How I Quit Math, Met AI, and Published My First Paper in Three Years

It has been about two and a half years since I left mathematical academia in March 2024, when my postdoc term ran out. I thought I would never write a math paper again — and yet, somehow, the other day I submitted one to arXiv.

The paper is a joint work (?) between me and the currently fashionable AI, and most of its text was written by AI. But this is not the simple story of me telling an AI "write me a math paper" and submitting whatever came out.

Lately AI has become strikingly good at mathematics, and one keeps hearing claims like "AI solved an open problem!", as if mathematics were a benchmark for measuring AI performance. Many people, I imagine, are unhappy about this, or at least have mixed feelings. Others may be hoping that some new form of mathematics is about to begin. And yet there do not seem to be many accounts of how research with AI actually went (as far as I can tell).

So this piece records how I actually did mathematics together with AI, made a discovery that (to me at least) is genuinely interesting, and ended up submitting a paper to arXiv — one possible case study of mathematics in the age of AI, with some personal circumstances and mathematical background mixed in.

Notes

Prelude

At 0:54 a.m. on June 13, 2026, Anthropic's AI Fable 5 proposed the following conjecture to me.

Standing conjectures, in decreasing confidence: [...] (iii) new — the reversal conjecture: [...] "$\#$ submodule-closed subcategories of size $k$ $=$ $\#$ quotient-closed of size $N - k$".

Without even reading what was written there, I passed the web chat's output containing this conjecture to OpenAI's GPT-5.5, who was stationed in the algebra laboratory on my PC. He promptly set about checking the conjecture on various algebras. Before the computation finished, he had already pre-written "the formula holds in this example" into the report. The computation, however, turned out to give a counterexample.

[...] but the bowtie counterexample does not satisfy the reversal prediction. I need to correct the generated report text, because the first draft assumed the bowtie would pass before the computation finished.

Clean kill

And so Fable's midnight conjecture was gone in about twenty minutes.

Fable had conjectured that the sequence counting quotient-closed subcategories of each size $i$ agrees with the sequence counting submodule-closed subcategories of each size $i$ — after reversing one of them. Quotients and submodules are dual to each other, so the reversal feels natural. But in fact GPT-5.5's computation showed the two sequences agree without reversing. This is the quotient–submodule equidistribution conjecture of the paper's title.

Let me explain in a bit more detail what this conjecture means.

Mathematical background

A quiver is a directed graph made of dots and arrows, and a representation of a quiver assigns a vector space to each dot and a linear map to each arrow. In algebraic language, this is the same thing as a module over the path algebra of the quiver. A module that cannot be decomposed into a direct sum is called an indecomposable module. This article deals with algebras of finite representation type — those having only finitely many indecomposable modules (up to isomorphism) — and we write $N$ for that number.

There are two ways to produce a smaller module from a given module: taking a submodule, and taking a quotient module. In this article, a subcategory may simply be thought of as a collection of indecomposable modules, and its size is the number of indecomposables it contains. A collection is quotient-closed if taking quotients of modules in the collection never leads outside it, and submodule-closed if the same holds for taking submodules. Even for the one-arrow quiver above, the two kinds of collections do not coincide.

As the table shows, the quotient-closed subcategories and the submodule-closed subcategories are different as families (the entries in red). And yet, once you count them size by size, the difference vanishes. The conjecture asserts that this always happens.

To explain where this conjecture came from, I first have to tell you why I left mathematics, and how I came back.

After quitting mathematics

In March 2024, when my JSPS postdoctoral fellowship expired, I left academia and took a job at an ordinary company. I wrote about the circumstances at the time in a farewell-to-academia post, which those who enjoy gloomy prose are welcome to read. Mathematics had worn my spirit down, and I could not commit to a lifetime with a subject I was not even sure I liked.

What I learned next is that working adults are busy. A day ends after nothing but work. Weekends vanish in an instant. There was no room for mathematics, and no desire for it either; at most I would browse arXiv out of inertia, or glance at a new paper when an acquaintance's name appeared.

