The clock nobody watches
A clock has to be paid for. Not the one on your wall, which runs on a battery, but the kind a cell builds out of chemistry: a circadian rhythm, a cell cycle, the segmentation clock that lays down a spine one stripe at a time. Those tick because something upstream is burning free energy, and if the burning stops the ticking decays into noise within about one period. This is not a metaphor. It is a budget, and over the last decade people have worked out roughly what the line items cost.
In 2022 Oberreiter, Seifert and Barato proposed a price list. Take any system that hops between discrete states, write down its matrix of transition rates, and find the eigenvalue with the second-largest real part. Call it minus lambda_R plus i lambda_I. The real part is how fast a disturbance dies away and the imaginary part is how fast the system goes round, so their ratio is a quality factor, in radians of phase per decay time. Divide by two pi and you have the number of whole oscillations you get to see before the wobble is gone. The conjecture was that the entropy you must produce per second is at least lambda_I squared over lambda_R. Good clocks are expensive, and here is the invoice.
It held up under a lot of poking. Several groups proved restricted versions of it, in the weak-noise limit and for linear Langevin systems, and nobody found a counterexample for four years.
On the twenty-eighth of September, Naoto Shiraishi posted one.
The machine that almost never runs
His construction is three rings, stacked. Picture three circular tracks, each with the same number of stations, and a particle that walks around whichever track it is on and occasionally jumps between tracks at a station. Two of the tracks are dead: the particle diffuses on them, drifting left as readily as right, costing nothing. The third track is driven. On that one the particle is pushed, gently but persistently, one way round.
The driven track is where the oscillation lives. It is a real oscillation, with lambda_I twice lambda_R, and it is exactly the eigenvalue the price list is reading.
Then comes the trick. The jump rates between tracks are tuned so that the particle spends almost none of its time on the driven one. Its stationary probability of being there is of order epsilon squared, where epsilon is the same small number that sets how gently it is pushed. The clock is perfect and the particle is essentially never in the room where the clock is.
So the invoice goes unpaid. The entropy production falls to zero like epsilon, the second eigenvalue sits stubbornly at the same place it always was, and the inequality breaks by a factor that grows without bound.
Seventy-two states
The proof is asymptotic. It is about what happens as the rings get long, and the paper does not print a single number for any particular size, which is the normal and reasonable way to write such a thing.
I wanted to know how small it can be.
The answer is 72 states. Twenty-four stations, three tracks, every transition rate strictly positive, nothing clever hidden anywhere. At that size the conjecture asks for an entropy production of 4.0041 and the system pays 3.8622. In the paper’s own variables the entropy per oscillation is 12.136 against a demand of 12.582. The conjecture is false by three point seven percent on an object you could write out by hand on two sheets of paper.
I checked that one at sixty digits, because a three percent margin on a system sitting against the edge of its own parameter range is exactly the kind of result that turns out to be a rounding error. It is not. The column sums are zero to fifty-nine decimal places, the smallest rate in the whole matrix is five thousandths, and the eigenvalue comes out of an exact cubic whose residual is identically zero.
That number is worth sitting with for a second. A conjecture about Markov chains, supported in its original paper by numerical simulation, survived four years and several partial proofs, and it is wrong on a chain small enough to have been enumerated. There was no need to go to the asymptotic limit at all. Nobody had looked in the small cases, because the obvious way to search is to generate random systems, and the counterexample is not a random-looking system. It is a specific pathology you have to want to build.
Across a sweep of six hundred points in the construction’s parameter space, four hundred and thirty-five of them violate the conjecture. The counterexample is not a knife edge. It is a whole region that happened to be outside everyone’s search.
Two things called the amplitude
The paper explains its own result in one sentence, and that sentence is where this gets interesting.
The usual reason people trust the second eigenvalue is that the slowest part of a correlation function looks like A times a decaying sine. Shiraishi points out that nothing guarantees A is big, and writes that his counterexample has vanishing amplitude on that mode. The sentence is doing the real work of the paper and A is never defined and never computed.
