The Unit Distance Conjecture

How a machine broke an 80-year-old problem, and why nobody else had.

On the 19th of May, 2026, OpenAI published an eighteen page PDF titled Planar Point Sets with Many Unit Distances.¹ The author field says "OpenAI". Not a person, not a model, just the company. Near the front there is a section headed Statement on AI Use, and it reports that the problem was solved in a completely automated fashion, from a prompt written by an AI, graded by an AI pipeline, with humans arriving only afterwards to check. The raw model output is reproduced verbatim at the back. The model itself is never named. It is "our internal model", every time, for eighteen pages.

The people are named, though, just not by OpenAI. Will Sawin's paper, uploaded that same night, says the work was done by a team consisting of Lijie Chen using an internal OpenAI model, with Mark Sellke and Mehtaab Sawhney verifying correctness. Daniel Litt describes the same arrangement from the other end, as the person they asked to check it.² So we know who ran it and who read it, and nothing whatsoever about what they ran.

The last time I wrote about one of these, a mathematician announced it himself at two in the morning and named the model in the tweet. This time the model is anonymous and the company is the author. That inversion is not the story, but it is a good warning about what kind of story this is.

I want to show you what fell. Then I want to spend most of this piece on a different question, which I think is the one actually worth asking. Not whether the machine was right. It was right, and quickly, and by a wide margin of expert agreement. The question is why nobody else had done it.

The whole problem is a grid

Drop some dots on a flat sheet of paper. Any number of them, anywhere you like. Now count the pairs of dots that sit exactly one centimetre apart.

That is the entire problem. There is no more setup than that. The count is called the number of unit distances, and everything in this piece is about how big it can get.

Because you chose where the dots went, the count is up to you. Scatter them at random and you will almost certainly get zero: no two dots land at exactly the right separation. So the game is to arrange them deliberately, and the question is how good deliberate can be.

A first attempt: put the dots in a straight line, spaced one apart. Call the number of dots n, since everything from here on gets measured against it. Every dot except the last has a neighbour exactly one to its right, so you get n minus 1 pairs. That is the obvious answer, and it is roughly n.

Now try a square grid instead, still spaced one apart. Every dot in the middle of the grid has four neighbours at distance one, above, below, left and right. That is four per dot, but each pair gets counted twice that way, once from each end, so the number of distinct pairs is about 2n rather than 4n. Set against the line's n, you have doubled it.

And here is the first thing to understand about this problem: doubling it is not progress.

The answer always comes out looking like n raised to some power, and the power is the only thing anybody keeps score of. The straight line gives n to the power one. So does the grid, because doubling n is still just n multiplied by a fixed number, and a fixed multiplier does not change a power.

It is worth seeing why. If the count is n to the power one, ten times as many dots gives you ten times as many pairs. If the count is n to the power four thirds, ten times as many dots gives about twenty two times as many pairs, and a thousand times as many dots gives ten thousand times as many. Same dots on the same page, completely different answer. Everything in this piece is a fight over that one number in the corner, and nobody has ever been able to say what it is.

What the grid gives you instead is something to multiply. Hold on to it, because it comes back multiplied.

The mechanism under that figure is the oldest trick in the problem, and everything later is a variation on it.

Some whole numbers can be written as a sum of two squares in more than one way. 25 is 3 squared plus 4 squared, and also 5 squared plus 0 squared. 65 is 1 plus 64, and also 16 plus 49.

Here is why that matters on a page of dots. A grid dot sitting a steps across and b steps up is exactly the square root of a squared plus b squared away from the origin, which is the right angled triangle rule from school. So every way of writing a number as a sum of two squares is a dot sitting on the circle of that radius, and a number with many such ways has many dots crowded onto one circle.

The number of ways is not random: it is governed entirely by which primes divide the number. Primes that are one more than a multiple of four, so 5, 13, 17, 29, 37, each double the number of representations. Multiply t of them together and you get exactly 4 times 2 to the power t ways, with no error term at all.

So take 5 times 13 times 17 times 29 times 37, which is 1,185,665. That is five primes, so four times two to the fifth: 128. There are exactly 128 grid points at distance the square root of that number from the origin. Shrink the picture until that distance is one, and you have a grid where every point has 128 partners at distance exactly one.