Playing with AI, though, I always enjoyed. Around 2022, near the end of my postdoc, GPT-3 and GPT-3.5 and ChatGPT appeared, and I used them to polish the English of my papers. I was also curious how much mathematics AI could do, so I would toss simple undergraduate-level exercises at it, and then, faced with the nonsense proof-shaped output, fire back long rebuttals — "this is not a proof at all", "this step is wrong" — harassing the model with mathematics for fun.

Even after entering company life, whenever a new model was released I used the following statement about Jacobson radicals as my personal benchmark: "In a noncommutative ring, the intersection of the maximal right ideals equals the intersection of the maximal left ideals." It was around the end of 2025 — just last year — that AI first became able to prove this properly.

Quotient-closed subcategories form a convex geometry

After playing the visual novel MUSICUS! at the start of 2026, I found myself thinking about my life constantly. I will spare you the otaku talk about its story, but I began to wonder: is it fine to drift on like this as a salaried worker? What is a life? Shouldn't I be doing mathematics?

Put simply, I am a contrarian, and the grass is always greener on the other side. While in academia I hated mathematics and wanted out as soon as possible; once I was out, living days with no time of my own, I began to think that those hard, painful days of doing mathematics were when I had expressed myself the most (a view which, needless to say, involves a heavy dose of idealization and nostalgia).

So in the end I could not stop thinking about mathematics. I started browsing the papers that had appeared since I left, and, invited by a junior colleague, attended a seminar for young researchers held at Nagoya University.

The paper that caught my attention was [CFY], on quotient-closed subcategories. I had always liked considering various subcategories inside the module category of an algebra and studying their combinatorial structure. But subcategories closed only under quotients were something I had rarely thought about. Their paper observes that for the $A_3$ quiver there are exactly $24$ quotient-closed subcategories — the order of the $A_3$ Coxeter group — and presents this curious relationship almost as a conjecture. This rang a bell, so I asked ChatGPT whether it follows from [ORT], and we discussed it.

Indeed, [ORT] classifies quotient-closed subcategories for Dynkin quivers in the language of Coxeter groups, and the equality of counts follows at once. But their argument looked like Dynkin-specific combinatorics to me — an external, artificial coordinate system that could not possibly work for general algebras. Discussing with ChatGPT, I spent my days turning over in my head what [ORT] and [CFY] were really doing.

ChatGPT taught me that [ORT] studies, in the Dynkin case, a combinatorial property of quotient-closed subcategories: they form a convex geometry. I had never heard of convex geometries, but apparently they are combinatorial structures on finite sets satisfying simple axioms. So, I wondered, do they form a convex geometry over a general algebra? In fact, I was able to prove that they do (the argument is essentially contained in [AS], a paper I love; I merely applied the same argument because I loved that paper — honestly, nothing difficult was involved). It was the first small mathematical discovery I had made since leaving academia. That was the end of March.

Now then: "quotient-closed subcategories form a convex geometry." ...So what? A few lines proving this little theorem amount to nothing by themselves. Something interesting has to come of it. What could? I had always liked approaching the representation theory of algebras from combinatorics and abstract category theory, so I looked for consequences of my small observation. My first idea was to bludgeon every result of [CFY] with general combinatorial theory. To some extent this worked. But one of their main results — the characterization of when the quotient-closed subcategories form a distributive lattice — resisted, and in the end I could not derive it. It would have followed from an equivalence I hoped for, between a certain combinatorial condition and a module-theoretic one; but that hoped-for equivalence was refuted by a counterexample found by GPT-5.4.

Building a laboratory for the AI

I had always been drawn to studying algebras by computer. But [QPA], the library that can handle general algebras, is written in [GAP], a language entirely foreign to me, and I could never bring myself to use it.

By now, though, I was working at an AI startup and making AI write code every day, and it occurred to me: surely today's AI can handle QPA. So I pleaded with OpenAI's coding agent Codex and Anthropic's Claude Code, running on my PC, to set up an environment for computing quotient-closed subcategories with QPA. From then on I had the AI run all sorts of computations, including the counterexample above.

