Why an 80-year-old geometry problem, a curse of dimensionality, and a Neolithic campfire suggest we’re misreading the most interesting instrument we’ve ever built, and why the fear and the wonder turn out to be the same evolutionary bug.
TL;DR:
in May 2026 an AI cracked an 80-year-old maths problem by welding together two corners of mathematics nobody had thought to weld. The case: AI is not a mind, it’s an instrument, a «cognitive electron microscope» that explores the space of what we already know and reveals doors we never walked to, and that, structurally, it can’t step outside that space. The Skynet panic and the digital-messiah rapture are the same evolutionary bug, an over-eager predator-detector misfiring on a fluent text box. The real worry is who’s holding the microscope, not a ghost in the machine. Right, off you go. The rest is for people who like the scenic route.
I’m a chemist, so not a mathematician (sometimes we’re not friends with that field), and not an AI person. So I want to say that up front because most of what follows is well outside my comfortable cottage. However, my brain is a compulsive interface-hunter, as soon as you give it two fields. It’ll spend the following hours poking at the seam between them, looking for a door. No surprise, it’s literally why this blog is called Between Systems.
So when I read, back in May 2026, that an AI had cracked an 80-year-old geometry problem by welding together two corners of mathematics nobody had thought to weld together, I immediately wanted to know more. The machine was doing the thing my brain does for fun, and that’s maybe the key to the whole public confused conversation about what these things are. Let me try to make the case, carefully, knowing I’m a guest in someone else’s field.
An 80-year problem, cracked in one shot?
First, what actually happened, because headlines oversold it and the precision matters.
In 1946 Paul Erdős asked a deceptively simple question: take n dots on a flat sheet, how many pairs of them can sit exactly one unit apart? For nearly eighty years the working belief was that arrangements looking roughly like square grids were essentially the best you could do. In May 2026, an internal OpenAI reasoning model didn’t «solve» the problem, it disproved that belief, producing an infinite family of point arrangements that beat the grid by a polynomial factor. (Will Sawin at Princeton later pinned the improvement to an explicit exponent, δ ≥ 0.014.) The exact answer to Erdős’s question is still open. So a long-standing conjecture fell but the problem itself didn’t and we must keep that distinction in mind.
But the detail that matters for everything below is not what fell, it’s how. The proof reached into algebraic number theory, Golod–Shafarevich, infinite class field towers, machinery that specialists (not me) knew perfectly well but had never thought to point at this problem. Notice the structure of that: nothing new was invented. Two rooms of the existing house got connected by a door nobody had opened.
The mathematicians who digested and checked the proof wrote something, in their companion paper, that I find beautiful: that the AI is helping us «explore the cathedral of mathematics we have built over the centuries», and that what it keeps turning up are connections we’d simply never thought to look for.
A map, not a mind
My “blog message”, in one line: a large language model is an explorer of the space of things humans already know, including the enormous regions of that space we’ve written down but never visited.
To picture the space, let me steal a tool from my last post. There I argued that patients don’t fill up the high-dimensional space of their biomarkers; they live on a thin, low-dimensional submanifold inside it, and the geometry is upstream of everything. Same picture here, one floor up. All of human knowledge, every paper, every proof, every recipe and argument, is a fantastically high-dimensional but extremely sparse structure: a manifold of the known, floating inside a much bigger space of the possible. We’ve nailed down points on it but between the points run connections we never yet thought to draw, doors between rooms, exactly like Erdős × algebraic-number-theory.
I’d love to call the LLM an «interpolator», but I have to be careful, because, and here’s a delicious irony, Yann LeCun himself (with Balestriero & Pesenti, 2021) showed that in high dimensions «interpolation», in the strict geometric sense, essentially never happens: almost any new point you pick lands outside the convex hull of your data. So I don’t mean it that way. I mean something more like: the model travels the manifold of the known and lights up the hollows, the regions that were always reachable from what we’d written down, but that no human trajectory had ever passed through. The Erdős door was there the whole time. The model just walked through it.
If you happen to be a chemist like me, you already own this intuition, a calibration curve is the whole argument in miniature. Your calibrators are the known points; a sample landing inside the calibrated range is interpolation, and you trust it; one landing outside is extrapolation, and you don’t, you redilute and run it again. With a single curve, one dimension, there’s plenty of room to fall inside. Add a second analyte and the sample now has to sit in range on both axes at once. Add a few hundred and «inside» shrinks to almost nothing, our old friend the curse of dimensionality. The manifold of the known is just that calibrated region, drawn in a space with a preposterous number of axes, and a model «interpolating on the manifold» is doing what you do every time you refuse to trust a point past your top standard, only in a space too big to ever plot.