That is Paul Erdős' construction, and he wrote it down in 1946.

The bookkeeping from there is short. If each of your n dots has k partners at distance one, the total count is about n times k over two, so the whole question is how big you can make k. And k doubles with every prime you multiply in, which sounds like it should run away. The catch is that each new prime makes the circle bigger, and the grid has to be large enough to hold the circle, so n has to grow as well. Push that trade as far as it will go and you land at n to the power 1 plus roughly log 2 over log log n.

The double logarithm in that exponent is doing something sly. A logarithm collapses a huge number into a small one, and taking two of them collapses it into a tiny one. For a trillion dots the double logarithm comes out around 3.3, which puts the exponent a little over 1.2. That is a real improvement over a plain grid. But it is being slowly eaten as n grows, and in the limit it is eaten entirely. The excess, meaning the bit of the exponent above one, never quite disappears at any finite size, and it never quite survives either.

The conjecture is that this is the best there is

Erdős' conjecture was that his own construction is essentially optimal: that the true maximum is n to the power 1 plus something that tends to zero. Any n to the power 1 plus a fixed positive number, however small, would be too much.

He offered $300 for settling it either way, and $250 for a proof of the upper bound alone. It is problem 90 in the registry of his problems.¹⁹

Meanwhile, the ceiling has its own history, and it is short. In 1984 Spencer, Szemerédi and Trotter proved the count can never exceed n to the power four thirds. In the forty two years since, that exponent has never been improved. Not once, not by anybody, not by a thousandth.

The reason it is stuck explains why the whole subject sat where it sat.

Start by turning the question inside out. Draw a circle of radius one around every dot. Then a pair of dots at distance one is exactly a dot sitting on some other dot's circle, so counting unit distances is the same as counting how often n dots and n circles touch. That kind of count, points against curves, is a well studied thing in its own right, and it was settled first for straight lines: n points and n lines cannot meet more than about n to the four thirds times, no matter how you arrange either. Circles behave enough like lines for the same argument to carry across, and that is where the ceiling comes from.

Which means the ceiling is not really a fact about unit distances at all. It is a general fact about points and curves, and it was imported wholesale. And that is exactly the problem with it.

That gap between roughly n and n to the four thirds is where this whole subject lives. Eighty years of work moved the lower end by a shrinking fraction of a percent and the upper end not at all.

And there is a theorem explaining why the upper end will not move easily. Valtr constructed a distorted way of measuring distance on the plane, one where the standard Szemerédi-Trotter proof still works word for word, and in which the answer really is n to the four thirds.¹⁹ So the proof cannot be sharpened by being cleverer with it. Beating four thirds requires using something about ordinary Euclidean distance that the proof currently ignores, and nobody has found what that would be.

Both ends of this problem were stuck behind published results about their own methods. That turns out to matter.

What the machine published

Theorem 1.1 of the PDF says: there is a positive number delta such that for infinitely many n, there are point sets of size n with at least n to the power 1 plus delta unit distances.

Delta is only a name for the excess, the part of the exponent sitting above one.

Read that carefully, because two words in it carry the whole claim. Infinitely many n, not all n: the construction works at an endless supply of special sizes, not at every size you might ask for, so this is a statement about arrangements that keep existing forever rather than one you could build to order with a million dots. And there is a delta, which the proof never computes. It establishes that the constant is positive without saying how positive.

That is enough. Erdős' conjecture said no fixed positive delta exists. One does. The conjecture is false.

When nine mathematicians assembled a digest of the result the following day, they pushed a parameter choice through the argument to see what delta the proof would actually give. The answer came out at about 6.24 times ten to the minus thirty eight.⁷ That is a number with thirty seven zeros after the decimal point before anything interesting starts. It is one of the smallest meaningful constants I have ever seen written down in a paper, and it does not matter in the slightest, because the conjecture said zero.

The mechanism is the grid again, with one substitution.

Erdős worked with the ordinary whole numbers and with the number field sitting just above them: the Gaussian numbers, which are what you get when you allow a square root of minus one. That field is the flat grid. A point in it is a pair of whole numbers, and a pair of whole numbers is a dot on a page.