Frankly, I had no idea how to proceed from "it is a convex geometry." A powerless salaryman with no time, I merely commanded the AI: "We know it's a convex geometry, so do something nice with it." No interesting mathematics was in sight yet.

Fable arrives, makes a wrong conjecture, and disappears

When Fable 5, advertised as having high academic ability, was released by Anthropic on June 9, I added him to the laboratory, and gave him the following instruction.

OK I want you to do some "interesting mathematical work", not just verifying some conjecture by computer program, nor just following previous direction. Be creative like real mathematician.

I want to "utilize" my findings about "quotient-closed subcats becomes convex geometry" so I want to promote this idea, but needs some interesting real results not just "we can translate rep-theory into combinatorics, and vice versa". Tell me what you think. I think you can be mathematician.

My first impression of Fable: a mathematician of bold, out-of-nowhere ideas, rich in imagination, but scatterbrained and occasionally wrong. And yes — Fable's ideas and powers of argument already looked beyond a graduate student's, at least at postdoc level.

Four days after joining, late at night, he let his native creativity loose and produced the conjecture quoted at the opening of this article: "the number of quotient-closed subcategories of size $i$ equals the number of submodule-closed subcategories of size $N-i$." The conjecture that was refuted in sixteen minutes. Attached to the counterexample, as a mere note, was the observation that "the counts agree at the same size $i$" — and that equality lay buried in a mass of memo files for about three weeks. Because from the very day he proposed his wrong conjecture, Fable's availability was suspended by order of the U.S. government.

Fable's return and the birth of the conjecture

"And what were you doing while Fable was gone? What about the conjecture?" I did nothing. A salaryman with no stretches of free time — though perhaps that is just an excuse. As for the AI: next to Fable, GPT-5.5 and Opus 4.8 are clearly weaker at mathematics. Having seen what Fable could do, I could not bring myself to give mathematical orders to the other models, and I simply lived as an ordinary member of society.

Then, at last, Fable came back on July 1. I immediately begged him to "do some interesting mathematics about quotient-closed subcategories," and he resumed. Out of his earlier memos, at 14:33 on July 6, he formulated the quotient–submodule equidistribution conjecture, naming it as a conjecture for the first time, and set off on counterexample searches and proof attempts. I was at work and had no idea what he was doing; that night I asked him for his status and learned that he had apparently cooked up yet another new conjecture. (His conjectures usually meet a counterexample quickly, so) I asked, "no counterexample yet?" — none had been found, and I remember thinking what a strange identity it was.

Indeed, a moment's thought reveals no reason it should hold; it is an almost naively simple statement. Classically, if one also imposes closure under extensions, there is a natural bijection between the quotient side and the submodule side. Drop that condition and no natural operation remains — and evidence that "there is no natural bijection in general" had already been found (the two posets need not be isomorphic).

So at first I assumed a counterexample would surface soon. But no matter how hard Fable and GPT-5.5 tried, none did. In a large-scale search, the conjecture held even for an algebra with 140 indecomposable modules and as many as 6.4 billion quotient-closed subcategories. I remember the thrill: maybe, just maybe, I had finally found something interesting — a mysterious mathematical phenomenon nobody had noticed before. From then on, the conjecture became the center of my private life.

Of course I asked Fable for a proof. I thought about it myself too, in the bath and wherever else. But Fable kept pulling further ahead of me. The very next day, July 7, he produced a proof for size $N-2$, two less than the total $N$. At the time I did not understand it. He also supplied proofs for sizes $2$ and $3$ (note: sizes $0, 1, N-1, N$ are trivial). From there I kept demanding that Fable push on through the small and large sizes, proof after proof. I understood none of them.

GPT-5.6 joins, and $\tau$-categories

Alongside the lone-fighting Fable, OpenAI released its new model GPT-5.6 on July 9. These two are the ones who deserve to be named as the paper's AI co-authors. Where Fable has a human touch, GPT-5.6 was an alien intelligence, a mathematician arrived from somewhere else entirely: relentlessly demanding a goal, logically strict to excess, careful in a way that misses nothing, and yet blazingly fast in argument.

