Water turns to ice at zero degrees — sharply, with no intermediate states. Something similar happens to complexity when we take a random quantum state of an entire system and look only at a part of it. As long as that part occupies less than half the system, its complexity grows smoothly. But as soon as the fraction crosses 50%, a jump occurs.
This transition has been dubbed "Page-like," after the Page curve for information in black holes. The authors go further and explain the cause. Before the midpoint, complexity is dictated by the energy spectrum (the set of available levels), and after, by the specific choice of basis: that is, how we describe the states. In a small town, order might be set by general laws, but in a large one, you also have to account for who lives in each house.
Then the idea was applied to an evaporating black hole in anti-de Sitter space coupled to a thermal bath. Using a toy model, a quantum circuit with random evolution, physicists showed that the complexity of the radiation also undergoes a sharp transition. In the holographic description, this transition is linked to the emergence of an "island": a region inside the black hole that suddenly starts contributing to the volume.
This resolves the apparent paradox: the complexity of the radiation is not lost but rearranged as the black hole ages. And again, half turns out to be the threshold, just like water at zero degrees.
The complexity transition at half the system is a direct relative of the "Page transition" for entanglement entropy, which shows that information about a black hole starts leaking out exactly at the midpoint of its life.