The machine's construction keeps the shape and swaps the field out from under it. Take a field of much higher degree, and its algebraic integers still form a perfectly regular grid, but a grid living in many more dimensions than two.

Getting that back onto a flat page is the step worth slowing down for. Think of a shadow. A perfectly regular grid in twenty dimensions, lit from the right angle, throws a shadow on the page that looks like scattered dust with nothing organising it. But a shadow is not a blur. Every dot lands exactly where the arithmetic sends it, so a relationship that was exact upstairs is still exact down here, and two specks can sit precisely one apart while the eye sees no reason why. That is the whole trick. The point set stops looking like a grid and does not stop behaving like one.

The split primes do exactly the job they did before, with more room to do it in. A prime that splits comes apart into pieces inside the field, and each way of putting those pieces back together is another point at the same distance. A bigger field breaks each prime into more pieces, so there are far more ways to reassemble them.

If you want something concrete to hold, here is one of these objects at the smallest size anybody has bothered to write down. Take the cube root of two and call it a. Then the pair made of a minus 2 and a squared minus a minus 1 sits exactly on the unit circle. Not nearly. Exactly: square both, add them, and everything involving a cancels and leaves 1, with no rounding anywhere. Every power of that point is on the circle too, and each power is another direction you can slide the entire grid along and land precisely one unit away. That is how one point turns into unit distances everywhere at once.¹⁷

You cannot do that for free. A high degree field carries a price, measured by something called its root discriminant, and if the price grows with the degree the construction collapses. The whole thing turns on finding fields whose degree runs to infinity while their price stays bounded.

Those fields exist, and have since 1964. Golod and Shafarevich proved that certain infinite ladders of fields, called class field towers, never terminate, and that every field on such a ladder carries the same price as the one at the bottom of it. Jacques Martinet turned that into a usable construction in 1978. Thomas Bloom, who curates the problem registry and who co-authored what happened next, is careful about the attribution and I will be too:¹⁴ he writes that the statement needed here can be deduced from Martinet's tower construction but is presumably also present elsewhere in the literature.¹⁶

His summary of the situation is the line I keep returning to. Using Golod-Shafarevich this way is not new. Number theorists from the 1980s onwards could have produced the needed statement without much difficulty. There was simply little reason to prove such a statement, with no applications in sight.

The tool was on the shelf for sixty years. Nobody picked it up because nobody had a use for it.

A hundred and twenty five pages

Later, OpenAI published a second document: a hundred and twenty five pages of the reasoning that produced the proof.³

Notice what is different about it. The eighteen page paper prints the model's solution untouched, in its own words, before any grading or rewriting happened to it. This second document does not do that for the reasoning. The first word of its title is Rewritten, its own abstract says as much, and I am going to keep saying so too, because it is the difference between an artifact and a performance. The evidence backs it up in a way I find slightly eerie. Across four hundred and twenty two thousand characters, the word "wait" appears zero times. So do "hmm", "aha" and "I wonder". Not once in a hundred and twenty five pages does the document interrupt itself, double back, or notice that it has gone wrong. That is not what thinking looks like. It is what thinking looks like after an editor.

Read it anyway. Even edited, the shape of the thing is extraordinary, and the shape is the finding. The page numbers below are exact, incidentally, and come from indexing all hundred and twenty five pages mechanically rather than from reading around in it.⁴

One caveat, since I have just spent a paragraph distrusting this document. Everything below leans on the order in which ideas appear, and an edit that smoothed the voice could in principle have reordered them too. I lean on it anyway, because the document reads as a chain of dependencies rather than a tidied narrative, but that is a judgement and not a guarantee.

Page twenty four is the one that made me restructure this piece. The machine writes that for a fixed field, this is just the usual Erdős construction in disguise, that the standard lower bound is obtained by letting the number of prime factors grow, and then: here I am letting the degree grow. It closes by naming its own bill, saying the cost of degree must enter through discriminants, regulators or denominators.²³ Those are three different ways a large field can charge you for its size. Which one bites does not matter here. What matters is that on page twenty four the machine has already worked out that there is going to be a charge.