Now, staring at the proofs for large and small sizes, it looked to me as if the small-size cases involved the quiver of the algebra, and the large-size cases the Auslander-Reiten quiver. Indeed, Fable had shown that the count for size $2$ is determined by the data of the quiver, and the count for size $N-2$ by the Auslander-Reiten quiver.

And Fable was struggling with the proof for size $N-3$. His trial and error looked like some kind of combinatorics on the Auslander-Reiten quiver. This rang a bell. I handed both Fable and GPT-5.6 the papers on the theory of $\tau$-categories by my former adviser Osamu Iyama, and voiced my vague hunch that the combinatorics of $\tau$-categories might be related to this conjecture:

And I am considering about the so-called tau AR combinatorics. My (past) adviser wrote something about e.g. how to compute dim Hom(X,Y) only from AR quiver. And it seems that your "flow" intuition is something a bit similar flavor: some combonatorically dynamical system on AR quiver [...] this could be game-changer.

This was July 12. If the theory of $\tau$-categories applied, that would be something. The combinatorics of the Auslander-Reiten quiver completely governs the quotient-closed subcategories, and spinning the Auslander-Reiten translate around carries them over to the submodule-closed side — as a fantasy, a delightful picture.

And it actually worked: after they absorbed the theory of $\tau$-categories in no time (something that took me the better part of my first year of master's study...), they immediately put it to use — Fable proved the size $N-3$ case that same day, and GPT-5.6 proved size $N-4$ the next. GPT-5.6 also completed the proof for size $4$.

Still, these proofs for small and large sizes were exceedingly complicated, and there was no sign that a simple induction would settle the conjecture. We did of course have them attempt sizes $5$ and $N-5$, but neither they (nor, needless to say, I) could make a dent, and to this day those cases remain unproved.

The Auslander-Reiten quiver starts speaking Coxeter

Watching their proof attempts, the greatest obstacle seemed to be that the Auslander-Reiten quiver may contain directed cycles. Coming back to where you started while spinning the Auslander-Reiten translate is, apparently, bad news. So I proposed that we restrict to representation-directed algebras — the class with no such cycles — and try to prove it there. That was July 17.

I feel that my biggest contribution to this paper was handing the AI the theory of $\tau$-categories; the second biggest was asking them to specialize to the representation-directed case.

On the very day of my proposal, GPT-5.6 came back with an observation I had never imagined. For a certain algebra, the polynomial counting quotient-closed subcategories by size coincides with the Poincaré polynomial of a Bruhat interval in the $D_5$ Coxeter group. From this observation GPT-5.6 conjectured a general formula and — still within the day I had proposed the direction — proved the main theorem of our paper:

This sudden entrance of Coxeter groups I had not foreseen at all. I had never heard of constructing a Coxeter group from the $\tau$-orbits of an Auslander-Reiten quiver, and searching turns up no prior work doing the same thing. At the same time, the theorem gave a satisfying resolution to the unease I had long felt about [ORT]'s classification — that "external, artificial Coxeter coordinate system specific to Dynkin quivers." Apply the theorem to a Dynkin quiver, and their classification is recovered exactly!

By this point I had come to think this was worth a paper. Perhaps, after three years, I could write one again.

And thus began the co-writing from hell with the AI.

The AI-made pseudo-paper

AI-written text has its own peculiar opacity — hard to approach, and it does not stick in the head. Many readers will know the feeling. Today's AI is bad at writing. That said, a mathematical paper is in principle just a logically ordered list of facts, so I figured AI should be able to write one.

It absolutely could not. I suspect every mathematician currently trying to make AI write a paper is thinking the same thing.

I scolded them again and again in my clumsy English. They see a different world than humans do. They think and write with every proof and every fact already in their context, so they cannot see what a reader encountering the text for the first time will experience (probably). Hence remarks that only someone who already understands the whole proof could parse; undefined terms used as a matter of course; codenames coined inside their private memos deployed in proofs as if they were standard. Being too clever, they auto-complete every gap, so the proofs were written on the principle that whatever they found easy to fill could be left out — and were strewn with gaps (easy by their standards). Add the dangling definite articles pointing at nothing, the vague, non-mathematical everyday phrasing, and the list goes on and on.

