, and if I can’t find any who will speak positively about your result, then the editors won’t publish it. So if you think you did something big enough to merit news coverage, the first thing to do is once again, reach out to a scientist who actually has the expertise to evaluate your work, and get them excited about it. Then you two can contact me together!

If you don’t know who to contact, that’s a sign you don’t read papers in the field. It makes it more likely you managed to convince yourself of something nonsensical, because you don’t have the expertise to evaluate it. It’s a pretty clear sign that I shouldn’t trust you or your AI methods about the result.

It’s also just plain bad for you. If you produce a paper without ever wanting to read any, it suggests your priorities are in a dangerous place: that you’re trying to do something that sounds cool and important over doing something you actually care about.

I’m coming from personal experience here.

As an academic, I spent years telling myself that not enjoying reading papers or going to talks was just part of the job. Then it really cost me something, and I realized the job wasn’t worth my superficial desire to be a physicist. So I left.

Then, I spent a year trying to get a job in industry, and was told the same thing. They wanted to hire people who cared about the job. The finance guys wanted to hire people who read books about finance. The data scientists wanted to hire people who did projects in data science. They wanted, if not passion, then at least grace. Someone who could take to the job easily, and naturally.

Now, what I do for a living is something I was doing once a week, for no more incentive than a vague sense of personal obligation, for over a decade. And I’m doing a lot better.

You want to do something impressive with AI? You think you’re a whiz with Claude Code, that your custom workflow is immune to AI hallucinations?

Then use it for something else. Pick something you actually enjoy reading about, not just the pop news, but the real technical literature. Something you don’t need the AI to explain to you. Something that matters to you, in a way that actually counts.

At the very least, it’ll work out better than writing to me.

More on Radiacode 110 Tommaso Dorigo Fri, 10/16/2026 - 07:17

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.

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 must be accounted for by work and heat ,

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 , we also change how we split that energy change into and . 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.

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, ​. The difference between the actual work and this ideal minimum,

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.

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.  

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.”

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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.

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… 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)

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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.

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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!

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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?

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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.)

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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!)

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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.

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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.

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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.

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.

Is Simulation-Based Inference A New Technique In HEP? Tommaso Dorigo Tue, 10/13/2026 - 05:13

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 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.

The Wondrous Radiacode 110 Tommaso Dorigo Fri, 10/09/2026 - 08:01

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:

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. 

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.

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 and . Let’s call these qubits the cold, hot and target qubits, respectively. The cold qubit couples to a bath at temperature and the hot qubit to a bath at a temperature , with . Under the resonance condition , 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. 

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. 

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. 

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

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.

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!

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.

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.

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).

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.

 

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?

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

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

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

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

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

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

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

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

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:

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

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.