# The abelian sandpile of research mathematics

[*Written without AI*](https://blog.danromik.com/on-provably-writing-without-ai)

Everyone is still struggling to understand what exactly happened on October 6 when OpenAI dropped its 372-theorem “[mathocalypse](https://scottaaronson.blog/?p=10169)”. Is AI "[the end of math as we know it](https://www.quantamagazine.org/is-ai-the-end-of-math-as-we-know-it-20261005/)"? Is mathematics research as a human enterprise dead, or is AI only going to kill the results that are [at distance one away](https://blog.danromik.com/the-feedback-loop-of-mathematics-research) from existing human knowledge? Should mathematicians [grieve](https://xenaproject.wordpress.com/2026/10/01/to-grieve-or-not-to-grieve/), or be excited, or both, or neither?

I want to offer a way of thinking about what happened.  
**Warning: it’s nerdy.** 🤓🤓🤓

**TL;DR: The frontier of mathematics research is an *abelian sandpile* exhibiting *self-organized criticality*. The OpenAI paper drop has triggered a *collapse to sub-criticality*.**

Woah, that’s a bunch of fancy buzzwords. Let me explain.

[The Abelian Sandpile Model](https://en.wikipedia.org/wiki/Abelian_sandpile_model) (ASM) imagines dropping grains of sand at random positions above an initially flat surface, one grain at a time. Some grains settle in place when they fall in relatively flat areas; others trigger small “avalanches” of cascading grains of sand when they fall in unstable places that cannot support one more grain; yet others trigger larger cascades of varying sizes. What sort of shape does the sandpile assume over time, and what properties do such sandpiles have? Bak, Tang, and Wiesenfeld who introduced this model in 1987 and studied it, discovered a remarkable phenomenon. The sandpile created by the dropping process is “at criticality”. What this means is a kind of fragility where even a small event of dropping a single grain of sand can trigger a chain reaction that sends sand grains flying in all directions, sometimes a very large distance away, as the sandpile loses its current equilibrium state and finds a new one. A power law governs the relationship between how far away the cascade propagates and the frequency of cascades propagating for that distance.

Here is what a sandpile looks like (source: [Wikipedia](https://en.wikipedia.org/wiki/Abelian_sandpile_model); in this simulation 30 million grains are dropped at the origin):

![An Abelian sandpile (source: Wikipedia)](https://cdn.hashnode.com/uploads/covers/69f6f0080ab374db99854035/7483868b-fa7d-4d79-8fc1-10139cd30a5e.jpg align="center")

The discovery of the abelian sandpile model triggered an avalanche(!) of research activity — there were 1000 papers citing the BTW paper within 10 years, and over 7500 citations at present. The model's characteristic feature of so-called “self-organized criticality” is now regarded as a very widespread and broadly applicable phenomenon.

Let’s talk about scholarly research in mathematics before AI came along. Think of every new paper being published as a grain of sand dropping on a sandpile. Many papers settle down in place without causing any disruption. Occasionally a paper triggers a minor avalanche, leading to 5-10 new papers being written over the course of (typically) a few years. If you’re a successful research mathematician, you might write a few papers over the course of your career that make a bigger impact, receiving more than 100 citations. Once in a a blue moon a major paper is published that disrupts an entire field, triggering a major avalanche.

The corpus of published literature in mathematics is the sandpile. Having existed for a long time, at the dawn of the AI age it was at criticality, just like an abelian sandpile in the ASM.

Then AI came along.

The effect of AI as we are presently experiencing it is that it changes the parameters of the sandpile dynamics. Grains of sand are being dropped at a much faster rate than they did before. More importantly, the places *where* they’re being dropped are changing: local critical sandpile peaks that were previously resistant to human attacks — representing major open problems such as the complexity of integer multiplication and the location of the zeros of the Riemann zeta function — just had grains of sand dropped on them, triggering [spectacular avalanches](https://app.getsupers.com/auto-research) that are likely just beginning. The system has been whacked way out of its previous equilibrium.

What will happen next? With its changed dynamical laws and following OpenAI’s bombshell October 6 release, the sandpile of mathematics research is currently in a *sub-critical* state. That means that the research landscape is now full of low-hanging fruits that are ripe for the picking by whoever comes along and has the patience to study the OpenAI papers (not an easy task, admittedly) or even direct an AI model to try to improve them.

I predict that we are in for a head-spinning period of time when the wealth of new techniques discovered by OpenAI’s internal model is studied, optimized, and deployed to maximal effect. (Actually I predicted this already in an [earlier post from August](https://blog.danromik.com/some-recent-ai-produced-mathematical-advances-and-their-significance); see my discussion in that post about an “optimization phase”.) This is [already happening](https://gtimg.github.io/QuasiRiemannTracker/) and will result in the creation of much new mathematical knowledge, even without AI models getting any more powerful than they currently are. And, of course, no one believes they will not get more powerful soon.

## The emotional lesson

One of the remarkable things about recent events is that we are seeing some of the leading mathematicians in the world reacting quite emotionally to the OpenAI papers release. Fields Medalists and well-known mathematicians are [sharing their feelings](https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/) and [behaving in public like, well, human beings](https://www.lemonde.fr/en/science/article/2026/10/12/openai-s-latest-blow-to-mathematicians-they-have-annihilated-my-field-of-research_6758515_10.html).

Does the above analysis tell us anything about how we should *feel* about the recent developments? I’d argue yes. I think this picture points more strongly in the direction of excitement than of frustration and grief. Did OpenAI solve a problem you’ve been working on for the last 10 years? So what! For every problem OpenAI solved, it opened 20 amazing new research directions we could now go in. To me it seems baffling to be so upset. Something has been lost, yes; but something very big has been gained as well. Even if all you care about is having interesting things for you and your students to work on, it’s a stretch to claim that OpenAI has taken that away from you.

I’ll frame it differently. If you are a mathematical explorer looking to make new discoveries, would you like to begin your quest for knowledge in a research sandpile that is at criticality? Or are you better off being dropped off onto a sub-critical sandpile? Which one of those sandpiles offers better opportunities for striking proverbial gold?