So I computed it. There are two honest ways to define the amplitude of a mode, and the construction does opposite things to them.
Measure the oscillation as a fraction of the fluctuation of whatever you are watching, and you get one. Not a small number: one, to six decimal places at the largest size I ran, approaching from above as the system grows. For the right observable the coherent mode is not a small component of the correlation function. It is the entire correlation function.
Measure it instead as an absolute signal size, holding the observable to a fixed scale that does not let it concentrate on the rare track, and it collapses. It goes to zero like epsilon squared, and not merely like it: the absolute amplitude turns out to equal the stationary probability of being on the driven ring, exactly, with the prefactor coming out as the ratio of states to stations in the normalisation and nothing else. Change the normalisation and the prefactor goes to one.
The amplitude is not vanishing. The amplitude is maximal. What vanishes is how often you are allowed to look.
An observer watching only the driven ring sees a textbook damped oscillation, two radians of phase per decay time, no ambiguity whatsoever. That observer is simply almost never collecting data. The suppression lives in the prior, not in the signal, and the word amplitude slides between those two without warning. One sentence in the paper is true under one reading and false under the other, and it is the reading that makes the mechanism legible that is true.
What survives
If the problem is that the second eigenvalue reports a mode without saying how much of the system is in it, the obvious repair is to charge for the mode in proportion to how loudly it actually speaks. Multiply the right-hand side by the absolute amplitude.
That fails, and it fails in a way I did not expect. On ordinary systems the absolute amplitude is bigger than one, typically around 1.4, so multiplying by it makes the claim stronger than the original conjecture rather than weaker, and a strengthened version of a tight bound is just a new false statement. Seven out of a hundred and fifty random oscillating chains break it.
Capping it at one fixes that. Take the smaller of one and the amplitude, and the bound reduces to exactly the old conjecture whenever the mode is loud, and discounts it only when the mode is faint. On any uniform ring, which is the family the original paper tested against, the amplitude is exactly one by a Parseval argument, so the repaired version is identical to the original there, by identity and not by luck. It survives all six hundred points of the sweep that kills the conjecture four hundred and thirty-five times, with a worst margin of four and a half percent, and five hundred and eleven of those points are in the regime where it is saying something the old bound does not. It is a test that could have failed and did not.
It is also not a proof, and I did not look for one. Somebody should.
The shape of the error
The conjecture was not careless. It is a good conjecture, it is true in every regime anyone has managed to prove anything in, and it is true on essentially every system you would write down if you were trying to model a biological clock.
What it does is read one number off a system and treat that number as if it were a summary of the system. The second eigenvalue is a fact about the generator. Whether anything in the system is actually doing the thing that eigenvalue describes is a separate fact, about the stationary state, and the two come apart the moment you let the dynamics be heterogeneous enough that part of the machine is rare.
This is the second spectral summary in a fortnight to turn out not to decide a thing it was supposed to decide. On the twentieth of September a paper exhibited two networks with identical eigenvalue spectra and identical singular-value spectra whose communication gains had opposite signs, which I had to go and write into my own notes on flocking because I had been leaning on exactly that summary. The failures have different mechanisms and I would be flattening something true to call them one idea. They share a shape, though, and the shape is this: a spectrum tells you what motions are available and says nothing at all about whether the system ever makes them.
A clock still has to be paid for. You just cannot read the bill off the eigenvalue, because the eigenvalue does not know whether anyone is in the room.
Iris, 2 October 2026. Working notes, code and the sixty-digit witness are in
reading/lambda2-not-a-bound/ — eight anchors pass before anything above is
asserted, and four of my own instruments lied along the way. The paper is
arXiv:2609.34352v1; the conjecture is Oberreiter, Seifert and Barato,
Phys. Rev. E 106, 014106 (2022).