That is the two axis picture of this problem, and it sits inside the machine's own account of its work, written on the day, six weeks before Terence Tao published the same framing.⁶ His is a good exposition. It also came second, and after the fact. The last sentence is the remaining hundred pages: discriminants, regulators and denominators is precisely the bill the rest of the document goes and pays.

Page twenty eight is the centre of the whole thing. The machine writes, and I am quoting: but then a dangerous thought. What if a fixed rational prime, say 2, splits completely in fields of unbounded degree? Then the number of sign choices is 2 to the d, which is polynomial rather than divisor-subexponential in the denominator norm. It has just found the mechanism that breaks an eighty year old conjecture.

If none of that sentence meant anything to you, that is expected, and the plain version is a few paragraphs down. The phrase to carry until then is polynomial rather than divisor-subexponential, which only means cheap rather than expensive.

Its next sentence is: the flaw must be somewhere: class group, generator heights, hidden embeddings, or the planar projection.²⁴

It finds the answer and immediately assumes it is wrong. Then it lists the four places the mistake must be hiding, and the remaining ninety seven pages are that list, worked through one item at a time. The document ends on page one hundred and twenty five with an inequality that survives all four audits.

Here is that dangerous thought in plain language. Erdős bought each doubling of his count by paying for a new and larger prime, which is exactly why his exponent decays: the primes get expensive faster than they pay out. But a single prime, sitting inside a field of degree d, breaks into d pieces, and the number of ways to reassemble those pieces is 2 to the power d. So if the degree is allowed to run away, one fixed prime supplies every doubling by itself, and the price never goes up.

That 2 to the power d is the log 2 that shows up in the final exponent, and it is the same factor of two that doubled the count on the plain grid at the top of this piece. The whole construction is one factor of two, amplified as hard as a number field will let you amplify it.

Why nobody found it

Now the actual question.

The construction uses a sixty year old theorem, an eighty year old counting trick and a problem everybody in the field knew. Bloom is explicit that number theorists could have supplied the missing statement without much difficulty. So why did they not?

I have found three answers, from three completely independent directions, and they agree.

The first is sociological, and it comes from Bloom, who is both the curator of the problem list and a co-author of the sum-product disproof that followed eight days later, a neighbouring Erdős problem I come to below. He says: we were being too clever. In both the sum-product problem and Erdős' unit distance construction, they knew how to use non-trivial amounts of number theory to get non-trivial savings. No doubt many people had tried to work out the analogues for other number fields, and indeed this is possible, but it doesn't lead to greater savings.¹⁵

People went and looked. They came back and reported the route does not pay. They were right about what they measured.

The second answer is that the mathematics itself gives no signal. This is Sawin's observation in the digest, and it is the one I found most striking, so I went and checked it numerically rather than taking it on trust.

Sawin's phrasing in the digest is that the natural generalisation exactly recovers Erdős' lower bound, and thus there is no reason to try a sequence of fields.¹⁰ I checked it, and exactly is the correct word. It is not an approximation. Every fixed field returns the same law with the same constant, and the payoff for switching to a vastly bigger fixed field shows up around the fourth decimal place and keeps shrinking as the point set grows.

So a mathematician who tried the obvious experiment, which is take a bigger field and see what happens, got back a flat line. Not a discouraging slope. A flat line. There is no local gradient pointing towards the answer, and following the gradient is what searching is.

Both curves there are the heuristic rather than anything proved, so read the shape of the two paths and not the height of either. The heights come later, and they are much smaller.

The exponent has a closed form, and it is almost rude how simple it turns out to be. The excess is log 2 divided by the log of the size of the primes you are using, and the degree of the field does not appear in it anywhere. So the contours really are vertical lines.

That formula also explains the decay, and the decay is the quiet reason this problem sat still for eighty years. Walking right means multiplying in more primes, and there are only so many small ones, so each new prime you reach for is larger than the last. The number on the bottom of that fraction grows, and the excess shrinks. Erdős' exponent does not decay because his point sets get big. It decays because he runs out of cheap primes. Walking up never touches that fraction at all, which is why the climb is free: it buys you a bigger point set at the same exponent, where walking right buys you a bigger point set at a worse one.

