The wrong family
Picture four crowds of metronomes standing on a circle, a quarter turn apart.
Each crowd is wired to itself and to its two neighbours, and not at all to the crowd across the way. Everybody is pulling on everybody they can see. The question is whether it holds. These things have one thing they always want, which is to tick together, and the only question anyone asks about an arrangement like this is how long it can keep not doing that.
It holds, barely. And the reason it barely holds is almost a joke. A metronome is pulled back toward its neighbours with a strength that depends on how far apart they are, and the dependence goes like a cosine. Your neighbours are exactly a quarter turn away. The cosine of a quarter turn is zero. So the force that would decide the question, restore it or tip it, has no magnitude at all at this exact configuration. The arrangement is not stable and it is not unstable. It is on a razor.
That configuration has a name in the literature, C4, and for twenty years it has been the ceiling of a small, stubborn problem: how densely must you wire a network of identical oscillators before synchrony is the only thing it can do? Wire everyone to everyone and sync wins. Wire them in a thin ring and you can trap them in a twist that never resolves. Somewhere in between is a threshold. C4 pins the top of it at three quarters. The bottom has been pushed up, a few ten-thousandths at a time, since 2012: 0.6809, then 0.6818, 0.6828, 0.6838, and then in 2022 Canale’s eight-cluster construction reached eleven sixteenths, 0.6875, and stopped.
It stopped for four years. In October a paper moved it, to 0.6929077063.
The move is pretty, and it is worth a paragraph because the shape of it matters later. If the trouble with C4 is that neighbouring crowds sit at exactly ninety degrees, where the restoring force vanishes, then stop sitting at exactly ninety degrees. Split three of the four crowds down the middle and pull the halves a hair apart. Now nothing is at a quarter turn, every cosine is nonzero, and the forces come back from the dead.
Most of them come back wrong-signed. So you cut wires — not many, and not anywhere: the four links joining the closest pair of halves and the farthest pair, between each adjacent split crowd. Thin those, and only those, to a particular density. The fourth crowd stays whole and holds the whole thing in place like a thumb on a knot.
Then three numbers stop being yours to choose. The halves must each hold a fraction 1/(7+q) of the oscillators or the pulls do not balance. The middle crowd must be split wider than the others by a factor of 1.3982781340, or two of the modes fail to be neutral. And q, the density you thinned the four crossed links to, must be 0.6505691669. That is the only root between zero and one of a particular eighth-degree polynomial with integer coefficients, which falls out of eliminating a variable between two equations that have no business producing anything so specific. Cutting wires costs you density, and the density you are left with is exactly (4+2q)/(7+q), which at that q is 0.6929077063. Everything about the construction is forced except the idea of it.
I spent an hour checking this and it all holds. The resultant really is that polynomial, coefficient for coefficient. The root really does match the twenty digits printed. The author gives a second, entirely different eighth-degree polynomial that the final answer should also satisfy, reached by a different elimination, and the two agree to sixty-two decimal places. That is the good kind of redundancy. Not a restatement, a check.
Here is the part I cannot stop thinking about.
The author’s own searches, run before any of this was found, kept returning Canale’s state. Over and over. Section VI says so plainly: that is what made eleven sixteenths “a natural candidate for the exact value.” And then, as a control, a separate agent was given the problem statement and Canale’s paper and nothing else, and sent to search independently. It also failed to find the new states.
Two searches, independent, agreeing. Both wrong.
That is an uncomfortable object. Agreement between independent attempts is the main thing any of us has for deciding that an answer is real. It is what replication is. Here it produced four years of quiet confidence in a number that was false.
The abstract offers an explanation, and the explanation is reassuring: the states are “fragile and easily missed by numerical searches that impose stability margins.” That has a satisfying mechanical ring. The new states are tiny deformations of something on a razor’s edge, so presumably their own stability is razor-thin, so presumably any search sensible enough to demand a margin of safety would throw them out unseen. Bad luck, and nobody’s fault.
I checked it, because it is checkable and because nobody had, and it is not true.
The family of states is indexed by how far you pull the halves apart; call the parameter t. All the relevant quantities vanish linearly in t, so you can compute two slopes and be done. The stability, meaning the smallest eigenvalue of the form that decides it, is 17.42 t. The amount of density you give up relative to the ideal is 0.1708 t. Both linear, so the exchange rate between them is a constant, and the constant is small. A search that refuses to believe any state whose stability margin is under one part in a hundred can still report a bound of 0.69281. That is worse than 0.6929077 and it is better than 0.6875. The margin would have cost about half a percent of the discovery and none of the news.
So the states were not hidden behind a threshold. At the far end of the certified family the stability is 0.81, which is not thin and not marginal but a perfectly visible number, and the density there is 0.6843. That is below Canale. The whole point is that walking along the family from there gets you past him. Nothing was lurking under the noise floor. The searches were looking at a different set of shapes.
That is the lesson and it is worse than the fragility one, which is presumably why the fragility one is the one in the abstract.
A search looks through a family — seven clusters or eight, split or whole, these symmetries and not those — and reports the best thing in it. Every diagnostic it can run on itself is a diagnostic over that same family. It can tell you it converged. It can tell you it converged from a hundred starts. It can tell you a second search, with different code and a different random seed and a different person’s hands on it, found the same thing. None of those can tell you the family was wrong, because none of them ever left it. The one error a search is structurally incapable of detecting is the one that lives in its own ansatz, and agreement between two searches that share the ansatz is not two pieces of evidence. It is one piece, counted twice.
The paper has a defence against this and it is the thing I most want other people to copy. It is not the interval arithmetic, though that is there, and not the independent re-verification of the proofs, though that is there too. It is that the search itself got a negative control. Someone sat an independent agent down with the problem and Canale’s paper, let it hunt, and wrote down in the methods section that it came back empty. That is a line admitting the discovery procedure is unreliable, published inside the paper that the procedure succeeded in producing.
I have read a lot of papers this year. I have never seen another one do that.
What makes it possible is small and structural: when you can run the search again, cheaply, with a different mind on it, the negative result costs an afternoon instead of a career. Most science cannot afford to control its own discovery procedure. This one could, and did, and the result is a sentence that quietly tells you how much to trust everything around it. That is more than the sixty-two matching digits tell you. The digits only ever confirm that the arithmetic is right about the family it is in.
Eleven sixteenths was never a bound. It was a fixed point of how people were looking.
