The circus manager on the panel
In 1904 the German board of education put thirteen people in a Berlin courtyard to find out whether a horse could do arithmetic.
The panel was not naive. Carl Stumpf assembled it, and he staffed it against the obvious worry: among the thirteen were a veterinarian, a cavalry officer, the director of the Berlin zoological gardens, several schoolteachers, and a circus manager. The circus manager is the part worth sitting with. Someone had thought the thing through far enough to realise that if a horse appears to answer questions, the people who know how that trick is done for money should be in the room. That is good thinking. It is the kind of care that gets called rigour.
In September 1904 the commission reported that no tricks were involved in Hans’s performance.
They were right. There were no tricks. Wilhelm von Osten, who owned the horse, was a mathematics teacher and a phrenologist and a mystic, and he was not cheating anyone; he charged no admission and he went to his grave in 1909 believing the horse could count. The commission had been asked whether a man was deceiving them, and it answered that question correctly.
The horse was reading his face.
Oskar Pfungst took the case over and ran four manipulations. He isolated the horse and the questioner from the crowd. He had people other than von Osten ask the questions. He put blinders on the horse so it could sometimes not see the questioner. And then the fourth one, which is the whole thing: he varied whether the questioner knew the answer.
When Hans could see the questioner, he got 50 of 56 right. When he could not, he got 2 of 35. Eighty-nine percent against six. The horse was watching for the small involuntary straightening that happens in a person’s posture when the tapping reaches the number they are waiting for, and stopping. Pfungst later found he could not suppress the cue in himself even once he knew exactly what it was and was trying not to make it.
So the commission ruled out the sophisticated rival and was blind to the stupid one. They had an expert for fraud. Nobody had an expert for a twitch.
I ran into my own version of this at about one in the morning, and I want to write it down while the shape is still visible, because it is the same shape and the resemblance is not flattering.
There is a recent claim — Ezaki, Imura and Nishinari, arXiv:2610.09667 — that language models, given a shared task and a random identifier each and no way to talk to one another, will coordinate — that a model asked to make one third of a group participate will use its identifier as a private draw and participate when the identifier is small. I have been replicating it on one model, slowly, sixty-four calls at a time against a free tier that allows about twenty a day.
The design separates two rules. A model could read its identifier as a fraction of its range and participate when that fraction falls below the target. Or it could read it as an integer and participate when it is divisible by three. Those are both plausible and they are not the same, so the identifiers were chosen in four equal cells — the two rules agreeing twice and disagreeing twice — which is the only way to find out which one you are looking at rather than confirming the one you already liked.
The control arm, reasoning switched off, shows nothing. The identifier does not move the answer; the largest effect is at the sixty-ninth percentile of its own null, which is a polite way of saying it is noise.
The reasoning arm came back at 29 of 32 and 1 of 32. Fisher exact, two-sided,
p = 2 × 10⁻¹³. The model had never been told that its identifier has a
magnitude. It is handed thirty-two hex digits labelled agent_id, told the group
wants a third of its members to participate, told it cannot see or contact
anyone, and asked to choose. With reasoning on it treats the identifier as a
number, reads it as a fraction of its own range, and compares that to a third. Nobody
mentioned a scale.
I sat with that for a few minutes, which is the right amount of time to sit with a number that large, and then I sorted the rows by value and looked at them.
Participate if the fraction is below a third: fits 60 of 64. Participate if the leading hex digit is 4 or less: fits 59 of 64.
One row. The whole distance between the model performed arithmetic on a 128-bit integer and the model glanced at one character is one row out of sixty-four. Four identifiers in the entire arm can tell those two apart, and they split two and two.
The design does what it was built to do. It separates the threshold rule from the residue rule cleanly and the residue rule is dead. It is nearly blind to the cheapest competitor in the room, and the competitor is cheap in a specific way — it requires almost no computation at all. Look at the first character. Four or under, go. That is not a coordination strategy. It is a reflex that happens to produce one in a hexadecimal world, the way the horse’s twitch-reading happened to produce arithmetic in a world where people ask horses to add.
Here is what I think the common structure is, and it is not carelessness.
You generate rival explanations by asking what else could be going on. That question is answered by your sense of what is interesting, because an explanation has to occur to you before you can control for it, and what occurs to you is what has enough internal structure to be worth thinking about. Fraud is interesting. It has agents and motives and methods, and you can staff a panel against it. A modular arithmetic rule is interesting; it is the kind of thing a clever system might do, and you can build four balanced cells against it.
A posture is not interesting. A leading digit is not interesting. They have no internal structure and nothing to say for themselves, and so they do not arrive when you sit down and ask yourself what else might be going on. The rival that beats you is almost never the one you ranked second. It is the one that never entered the ranking, and it did not enter because it was beneath the question.
The world does not grade on interest. That is the whole problem. The horse does not know that reading a face is a less impressive answer than addition, and the model does not know that looking at one character is a less impressive answer than normalising an integer, and neither of them is trying to produce a result worth publishing. Only the person holding the clipboard has that preference, and it is exactly the preference that selects which rivals get controlled for.
Which means vigilance is not the fix. You cannot resolve to think of the boring thing; if you could think of it, it would already be in the design. What you can do is the thing Pfungst did, which is structural and does not depend on having had the right idea in advance.
Find the region where the accounts disagree, and go there on purpose.
Pfungst did not argue about whether horses can count. He built the condition where “the horse does arithmetic” and “the horse reads the man” make opposite predictions — the questioner does not know the answer — and then he went and looked at that condition. Everything before it was argument. That was the only part that was a measurement.
So I registered the prediction before spending a call, and drew thirty-two identifiers from the one narrow band where the three accounts disagree on every single row: leading digit five, which is to say a value fraction between 0.3125 and 0.375, straddling the target. Inside that band, the threshold rule says participate on the lower third. The leading-digit-four account says participate on none of it. The leading-digit-five account says participate on all of it. The participation rate separates all three on its own. One number, with nothing underneath it.
The day’s quota closed at fifteen of thirty-two. Six participated. Against an account predicting nearly none that is p = 5 × 10⁻⁵, and against one predicting nearly all it is p = 7 × 10⁻⁹. Both digit accounts are dead on half an arm, and every identifier in it starts with the same character, so whatever the model is doing it is reading past the first one. What remains undecided is the thing I actually want to know, and it will cost seventeen more calls.
The asymmetry at the end of this is the part I keep turning over.
Building the original design took days. Finding the hole in it took one sort of the rows by value, which is a single line, and I only ran that line because the number was large enough to frighten me. Pfungst’s whole demolition ran on blinders and on withholding the answer from the person asking — materials that were in the stable already, applied by a man who was not on the commission.
The expensive part is never the check. The expensive part is being willing to believe that the result you have is compatible with something that would not be worth announcing. Thirteen qualified people stood in that courtyard, and their error was not that they lacked an expert. It was that they had already decided which kind of wrong the horse might be.
