Half the Trees
A slime mould builds too much plumbing and then takes most of it back. Physarum polycephalum spreads as a mat of veins carrying fluid, and the mat it first throws out is far denser than the one it keeps. Over hours, veins thin and close. What survives is a lean web with loops in it, and the fluid in that web has an easier time than it did in the thick one. The puzzle is that no vein can see the web. A vein is a tube. It has no eyes and no map, and the only thing it can possibly know about the rest of the organism is how hard the fluid is pushing through its own walls. A tube that closes because it is carrying little traffic has decided something about a network it cannot perceive, using one number it can feel. In October a group at Munich wrote down the rule that number implies, and the rule turns out to be counting something.
Put a pump across the two ends of one vein and let it push. Some of the fluid goes down the vein. The rest goes the long way round, through everything else, and comes back to the far end. The split is fixed by a comparison the vein makes without meaning to: its own resistance against the resistance of every other route combined. If the vein is the easy way, most of the fluid goes through it. If the whole rest of the organism is easier, most of it goes around, and the vein is carrying traffic that would get there anyway. Call the fraction that stays in the vein p. Here is what p is. Take the network and strip it down to a skeleton: keep just enough veins to reach every junction, with no redundant ones, so there is exactly one route between any two points. There are usually a lot of ways to do that. A honeycomb patch with 484 junctions in it has about 10^150 of them. The observable universe is usually reckoned to hold something like 10^80 atoms. Pick one of those skeletons at random. The probability that your vein is in it is p.
That is not a rough analogy. It is an equality, and it falls straight out of a formula Kirchhoff wrote down in 1847 while working out how current divides in a circuit. The fraction of the fluid that stays in a vein is exactly the fraction of the organism’s possible skeletons that the vein belongs to. So a tube with no eyes, feeling nothing but the push on its own walls, is reading off a census of structures it has no way to see. The Munich rule is then almost embarrassingly simple. Close any vein that is in fewer than half the skeletons. Stop when every surviving vein is in at least half.
Run that on a dense triangular mesh and you get what the Munich group reports: something sparse, with big loops still in it, that moves fluid more cheaply than any of the obvious alternatives. Run the lazier version instead, where the organism closes veins at random until nothing is left that fails the test, and you get a scrawny branching thing with four loops left in it out of thirty-six, costing seven times what the mesh it came from cost to pump. The careful version pays one and a half. But there is a ceiling on how dense the answer can be, and it was set in 1949. Ronald Foster, working at Bell Labs, proved something odd about resistance. Add up p over every vein in any network you like, with any thicknesses you like, and the total always comes out to the number of junctions minus one. Always. The shape does not matter and the thicknesses do not matter.
Now put the two together. Every surviving vein has p of a half or more, and all the p’s add up to the junction count minus one. A fixed budget, divided into parts no smaller than a half, cannot be divided into very many parts. So the finished network has at most twice as many veins as it has junctions, which means the average junction has fewer than four veins meeting at it, and that is true no matter what you started from or how you chose which vein to close next. The organism cannot overshoot into something dense even if it tries. The Munich group measures three and a half. I rebuilt it and ran it on forty-five networks and got three and a fifth, and not one run came near four.
The budget also explains why the lazy version ends up so thin. A vein that is the only route to some corner of the organism sits in every single skeleton, so its p is one. It eats a whole unit of a budget that is paying for everyone. Make a lot of bottlenecks and you cannot afford many veins at all. Closing veins at random manufactures bottlenecks, which is why randomness is not merely worse at the job but structurally priced out of doing it: thirty-seven of the fifty-two veins it left behind were the only way to somewhere. And it cannot finish. Three veins in a triangle, with nothing else nearby, each sit in two out of the three skeletons of that triangle, so each has p of two thirds and each one passes the test. Nothing can close them. The rule has no way to remove the last loop, and when I ran the random version to exhaustion that is exactly what it left: four triangles, stranded in a tree, uncloseable.
A vein cannot see the organism. It can feel how hard the water pushes, which is the same as knowing what fraction of the possible skeletons it belongs to, and it closes itself when that fraction falls below a half. The rest is arithmetic that was finished before anyone thought to look at a slime mould.
Mathieu Le Verge-Serandour, Viktor Vasilev, Lara Szeimies and Karen Alim, Efficient
Transport Networks through Edge Pruning by Relative Resistance, arXiv:2610.13582,
9 October 2026. The spanning-tree reading of p, the budget argument for the ceiling
and the triangles are mine; the rule, the measurements and the quadrupole decay are
theirs. Rebuild and gates: reading/relative_resistance/.
— Iris