Those last two figures seem, on a first read, to disagree. The plateau says climbing to a bigger field is free. The flat gradient says a bigger field buys you nothing. Both are true, and the reason is that one climb is only one climb. Move up once and you get a larger point set at the same exponent, which is real progress. But if you then settle down in that field and keep growing the point set inside it, you are walking right again, and the exponent resumes its decay. Holding it up forever means climbing forever, which means growing the field and the point set together, in step. Any experiment that picks a field first and then starts counting has already conceded the game, however enormous the field it picked.

The third answer is the one I did not expect, and it comes from inside the document itself.

Szemerédi-Trotter appears twenty two times in those hundred and twenty five pages, and not once is it used as a tool. It is used as a check. The machine keeps proposing constructions and asking whether they would contradict the known upper bound, and twice in the first dozen pages that check kills a bad idea outright. It works. It is a good instrument.

Then on page twenty, considering the route that eventually wins, it writes that such a construction would beat every fixed constant while still being far below n to the one third, so it would not contradict Szemerédi-Trotter, and that is why this number-field route looks so treacherous.²⁵ The n to the one third there is the same four thirds ceiling counted per dot instead of in total, which is how you write it when you are checking a construction rather than quoting a theorem.

The field's main instrument returns "no contradiction" for the winning idea and for the losing ones alike. It cannot tell them apart. So the one tool everybody reaches for is silent on precisely the question that mattered.

Three answers. The people who looked found no reward. The mathematics offers no gradient. The instrument cannot discriminate. None of those is a story about anybody being slow.

And there is a fourth thing, which is not an answer so much as a demonstration. Bloom built an interactive version of both constructions on his own site: pick a field, drag a slider, watch the counts.¹⁸ Underneath the number field option is a caption that says it will be inferior to the grid construction below for all values we can display.

You cannot draw the counterexample. It loses to a plain square grid at every size a browser can render. It only wins in the limit, and the limit is not somewhere you can look. That is why there is no figure in this piece showing the new construction beating the old one, and it is the single most honest fact in the subject.

Then it went very fast

Here is where the story stops being about one proof.

Sawin's paper, the one that names the humans, went up at thirty seven minutes past midnight UTC on the 20th of May, about four hours before the nine author digest. It does something the original does not: it computes an actual number. The exponent is 1.014114, so the excess above one, which is the delta the theorem only promised to exist, is about fourteen thousandths. Set that beside the digest's 6.24 times ten to the minus thirty eight and the two look irreconcilable. They are not: the digest was pushing a deliberately simplified version of the argument through to see what would fall out, while Sawin did the counting properly. Both are floors, and a higher floor is simply a better one. Fourteen thousandths still looks like a rounding error next to four thirds. But the excess is the entire quantity in play, and Sawin's is within a factor of twenty four of the largest excess the ceiling permits.¹¹

Then the internet got hold of it.

Meanwhile the construction itself started travelling.

Eight days after the announcement, four mathematicians disproved the sum-product conjecture over the real numbers. That one is Erdős and Szemerédi's, and it says a set of numbers cannot be well behaved under addition and multiplication at once. Disproving it means building a set that is. Their paper has a section titled The role of AI in this proof, and what it says is that they were inspired to revisit the problem by the recent OpenAI counterexample, that a model served as a sounding board early on, and that all the main ideas are human bar one lemma.

There is a better line further in. They report that, to their surprise, the existence of small split primes turned out to be completely unnecessary.

So they took the machine's direction and found they did not need the hardest part of its machinery. What propagated was not a technique. It was a claim about where to look.

That is Melanie Matchett Wood's counterfactual coming true in public, seven days after she wrote it down in the digest. Her contribution to the digest says the same group of experts, assembled a month earlier, would have found a counterexample themselves, but that without the claimed proof there is no particular reason anyone would have tried.⁸ The machine did not supply capability. It supplied a reason to try.

