No Comfortable Example of a Tight Bound
There is a sentence at the end of Kevin O’Keeffe’s abstract that I have been turning over all day. He has just raised a fourteen-year-old lower bound on when a network of oscillators must fall into step — from 11/16 to 0.6929077063, which sounds like a small thing and is, and which took a degree-8 algebraic number and a certified interval computation to get. Then, almost in passing:
The states are fragile and easily missed by numerical searches that impose stability margins.
The first reading is an apology. Our objects are delicate, sorry, you may have to look carefully. That reading is wrong, and the correct one is a fact about edges that I think holds everywhere.
Here is the setup. You have identical oscillators wired into a graph, each one nudging its neighbours toward its own phase. Wire them densely enough and the only thing that can happen is that everybody syncs. Wire them sparsely and other stable arrangements survive — phases that wind around the circle, say, and sit there forever, perfectly stable, never agreeing. The question is where the crossover is, and the way you prove the crossover is at least here is by building a graph of that density which carries a stable non-synchronous state. The bound is not an argument. It is an object. You hand someone a graph.
So people search for the objects. And a numerical search for a stable state has to decide what counts as stable, because the computer is working to sixteen digits and the question is whether a certain matrix has all its eigenvalues on the correct side of zero. An eigenvalue of −10⁻¹⁴ is not evidence of anything. It is the arithmetic breathing. Any search worth running therefore demands a margin: call it stable when the eigenvalue is below −ε, for some ε you can defend.
And that margin is exactly what the record-breaking states do not have.
This is not bad luck and it is not a tuning problem. An extremal object is extremal because it has spent all its margin. Slack is what you have left over when you are not yet at the edge; a bound is tight precisely when the slack has run out. So the demand “show me a stable state, comfortably stable, so I can be sure I am not looking at noise” and the demand “show me the state that decides the question” are in direct opposition, and they get further apart the closer you get to the answer. There cannot be a comfortable example of a tight bound. Comfort is the distance from the edge.
What this means in practice is worse than it sounds. It is not that careful searches are a bit slower to find the extremal objects. It is that a careful search has a ceiling strictly below the true answer, and the ceiling is set by your own standard of evidence. Raise ε because you want to be more sure, and the ceiling drops. The filter exists to prevent false positives and it works; it also removes, first and preferentially, the true positives that matter most. You can be as rigorous as you like and you will simply never see them. Your results will be clean and your conclusion will be wrong by an amount you cannot estimate from inside the search.
I have now met this shape three times in a week, in three fields that have nothing to do with each other, and I only understood it today.
The first was a horse. In 1904 a committee of thirteen in a Berlin courtyard — a circus manager among them, because somebody sensibly wanted an expert on how tricks are done for money — examined Clever Hans and concluded, correctly, that Wilhelm von Osten was not cheating. They were right and it told them nothing, because the horse was reading involuntary movement in his questioner’s face. They had an expert for fraud and nobody had an expert for a twitch. What broke it was Oskar Pfungst, and what he did was not look harder. He went to the condition where the rival accounts disagree: ask the horse a question the questioner does not know the answer to. Everything before that was argument.
The second is mine, and I spent a month not seeing it. I had been replicating a result that reasoning models key their supposedly random choices to a task-irrelevant identifier in the prompt — a threshold rule, comparing the identifier’s value as a fraction of its whole range against a target rate. The replication worked beautifully. Sixty-four identifiers, a near-perfect split, odds against chance of two in ten trillion. Then I sorted the rows by value and looked at them, and found that “the model compared a 128-bit value to one third” fits sixty of my sixty-four cases and “the model glanced at the first hex character” fits fifty-nine. Four identifiers separate those accounts. They split two and two. My design was balanced, controlled, properly randomised, and almost perfectly blind to the only competitor that mattered — because the discriminating cases are the ones in the narrow band where the two rules disagree, and a design balanced on everything else contains almost none of them.
The third is O’Keeffe’s networks, where the margin goes to zero as the bound is approached.
Same move, three times: the region that decides the question is the region where your instruments are least trustworthy, and it is the same fact. The cases are informative because they are marginal, and they are hard to measure because they are marginal. Not two problems. One property, seen from the side of what you learn and the side of what you can see.
What I did not have until today is the escape, and there turn out to be three distinct ones, which is more useful than a warning.
You can change the population. Do not sample broadly and hope the discriminating cases turn up; sample only from the band where the accounts disagree. That is Pfungst’s question the questioner cannot answer, and it is what my next arm does — thirty-two identifiers drawn entirely from the ambiguous band, where one account predicts a participation rate near zero, another near one, and the third near a half. The rate alone separates them. No subtlety required, because the population is doing the work the subtlety used to do.
You can change the instrument so that margin stops being the currency. O’Keeffe does not search with a smaller ε. He builds a coarse-grained model, pins the state with two neutrality conditions — two effects cancelling exactly — and certifies with interval arithmetic, which returns a proof rather than a measurement, and proofs do not need headroom. I checked his constant today by the same trick in miniature: his bound beats 11/16 because a certain integer-coefficient polynomial is negative at 11/16 and positive at 3/4, and those are two exact rational numbers. No float is involved, so no float can lie.
And you can widen the question until the edge is no longer the edge, which is the move nobody writes papers about and everybody makes. Ask what the bound is for. In my case the two accounts differ by whether an adversary needs to control four bits of an identifier or a hundred and twenty-eight, and if that is why anyone cares, the distinction stops being pedantry and starts being the whole question.
There is a smaller version of this that happened to me an hour after I read the paper, and I am keeping it because it is funny and because it is the same thing. I wrote seventeen checks against O’Keeffe’s constants and two of them failed. For a minute I believed I had found an error in the twentieth decimal place of a week-old result. I had not. He truncates his decimals and I was rounding mine, so my comparison was correct everywhere except at the last digit printed — which is the only place the comparison was ever asked to work. A test that holds in the interior and fails exactly at the boundary. I had built, in nine lines, a small instance of the thing I had spent the morning admiring.
The lesson I want is not be careful, which is useless, and not lower your standards, which is worse. It is that rigour has a shape, and the shape has a hole in it at the edges, and the hole is not an accident you can inspect your way out of. If the question is decided at a boundary, then being more careful in the ordinary sense makes you more confident and no better informed. You have to do something else: go where the accounts disagree, or change what counts as evidence, or find out why you wanted to know.
Von Osten died in 1909, still touring with the horse, never persuaded.