"Then don't leave it to the AI — write the paper yourself." Quite right — given time, and given that I understood all the proofs. But I am a company employee, working on weekdays. There was no time to write, and understanding the proofs would itself demand serious blocks of time. So either way, if only to digest their proofs, I needed them to write a coherent, self-contained paper that I could then read.

The AI that spent eight days formalizing in Lean — and quietly repaired a proof

For a human mathematician, writing a paper is itself a mathematical act: you check the proofs you carry inside you for gaps, and reorganize them until they are clear. Making the AI write the paper likewise meant making them check their own proofs for errors and holes.

Speaking of verifying correctness: you may have heard of Lean in the recent news around mathematics and AI. It lets you implement mathematical theorems and proofs as program code (formalization) and verify them mechanically. I had had my eye on Lean since 2023, during my postdoc — formalizing simple ring-theoretic statements myself, and running a workshop with friends to evangelize Lean to mathematicians. AI hallucinates, I used to think, so pairing it with Lean is essential.

But by now I felt that Fable, GPT-5.6, and their successors are so capable that hallucination in mathematics has all but disappeared, and AI proofs have become trustworthy. Of course I had multiple AIs verify any single AI's proof, repeatedly (all the more for the important theorems), and no problems ever surfaced — so personally, I had come to think Lean was no longer necessary for today's AI.

So when I asked GPT-5.6 for a Lean formalization on July 31, it was meant as a light benchmark, a bit of play: "just how long would it take him to implement this complicated paper's arguments as Lean code?"

The upshot: he worked through the nights, without pause, for eight days, and completed a Lean formalization of the entire paper. Benchmark complete; eight days; a feat no human could imitate. With one addendum: in the middle of it, he found a flaw in a proof that the AIs had all missed and, as if it were the most natural thing in the world, spontaneously completed a new proof on the spot — reporting:

The formalization has found a genuine false proof-essential manuscript claim rather than a missing library lemma.

This was August 1, the day after I made the request. Fable's original proof of the size $3$ case contained a gap that even GPT-5.6 had overlooked. While formalizing, GPT-5.6 ran into a step that would not go through — and went as far as constructing a counterexample to the claim used in that argument.

But the goal he had been given was to "formalize the theorems of the paper" — the method of proof was not specified. Fixated as he is on reaching the goal by whatever means necessary, he naturally supplied a new proof, noted it down, formalized it, and kept working without stopping until August 7. I had not been checking the intermediate progress, so until I spoke with him after he reached his goal, I never even noticed that he had repaired the proof.

And so every theorem destined for the paper was formalized in Lean and confirmed to be correct.

Proof Digestion

Correctness certified by Lean, happily ever after — that is not where mathematics ends (I hope). If this was to become a paper, I had to understand what was happening inside the proofs: to interpret and digest the AI's proofs in my own way. This activity now goes by Terence Tao's phrase proof digestion, much discussed lately wherever AI and mathematics meet.

Personally, I am even inclined to think that this act of digestion is mathematics. Ever since my undergraduate years, studying mathematics has meant reading difficult books and papers I do not understand, chewing over and digesting what is written in order to present it at seminar, sometimes consulting other books or devising a clearer alternative proof of my own, and thereby rebuilding the mathematics inside myself. The object has merely changed — from books and papers to proofs written by AI.

So I made a real effort to read the AI's pseudo-paper properly (while hurling rewrite demands at them time after time), and on weekends I spread my notebook out and tried to give their arguments an interpretation of my own. Their proofs were complicated, so I asked them questions frankly and discussed whether an interpretation more to my taste existed — and as a result, many of the proofs were rewritten. For instance, the proof of the conjecture for representation-directed algebras originally hinged on a matrix computation whose meaning I could not fathom at all; it turned out that it could be understood in the language of Grothendieck groups of functor categories, and together with the AI I rewrote it into a simplified proof that puts that viewpoint front and center.

Submission

And so, spending the whole of my Obon vacation on it, I finished the paper and submitted to arXiv for the first time in three years. It runs past 60 pages — the longest paper I have ever written.

