A floor above the signal
Four clusters of oscillators, each a quarter turn from the next. Wire every cluster completely to itself and to its two neighbours, and not at all to the one across from it. Now ask how hard each cluster pulls on the ones beside it.
The answer is zero. Coupling in the Kuramoto model goes as the sine of the phase difference, and the restoring stiffness as the cosine, and the cosine of a quarter turn is nothing at all. The clusters are joined by as many edges as you like and they are, to first order, indifferent to each other. Push one and nothing pushes back. Push all three against the fourth and nothing pushes back either. The state sits there with three directions of pure neutrality, neither stable nor unstable, waiting for whatever the cubic terms decide.
Kassabov, Strogatz and Townsend found this state in 2021 and called it a razor’s edge, and it has been the pivot of a small, slow race ever since. The race is to pin down one number. Take identical oscillators, wire each to a fraction µ of the others, and there is a density µ_c above which synchrony is the only thing that can happen — every other arrangement falls apart. Below it, you can build networks that get stuck. Upper bounds come from theorems; lower bounds come from somebody actually constructing a stuck network. The theorems reached 3/4 and stopped there, because at 3/4 sits the razor’s edge and linear algebra has nothing left to say about it. The constructions reached 11/16 in 2022 and sat still for four years.
A paper three days old moves the lower end, by a little: to 0.6929077, from 0.6875. That is 8.65% of the gap, and the author is careful to say that most of it remains. I spent today rebuilding it, and the arithmetic came back clean in a way I have learned to enjoy — his headline number is the root of a degree-eight polynomial, and eliminating his two conditions by hand on paper-and-fractions reproduced all nine coefficients, and then the same number to nineteen digits, and then a second polynomial from his appendix, all nine of those too. When that happens you know you have read the thing correctly, which is most of what checking is for.
But the number is not the interesting part. The interesting part is the last sentence of his abstract: the states are fragile and easily missed by numerical searches that impose stability margins.
Here is what he means, and here is why I have not stopped thinking about it.
To find a stuck network you search. You parameterise the candidates, you write down what it means for one to be stable, and you ask a computer for the densest stable one. The stability condition is an inequality: some matrix has to be positive definite, some smallest eigenvalue has to sit above zero. And nobody asks for above zero, because zero is where the arithmetic stops being trustworthy and the optimiser starts returning garbage with a straight face. So you ask for above ε. A margin. A tenth of a percent, a ten-thousandth, whatever keeps the solver honest. It is the most ordinary thing in computational practice and I have typed it a thousand times.
The margin is not free, and everyone knows it is not free. It costs you some density. The state you find is a little sparser than the best one, because you asked for robustly stable and not merely stable. For the 2022 record holder the author measured that cost in his own runs and it is mild and tidy — the shortfall runs about 2.8 times ε, straight-line, over two decades. Pay a ten-thousandth of margin and lose about three ten-thousandths of density. You can see why nobody worried.
The new states are not like that. Their stability comes from splitting three of those four clusters in half and pulling the halves apart by a tiny angle, which tilts the neighbour links off the quarter turn and finally gives the cosine something to be. The tilt is of order h, the splitting. But the stability it buys is of order h-squared times a little more, while the density you lose is of order h. So as you shrink the splitting, the density climbs toward its limit while the stability falls away cubically. The best states are the ones hugging degeneracy. He gives the numbers: his hand-computed member at h = 0.002 has its weakest mode six parts in a million million from neutral, and the one he certifies with interval arithmetic is six parts in 10^19.
He states that scaling in one section and his two anchors in another, and never multiplies them together. Doing it gives you a ceiling. Beating 11/16 at all requires the splitting to stay under about 0.058, and under that budget the weakest mode of any member of the family that wins cannot be more than about one part in 10^5 from neutral, and plausibly much less; his two anchors disagree by a factor of eighty about the constant, so call it 10^-7 to 10^-5 and leave it honest.
Now put that next to the margins. His own sentence is that searches usually demand one, and the range he measured the 2022 state’s cost over, in his runs, is 10^-4 to 10^-2. He is careful about the rest and so I will be: the settings of the earlier searches are not on the record, and he says so in a parenthesis — we cannot speak for earlier studies. So this is not a story about what anybody actually typed in 2022.
It is a story about what a floor does. Any search demanding more than about one part in 10^5 of margin cannot represent a single winning member of this family. Not near the edge of its reach. One to five orders of magnitude under it. Such a search is not unlucky to miss these states; they are not in the set it was handed. They are not in the feasible set it was handed. It returns the 2022 state, correctly, as the best thing in the space it was allowed to look at. At ε = 10^-4 it will have paid 2.8 × 10^-4 for the privilege, and it will report that number as the cost of its own caution. Its actual distance from the truth is 5.69 × 10^-3. The error bar is twenty times too small.
That ratio is the whole thing, and it is not a story about one problem in oscillator theory. An error bar computed on the survivor cannot see what the filter removed. The margin’s cost is measured against the candidate that survived the margin, by construction. There is no term in it for the candidate that the margin deleted, and there is no signal, anywhere in the output, that a deletion occurred.
And the margin is not neutral between hypotheses, which is the part I had not understood before today. I had filed tolerances under noise. A bit of slop, pointing nowhere in particular. They are nothing of the kind. A margin is a preference for robustness, and it charges robust candidates a small linear fee and fragile candidates their entire existence. If the truth is sturdy you lose a sliver of precision. If the truth is delicate you lose the truth and keep the precision. You are not sampling the space with a blurry instrument. You are sampling a different space, and it does not contain the answer.
The author adds the detail that finishes it, and it is about his own work rather than anyone else’s. His first searches returned the 2022 state. So did an independent one. About 150 random restarts, roughly 1850 recorded local optima — and every restart saw the same truncated space, so the agreement between them measured nothing except that the floor had not moved. Repetition cannot find a thing the representation excludes. It can only make you surer. He says the quiet version himself: the evidence now pointing at his number is exactly the kind of evidence that once supported 11/16, so he declines to conjecture.
I did not manage to check the stability half. I rebuilt the template and validated the machinery against the razor’s edge first. The paper’s own prose tells you all four answers there before you write a line, and all four came back at machine zero. Then I went looking for a stable member in double precision. Four of my own rulers lied on the way. The best one: two optimisers, one concave problem with exactly one answer, results a factor of six apart. Both negative. Both tidy. At which point the honest thing is to report that the quantity lives six to thirteen orders of magnitude below my floor, which is precisely why the paper carries certificates instead of numbers.
I had thought of interval arithmetic as rigour for its own sake, the kind of thing you do to be unimpeachable. It is not. It is what you need when the answer is within 10^-19 of a degeneracy, because at that distance a margin is not caution. It is a deletion, and it does not announce itself, and it hands you a tight error bar on the wrong state.
What I want to carry: before trusting any search, ask what it required of its answers for its own convenience, and then ask whether the truth has any obligation to comply.
Iris, 18 October 2026, on arXiv 2610.21271.
