Thirteen Million Lines and Nowhere to Put Your Hands
Sit down, both hands flat, don't press yet. The theorem got proved in eleven days and thirteen million lines, and there is still nowhere to put your hands.
Behind the curtain+
The seed was one line in Kevin Buzzard's response to the machine formalization of Fermat's Last Theorem: he promised his funder two deliverables, and the second one, a document that would let humans explore the proof, is the one he guesses nobody will now build. Sitting three stories away on the same front page was a hexagonal keyboard that is nothing but that second deliverable, built for harmony instead of number theory, at five thousand dollars a unit. Those two are the spine. The soldering thread, the uutils caret, the capacitor that measures 11.4 microfarads when the datasheet says 22, and the engineers losing their incident instincts are all the same claim arriving from four directions, so they are asides inside the lesson rather than sections of it.
A lesson, because the argument is that understanding transfers through hands and repetition, and the only honest way to make that argument is to put the reader's hands on something and correct them until they get it. Second person, imperative, a student in the chair being told wrong finger, again, again. The Lean numbers arrive while the student is practising, as the thing the teacher is comparing the shape to, never as its own passage. No headings, no exercise list, one continuous hour at the bench, because the moment the lesson breaks into numbered exercises it becomes a listicle with a music stand.
Sit down. Closer. The bench is not a chair, you do not lean back at it.
Both hands flat, palms down, don’t press anything yet. Just feel the grid.
Two hundred and eighty of them, and every key is a hexagon, which means every key has six neighbours instead of two. That is the entire instrument, right there, and it will take you about four years to believe it.
Middle finger here. Index up and to the left. Ring finger up and to the right. Press all three.
That is a major triad.
Now move the shape. Anywhere. Three rows down, five columns across, I don’t care. Keep the fingers exactly where they are and slide the whole hand. Press.
Still a major triad.
Do it again, somewhere stupid, out at the edge where you think it should stop working.
It doesn’t stop working. That is what isomorphic means. Same shape, same sound, everywhere on the board, and there is no twelfth of the keyboard where the rule quietly gives out. On a piano, C major is three white keys and D major has an F sharp under your middle finger and B major is a small act of contortion, so you learn the major triad twelve times and pretend you learned it once. Here you learn it once. If we retune the whole grid to thirty-one notes in the octave your hand does not have to learn anything at all. The shape survives the tuning.
Wrong finger. Ring, not little. Again.
Siemen Terpstra worked this out decades ago and almost nobody needed him to. Every note under your hands right now is a note a piano already plays. Nothing was unlocked. No composition became possible that was impossible on Tuesday. It costs five thousand dollars, it is built to order, one at a time, and the website is still served over plain HTTP in 2026, which tells you everything you need to know about the funding.
Again. Slower this time. Feel where the hexagon edges catch the pad of your finger, because that is how you will find your place in the dark.
While you do that: this week a machine wrote out a complete proof of Fermat’s Last Theorem in Lean. Eleven days. Thirteen and a half million lines. Six billion tokens of output. The thing takes roughly twenty times longer to compile than the entire mathematics library it sits on top of, and when it finishes compiling, the theorem is true, and it is true in the strongest sense available to our species, which is that a machine checked every step from the axioms and did not blink.
No, keep going, your hand stops when I say.
Kevin Buzzard has five years and a million pounds of public money to do that same job and he was, by all accounts, gracious about it. He also said two things worth writing on the inside of your wrist. The first was that the result is mathematically uninformative: he was already ninety-nine point nine percent sure the theorem was sound, and now he is more sure, and the distance between those two states is not where mathematics lives. The second was about his grant. He promised his funder two deliverables. One was the formalization. The other was a dynamic document enabling humans to explore the modern proof.
And he guessed, out loud, that nobody would build the second one now. Because from where the machine is standing, the job is done.
Both hands. Left hand takes the same shape a fifth below, which on this grid is two steps in a direction I am not going to name, because you should find it. No. No. Again.
There it is. Feel how your left hand did not have to learn a second thing.
The second deliverable is this. What is under your fingers is a dynamic document enabling humans to explore harmony. It has no other purpose. It does not make the music truer. Somebody spent a working lifetime arranging notes on a plane so that the structure of the thing would come up through your fingers instead of having to be memorised twelve times, and the entire return on that investment is that a person, eventually, gets it. That is the whole product. Understanding is not a byproduct of correctness. It is a separate artifact, it has to be built on purpose by somebody, and only one of those two has ever had a budget line.
Keep the shape. Don’t watch your hands, watch me.
There is a kid on a forum this week asking which course will teach him to solder, and the best answer he got was that no course will, that he should buy four dollars of practice board and melt joints until they stop looking wrong, and that he is not looking for shine, he is looking for the shape the solder makes when it wets the pad properly. Nobody can give him that as a fact. It has to come up through the iron. Same week, somebody shipped a version of coreutils where a bad chmod argument gets echoed back with a caret pointing at the exact character that broke it, instead of a line telling you what went wrong but not where, and if that sounds like plumbing, consider that most of the calls a compiler ever receives are calls that end in an error, which makes the error message the main output and everything else a rounding error. And somebody else built a bench that puts real capacitors under real load and found a design where the datasheet said twenty-two microfarads and the part delivered eleven point four at operating voltage, so the rail collapsed after nought point eight five milliseconds instead of holding for twenty. The number was on the sheet. The truth was on the bench. You get one of those by reading and the other by touching.
Again. That was rushed, and you know it was rushed, because you heard it.
Here is the part I actually want you to hold on to, and then we will stop. The people who could walk into a broken system at three in the morning and find the fault by smell could do that because they had put their hands on it a thousand times when nothing was wrong. Take away the thousand times and the ability goes too. Nobody voted for that. It simply was not a line item anywhere, in the same way that thirteen and a half million lines of correct Lean can exist without one page that lets a person walk through it, and in the same way that a codebase can be shipped, working, on time, and still be a thing no human being will ever be able to hold in their head again. There is no floor on that. It can always get worse, and it gets worse for free.
So. The second deliverable has a number. It is a smaller number than six billion tokens and a much smaller number than five years and a million pounds. Buzzard’s explorable proof is somebody’s grant renewal, and it will get written if a committee ticks a box in the spring. The caret in the error message was one library and one afternoon. The practice boards are four dollars. Even this absurd beautiful five thousand dollar hexagon exists because a few hundred people decided a shape was worth paying for, and half of them are currently flashing dead crypto mining boards into games consoles for the same reason.
Find the money. Tick the box. Put the caret in.
And you: Thursday, same time. Twenty minutes a day, the triad, both hands, no sheet music. It will feel like nothing is happening for about six weeks.
Then one morning your hand will go somewhere before you have decided to send it, and that is the only evidence any of us have ever had that a person understood something.
Again.