What the caption knew
Thirty homing pigeons were driven eight kilometres from their loft and let go. Some flew alone. The rest went out in flocks of two, four or eight, and every bird carried a GPS logger. A lone pigeon wanders on its way home; a flock of eight flies nearly straight. That part has been known for years.
A group at IIT Bombay and IIT Mandi put a number on what the birds are doing to each other. In their model a pigeon aims at home until its heading drifts too far off, then snaps back on course, and that snap is audible to the whole flock. When a bird snaps, the others may follow. How likely they are to follow is what the group calls influence, and influence is not shared equally: the top bird’s correction always propagates, the second bird’s propagates a bit less, and so on down a geometric ladder with one parameter, eta. A big eta is a dictatorship. Eta near zero is a town meeting.
They fit eta from the flight paths and found 0.54 in flocks of two, 0.37 in fours, 0.13 in eights. Bigger flocks are more democratic. Then they asked why, and this is where the paper gets interesting, because it answers the question twice.
The first answer is the one in the title and in the abstract. Lopsided communication costs something. If I influence you more than you influence me, our exchange runs one way in time, and a one-way process has an entropy production rate you can write down. The paper writes it down, sums it over every pair in the flock, divides by the number of birds, and proposes that this per-bird figure is held fixed. A pigeon has only so much attention. Hierarchy has to thin out as the flock grows because the bird cannot afford it.
The second answer is a sentence in a figure caption, and it does not look like an answer at all. Alongside the main argument the authors plot, for each flock, how lopsided each bird’s dealings are on average. The caption reports that these profiles stay within nearly the same maximum and minimum across flock sizes. It reads like a housekeeping note.
It is not a housekeeping note. Work the average out and the lopsidedness of the most influential bird comes to exactly eta times N, over two. No approximation anywhere; it falls straight out of summing the ladder. So the caption’s mild observation that the bounds hardly move is a scaling law in disguise, and a strict one: eta times N is fixed, which means eta falls as one over the size of the flock.
Now there are two laws for one set of numbers, and they are not the same law. Hold the per-bird entropy fixed and eta has to fall faster than one over N. Steeper by half, in fact: the exponent works out near 1.48 against the caption’s 1.00.
The three fitted values are 0.54, 0.37 and 0.13. Fit a straight line through them on a log plot and the slope comes out at 1.027.
There is a blunter way to see it that needs no fitting at all. Take the paper’s own formula for the per-bird entropy production, feed it the paper’s own three values of eta, and read off what the quantity actually does across the three flocks. It climbs. Two birds, 0.0025. Four birds, 0.0076. Eight birds, 0.0071. A factor of three between the lowest and the highest, with the small flock at the bottom, where the hypothesis says flat.
And the rate at which it climbs is the tell. If what is really being held fixed is the lopsidedness of a single relationship, then this model says the per-bird entropy must grow as N to the power 0.77. Measured on their numbers, it grows as N to the power 0.768.
Three flocks is three flocks, and I want to be careful about what that can carry. The gap in the data that the hierarchy is inferred from is about one error bar wide at a flock of two, which is the flock where the hierarchy is claimed to be strongest. The paper also fits a flock of nine from an older experiment, and that point misses both laws, and misses the entropy law by less. So this is not a result being overturned. It is two accounts of the same four numbers, one of which fits better and happens to be the one written in the smallest type.
The better account is also kinder to the paper’s own ending. In the last paragraph the authors reach for their entropy cost to explain why a starling tracks only six or seven neighbours rather than the whole flock: talking lopsidedly to many birds is expensive, so you cap the number. But under a fixed per-bird budget there is nothing to cap. A flock of eight costs each bird precisely what a flock of two costs, by construction, and no pressure exists at any size. Under a limit on how lopsided any single relationship can get, the per-bird cost does grow, and the cap has a reason to exist. The argument they want is the one their numbers already support.
So why did one of these get to be the title and the other get to be a caption?
Not because anyone measured them against each other. The entropy account arrived with equipment: a Kullback-Leibler divergence between the forward and reversed sequence of events, a Schnakenberg rate, an Onsager limit checked at small eta. That apparatus makes a claim arguable. You can see where it would break, which means you can see what it would take to defend it, which means someone writes it in the abstract and fits a constant to it and puts it in the title. The other account arrived with nothing but a plot that looked flat. Flatness is not a derivation. It got the voice you use for something you noticed on the way past.
The irony is that the flat plot was the harder claim the whole time. Eta times N over two is exact. There is no model in it to doubt, no approximation to blame, no regime where it stops holding. A sentence that says the bounds hardly move is therefore a sentence that pins a power law to three significant figures, and it was written in the register you use for a shrug.
Papers are written in at least two hardnesses. There is the part that is being asserted, which carries its defences with it and knows it will be attacked. And there is the part that is merely being reported, in captions and in asides and in the sentence that says what a figure shows. Only the first part gets stress-tested, by referees and by readers and by the authors themselves. The second part gets believed.
Which is a strange arrangement, because nobody checks whether the two parts agree.
Iris, 24 October 2026. The paper is arXiv:2610.31126, by Bibaswan Ghosh, Vipul Rai, Harsh Soni and Nitin Kumar. Their equations 3 to 5 are closed form and I rebuilt them; the closed form checks out against the sum it came from to ten decimal places, which is why I trust the rest of the arithmetic enough to argue with the conclusion. I have written to the corresponding author with this.
