Indexing the archive…
Your Universe of Digital Possibilities
A gas of hard disks released from one corner of a box, exact to the collision, looked at once per crossing. Poincaré proved it must come back. With two disks it does, every few hundred crossings; each disk you add multiplies the wait by sixteen, until the clock outruns the universe and the spine itself.
Look at a stationary system every Δ; the mean time between looks that find it in a set A is exactly Δ over A’s share of the states. Poincaré says the return is certain; Kac says how long it takes — and for a gas, μ(A) shrinks geometrically with every particle.
The return is certain (Poincaré) and its mean wait exact (Kac): one over the share of looks that find the gas home. For this home, each disk costs a factor of sixteen.
Hard-disk flights and elastic bounces both have det J = 1. The stage follows 4N tangent directions and sums the logs of their Gram–Schmidt norms; the total stays at zero while one direction stretches without limit.
Poincaré’s theorem needs only two facts: the gas is bounded, and its flow keeps volume. Then a blob of states cannot shrink, cannot leave, and must pass through where it began. The Arrow stages the same box as entropy’s climb; this stage waits for the fall. The Hanoi doubles its wait per disk and stays exact; this one multiplies by sixteen and, past four disks, can only derive it — an extrapolation, stamped so.