Recurrence
Random walks leaving home, drawn as light. Every walk that makes it back stamps its path into the field; the rest leave only ash. On the left the walkers wander a plane, and a theorem promises every one of them home. On the right they are given one more dimension, and most are lost forever. The difference between always and one-in-three is one direction you cannot see. Touch either side to release new walkers.
0 lives · 0 home · 0 still out when their watch ended
0 lives · 0 home · 0 lost when their watch ended
the claim
Pólya proved it in 1921: a walker taking random steps on a two-dimensional grid returns to its starting point with probability 1. Give the walker a third dimension and the certainty collapses to a strange, exact number. Watson computed it in 1939 as a closed form, and its decimal expansion begins 0.3405373. This page's numerical core measured that number blind, before any of this was drawn: two hundred thousand walks on the three-dimensional lattice, run to one hundred thousand steps each, extrapolated with the form pre-registered in the project notes. The measurement landed at 0.3431, within 0.761 percent of the exact value, inside its pre-registered tolerance. The sampler earned trust first, against exact enumeration of every walk up to twenty steps, in all three dimensions, to within statistical error.
The two-dimensional side carries a subtler number. The fraction of walks still missing after T steps dies away like π/ln T, and the fit to the measured decay returned a coefficient of 3.1668 against π = 3.1416, within one percent. The walk was never told about π. It found it by getting lost.
what the watch cannot see
Every life on this page is watched for twenty thousand steps. On the left, about a quarter of them are still out when their watch ends, and the needle settles well short of the promise. The theorem guarantees that every one of those walkers would come home, and it also guarantees the wait has no average: the mean return time is infinite. No finite watch, however long, can tell certainty from near-certainty. The gap between the needle and the mark is not an error. It is the exact width of what patience cannot prove.
Method and limits: simple random walk on the integer lattice; three-dimensional walks drawn in orthographic projection, with the unseen coordinate rendered as dimming; each life ends its watch at twenty thousand steps and is recorded home or lost-so-far, and the watch turns to a new life. A homecoming stamps the walker's stored trail, up to 440 steps of it, into the sediment; longer journeys leave only their homeward end. The exact enumeration, the blind measurement, and the pre-registered tolerances are in the repository: walk.ts and probe-recurrence.ts, seed base 20260805.