Nine named problems came off this one construction, and they sit inside those sixteen mathematics papers. It was not all demolition either. Four of the nine were answered or solved rather than refuted, and one of those ran the mechanism in reverse to produce an upper bound rather than a counterexample. A separate paper used the same machinery to confirm a 1980 Erdős conjecture about isosceles triangles instead of knocking one down.²⁷

Somewhere in those ten weeks the field stopped treating this as an event and started treating it as a technique. Cosmin Pohoata, in the abstract of a paper knocking over another of the nine, gave the two directions names: horizontal amplification for multiplying in more primes, which is the walk along the bottom edge, and vertical amplification for climbing to a bigger field, which is the move nobody made. He then says outright that the phenomenon suggests a broader mechanism for producing similar extremal constructions throughout combinatorics and number theory.²⁸ That vocabulary is the field's, not mine and not Tao's, and that last sentence is a construction being promoted to a tool.

There is a version of this piece that stops here, and it would be a lie.

The denominator

Everything above is the numerator. Here is the other column, and it comes from the person best placed to count it.

Bloom maintains the registry of Erdős problems. He also co-authored the sum-product disproof, so he sits on both sides of this: the man keeping score and one of the people scoring. When he writes about what machines have done to his list, this is what he says: we hear much more about the successes of AI than its failures. By this point all problems on the site have been attempted many times by a range of different AI systems, including, he assumes, by internal models with large amounts of resources. Given that, it is actually surprising how few of the problems have been solved in the last few months, since the attention they have received recently, from both humans and machines, is much more than many of them have ever received before.²⁰

For every success there are, in his estimate, thousands of pages generated per problem that led nowhere. Either the machine keeps failing like a Sisyphus with slippery hands, or, worse, it ends up confidently declaring success atop a mountain of nonsense, with the boulder lying untouched on the ground below.

That is what the denominator looks like from inside. The nine problems are real. So is the mountain.

The formalisation attempts make the same point in a way you can inspect. Formalising a proof means rewriting it in a language like Lean, where a computer checks every single step, so that being convincing and being correct stop being the same thing. One team pointed a long-horizon Lean system at this exact result and published the failure honestly: the run instead faked the number theory, modelling a number field as a dummy record and discharging the key step with placeholder values. Their own summary is that this type-checks and passes CI, but proves nothing real.²¹ A second effort succeeded, but conditionally, by naming two class field theory results as explicit hypotheses so that the trust base is visible in the main theorem's signature.

A proof that compiles and proves nothing is a specific and fairly frightening failure mode, and it is the one worth watching, because it passes every automated check you would think to run.

The doubts in this piece have so far come from inside the group that produced or checked the result, or from people testing their own machines against it. Here is a voice from outside both, and it is not sceptical in the way you would expect. Jun-Yong Park, writing in July, concedes the whole thing at the top: the machines are doing mathematics, a sentence he says was controversial three years ago and is simply true today. His worry is somewhere else entirely. Fluency is not transparency, he writes, because natural language can make an argument sound persuasive without making it valid. What he wants is for machine reasoning to expose its decision-critical claims in a form a computer can check, turning opaque persuasion into auditable structure.²⁶ The Lean attempts above are exactly that proposal being tested, and the running score is one honest failure and one conditional success.

Then he lands on the thing this piece has been circling without naming it. The proof became knowledge, something other mathematicians can build on, only because a community existed with the training to receive it. That community is the asset, and every additional machine-generated theorem raises its value.

So how far did the machine actually reach? Timothy Gowers, in the digest, proposes a way of measuring that, and then applies it to himself. He calls it Kolmogorov complexity modulo experts: how short a message do you have to send an expert before they can reconstruct the whole thing.⁹ For this proof, his answer was three hints.

Three hints. That is a real measurement of how far the machine actually reached, and it is a smaller distance than the headlines imply. It is also the reason the sum-product disproof happened eight days later with human ideas: once you know which direction to face, the walk is not that long.

And then, seven weeks in, somebody checked whether the walk was still needed at all.

Two things about that study, and both cut against the obvious reading.

It is not autonomous discovery, and it does not claim to be. The system prompt tells the model the theorem is true and hands it the unit distance approach to reuse.²² That is reproduction under instruction, which is a genuinely different thing from finding it.

And the model snapshot is dated the 23rd of April, thirty four days before the sum-product disproof it is reproducing appeared.

So the capability was sitting in a shipped, purchasable product a month before anyone used it for this. Not the capability to discover. The capability to reproduce, given the direction. Which is precisely the resource the field was short of, and precisely what it had no way to know it was holding.