And now the limit, and the important word is structural, not technical. More parameters, more data, more compute: that explores the hollows faster, deeper, more thoroughly. It does not let the thing leave the manifold. You cannot interpolate your way to a point that has no coordinates in the space you were trained on. Whatever a genuinely new concept is, it’s off the map, and an instrument built out of the map cannot, by construction, point off it. It’s not «hasn’t yet», it’s «Can’t».
What would it take to leave the map?
This is exactly where people like LeCun show up waving non-LLM architectures, «world models» that learn structure from raw experience rather than from our text about the world (LeCun, 2022). In principle such a system could grow representations with no foothold in any human language at all: not hollows in our manifold but points on a different one entirely.
Which opens a lovely problem: translatability. If a machine builds a concept that has no anchor in math, chemistry, music, or speech, how would we ever read it? My optimistic guess is: with a translator AI in between, plausibly an LLM, doing for us what your own brain already does between its parts. You run constant translation between the module that does language, the one that does spatial intuition, the one that (in my case) does chemistry, you just never notice the customs checkpoint.
But, and this is the part I can’t wave away, some structures might be untranslatable not in practice but in principle. And here you have to be surgical, because there are two very different claims and only one of them survives.
Claim one: untranslatable to formal manipulation. That one is weak, a human can, in principle, follow any finite procedure and push any symbols around given enough patience and coffee.
Claim two: untranslatable to intuition, we might compute with a structure forever and never see it the way we see a triangle. That one is strong, and quantum mechanics is the standing proof: we work the formalism flawlessly, and, as Feynman liked to needle, nobody really understands it. A dog will never get quantum mechanics, however long you give it, not for lack of time, but for lack of architecture. The honest, slightly vertiginous question is whether there are structures that are to us what quantum mechanics is to the dog, except we won’t even have a formalism to fake our way through.
(David Deutsch would tell me I’m flat wrong here, that humans are «universal explainers» and nothing is forever beyond us. He might be right. I genuinely don’t know. Did I mention I’m a chemist?)
Fire before chemistry
Notice something we’ve quietly slid past: we’re already using outputs we don’t understand. The Erdős proof got checked, refined, and built upon before anyone had fully digested why the trick works.
This is not new. This is the oldest move our species has. We had fire for hundreds of thousands of years before we had the faintest clue what combustion was, no oxygen, no oxidation, no thermodynamics, just: this is warm, this cooks the meat, do not touch the bright part. The chemistry showed up tens of thousands of years late and didn’t make a single campfire any warmer. There is no alarming paradox in an instrument whose outputs are usable before they’re understood; that is the normal, ancient condition of every powerful tool we’ve ever picked up.
A cognitive electron microscope
So here’s the metaphor I’d like to put on the table, against the «it’s a new lifeform» crowd and the «it’s just glorified autocomplete» crowd, both of whom I think are missing it.
An AI is a cognitive electron microscope. *tada.wav*
A microscope doesn’t think and isn’t alive. It extends a perceptual capacity into a space we can’t reach with our naked senses, and what it shows us is really there, sitting in the sample, just below the resolution we were born with. The electron microscope didn’t invent the virus; it let us see one. This model didn’t invent the Erdős construction; it let us see a door in our own cathedral that had been bricked into the wall the whole time. And, look again, that’s the exact verb the mathematicians reached for — explore, reveal, rather than create. An instrument for seeing into the space of our own ideas, the overwhelming majority of which we have simply never gone and looked at.
The bush that wasn’t a tiger
So why does almost nobody talk about it this way? Why is the public conversation pinned between Skynet and salvation?
Reason 1: I’m completely wrong, okay, sorry for that.
Reason 2: Because of a bug.
A specific, ancient well-documented bug in the standard-issue human cognitive kit. The ancestors who heard a noise in the bush, assumed predator, and bolted lived to have descendants. The ones who shrugged «probably just wind» occasionally got eaten (oops, not this time). So every one of us is running an over-eager agent-detector, Justin Barrett literally named it the Hyperactive Agency Detection Device, that fires on the faintest whiff of intention. It’s the thing that sees faces in clouds and a will behind the thunderstorm (Guthrie, 1993). And it is exactly why a text box that answers you fluently feels like a someone.