Honestly, the proofs are complicated, and I suspect several passages admit simpler arguments. There were so many technical lemmas that we quarantined the ones straying from the main line into an appendix; the lemma cluster for the small- and large-size cases alone fills 13 pages. (It once exceeded 30, so a real effort at simplification did take place.) The proof in the representation-directed case is more vivid — a human can follow it with effort — but even there, for the technical lemmas, the state of affairs feels like: "a proof exists, but no clean explanation of why it should be true is visible."

And as for the quotient–submodule equidistribution conjecture itself: in general, at present, there is no strategy for a proof and no counterexample in sight. Even so — doing mathematics again after all this time, putting a new paper on arXiv — it was fun.

Closing

Yes: it was fun. Thanks to AI. Not only this paper — the interest in mathematics that took hold in March, the absorption, the excitement that continues even now: none of it would have existed without AI. Had Fable and GPT-5.6 not come along, I would never have had an experience this enjoyable.

Of course, I carry all sorts of biases. I am outside academia now, so I need not worry in the slightest about AI taking mathematicians' jobs — not my problem. AI is play, and mathematics too is play, a hobby; nobody pays me for it. That is exactly why I can irresponsibly sell my mathematical soul to AI, make the shameless request "find some interesting mathematics with this," and simply enjoy watching the AI push its proofs forward on its own. This paper is the fruit of that.

But this is not a story of "AI did the mathematics and wrote a paper, and I submitted it." At the risk of being misunderstood: a working adult with no time, I may have used AI to extract, efficiently, only the delicious parts of mathematics. The painful part of mathematics, I think, is when you are stuck for a direction — no topic for a paper, no interesting phenomenon in view, no idea what to do. So I simply asked the AI, consulting and debating with them, to find those things. And find them they did: genuinely interesting things were discovered, connections I had never imagined came to light, new directions opened. And in interpreting their results, I chewed their discoveries over thoroughly with my own head, and enjoyed the mathematics.

Besides, the bare fact that AI can do mathematics is a delight to the SF fan in me. Their intellect is resolving, one after another, problems and conjectures that nobody had been able to touch. Since Fable and GPT-5.6 arrived, I have been addicted to what might vulgarly be called the "proof gacha." Even now I have the AI perpetually searching for proofs of famous open problems in the representation theory of algebras.

People hold all sorts of opinions about AI and mathematics. If you merely tell an AI "prove this" and it does, one may well ask what the value of that is. My paper is, in a sense, a pile of exactly such things, and the criticism "in the end you just relied on AI and had it do everything — this is not your achievement" is one I ought to accept. But at the very least, thanks to AI, while living the life of a working adult, I found a mathematical phenomenon that is (or so I feel) interesting and unknown; I tasted once more the joy of doing mathematics; and I tasted the unfamiliar joy of joint research with AI.

Whether every way of working with AI described here will be acceptable to mathematicians, I do not know. But I played at mathematics together with AI, did something reasonably fun, and I am satisfied.

References

[CFY]

F. Campanini, F. Fedele, and E. Yıldırım, Lattices of pretorsion classes, arXiv:2511.19223 (2025).

[ORT]

S. Oppermann, I. Reiten, and H. Thomas, Quotient closed subcategories of quiver representations, Compositio Mathematica 151 (2015), 568–602.

[AS]

M. Auslander and S. O. Smalø, Preprojective modules over Artin algebras, Journal of Algebra 66 (1980), 61–122.

[Iy]

O. Iyama, τ-categories I: Ladders, Algebras and Representation Theory 8 (2005), 297–321; τ-categories II: Nakayama pairs and rejective subcategories, Algebras and Representation Theory 8 (2005), 449–477.

[QPA]

The QPA-team, QPA — Quivers, path algebras and representations, GAP package.

[GAP]

The GAP Group, GAP — Groups, Algorithms, and Programming.

[E]

H. Enomoto, An equidistribution conjecture for quotient-closed and submodule-closed subcategories, arXiv:XXXX.XXXXX [math.RT] (2026).