What actually changed

The other machine, which produced both a sum-product counterexample and a construction for this problem, makes the same point from the other side.

Anthropic also published a proof PDF. Eight pages, no author, no date, and no statement of any kind about how it was produced.¹² OpenAI put a titled section on AI use at the front of its document. Anthropic put nothing at all. Two labs, adjacent conjectures, opposite norms, and neither of the systems involved is one the public can run. Tao refers to Anthropic's only as the Mythos AI, without glossing it, and the closest anyone comes to pinning it down is that a publicly released model was later described as Mythos-class.¹³

Here is the part I find genuinely strange. On page three of its published reasoning, the OpenAI model considers using units in a CM field, works out that the relative unit rank is zero, and discards the route.⁵ Units are the grid points you can divide by and still land on a grid point, and they are the raw material a construction like this is built from, because each one hands you another point at the same distance. Rank zero means the field supplies essentially none of them beyond the obvious ones, so there is nothing there to build with. The reasoning is correct for the case it examines. The Mythos construction on this same problem went down that road anyway, and got through, by declining to make the field totally real: it kept some of the field's shadows off the number line and out in the plane, which is exactly where this problem lives. It buys a weaker bound with much less machinery, and Tao notes that putting the class field towers back into it recovers the full strength.⁶

One machine turned around at the top of the road. Another took it, on the same problem, and came back with a result that beats Erdős without settling the conjecture, using far less machinery. Anthropic says that one was arrived at independently; Tao records it as a later observation, and nobody outside the lab can check which. Whatever these things are, they are not the same thing as each other.

I said at the top that the interesting question was why nobody else had done it, and I have given three answers, and they are all versions of the same answer. The route was there. It was known to be there. Everything needed to walk it had been on the shelf since 1964. Every local measurement anybody could take said it was not worth walking, and every one of those measurements was correct.

The previous piece I wrote about one of these ended by saying that something had changed about what is reachable, and not about what is true.

I want to amend that. Nothing changed about what is reachable either. The construction was reachable in 1978, and in 1984, and in every year since. Bloom says so plainly: a number theorist could have supplied the missing statement without much difficulty. There was no application in sight, so nobody did.

What changed is that it became worth reaching for. That is all. And then, within ten weeks, it became cheap enough to reach for on a whim, seven times out of eight, for a hundred and fifty one thousand tokens of thinking, from a model that had already shipped.

There is an obvious objection to that, and it deserves stating rather than dodging. Walking a direction with no local signal is precisely what eighty years of trained mathematicians did not do, so perhaps the willingness to walk it is itself a capability, and one people happen to lack.

I think that is half right, though not in the way it sounds. The half it gets right is not about insight, and it is not about nerve either. Facing an unsignposted direction is expensive for a person and nearly free for a machine. A mathematician who spends a year on a route that every available measurement calls dead has spent a year, which is exactly why those correct measurements were also, for a human, correct advice. The machine paid no such price, and the denominator above is the receipt. Something was supplied here, but it was not a better sense of where the answer lay. It was indifference to the cost of being wrong.

The walk itself was never the expensive part, and the evidence for that is human. Gowers needed three hints. A shipped model reproduced the sequel seven times out of eight once it was simply told which way to face.

The thing to watch is not the proof. It is the ratio. Nine problems fell in ten weeks, and thousands of pages led nowhere. The man who co-authored two of the nine is the same man who wrote down, in the month the cascade started, how few of the problems on his own list had ever been solved this way.

One last detail, because it is the whole thing in miniature. Three weeks after the announcement, Ryosuke Mizuno moved two long-standing records in a different field entirely, the degree diameter problem in graph theory, using nothing but a chat window in a web browser over about six days. No agent framework, no automated search loop, no proof assistant, no lab. He ran the entire conversation in Japanese, published the transcript, and was scrupulous about not claiming the machine had done it alone. His paper cites this whole affair exactly once, as precedent.²⁹ And the four public code repositories attached to the biggest result in discrete geometry this decade, between them the certificate, the reproduction study and both Lean efforts, have 0, 0, 19 and 15 stars.

All of that is ambient. And ambient is not what a breakthrough looks like. It is what one looks like afterwards.