We have known this precise failure mode for sixty years. In 1966, Joseph Weizenbaum wrote ELIZA, a program of a few hundred lines that did essentially nothing but bounce your sentences back to you as questions, and people poured their hearts out to it, convinced something on the other side understood. Weizenbaum was horrified for the rest of his life. The «ELIZA effect» isn’t an analogy for what’s happening now; it’s the same phenomenon, scaled up by about nine orders of magnitude of fluency.
And the bit I think goes under-appreciated: the fear and the wonder are the same bug. Skynet-panic and digital-messiah-rapture both come from the agent-detector misfiring on an instrument. They feel like opposite reactions but they’re one reflex, pointed at one illusion, the imagined mind behind the text. (Epley, Waytz & Cacioppo, 2007 give the cleaner account of when we anthropomorphize: roughly, when a thing is unpredictable, when we’re lonely, and when we badly want to make sense of it. For a chatbot in 2026: check, re-check, and re-re-check.)
What the bush actually hides
Now I have to turn the knife on my own argument, because this is the exact spot where a «relax, it’s only an instrument» take usually cheats, and I’d rather catch myself than have you catch me.
Debunking the agent-detector does not debunk every worry. It debunks one worry: that the model wants something. It says precisely nothing about the perfectly real, perfectly agent-free risks, that a planetary-scale cognitive instrument concentrates power in whoever owns it, reshapes who gets paid for what, can be aimed by ordinary humans at ugly ends, and breeds dependencies we don’t fully see yet. A microscope has no agenda but the institution holding the only microscope in town very much does. So I won’t tell you «nothing to see here». The thing to worry about is us, what we do with the instrument, not the ghost we keep hallucinating inside it. Pretending the second reassurance follows from the first is the sleight of hand, and I refuse to perform it even though it would make for a tidier ending.
So what I’d actually like
What I’d like, and again, chemist, off-piste, take it for what it’s worth, is a public conversation that runs on the microscope frame instead of the predator frame — not because it’s «superior» in some hierarchical, I-know-better way, but because it’s more operational: it tells you what to do, point the instrument at good problems, and watch closely who owns it, instead of what to feel, which is terror or rapture, take your pick of the same misfire.
The telescope and the microscope didn’t change what science is. They shoved out the edge of what we could see, and the maps quietly redrew themselves around the new view. I suspect this is the same species of moment, with the instrument turned, for the first time, on the space of our own ideas. We’d do well to be a lot less interested in whether it’s alive, and a lot more interested in where, exactly, it’s about to show us a door.
Header image by Cash Macanaya
Postscript — how this was made
I’m a chemist, and most of the above lives outside my field. My job teaches me to hunt for meaningful interconnections, and over time I’ve built a «gut-feeling» for which ones might actually matter. So, based on that instinct, I built this post in dialogue with an AI — the very instrument it’s about — which I used exactly as I describe: to trace connections out of my domain, and to stress-test my reasoning until it stopped leaking everywhere. This post, the judgment, and any surviving mistakes are mine; it mostly handed me references and poked holes. Make of the recursion what you will.
The maths. OpenAI’s announcement; the human companion paper «Remarks on the disproof…» on arXiv; Gil Kalai’s write-up for the mathematician’s-eye view; Will Sawin’s note pinning the exponent to δ ≥ 0.014.
Why «interpolation» is the wrong word in high dimensions. Balestriero, Pesenti & LeCun (2021), Learning in High Dimension Always Amounts to Extrapolation.
World models / leaving the map. LeCun (2022), A Path Towards Autonomous Machine Intelligence. And, for the opposite bet, David Deutsch, The Beginning of Infinity (2011) (book, ed.: PenguinRandomHouse) on humans as «universal explainers».
Discovery from data as a mode of science. The «fourth paradigm» — Hey, Tansley & Tolle (2009), The Fourth Paradigm: Data-Intensive Scientific Discovery.
Why a text box feels like a someone. The ELIZA effect — Weizenbaum (1966); the Hyperactive Agency Detection Device — Barrett (2004), Why Would Anyone Believe in God?; Guthrie (1993), Faces in the Clouds; Epley, Waytz & Cacioppo (2007), On Seeing Human.
The geometry of «the space of the known» is the same submanifold picture from More biomarkers, better diagnosis?
