Depart

what this is

A walker takes steps of one fixed length in directions chosen blindly, each forgetting the last. On a flat page this rule is famously homesick: Pólya proved in 1921 that such a walk returns to its neighborhood of origin not once but endlessly, and the flat world here shows it, the trails folding back through the source over and over. Curve the plane the other way, so that circles grow exponentially with their radius, and the same blind rule changes fate entirely. The walk departs. It crosses each circle around home a last time without knowing it, settles into one direction, and holds it forever, because out here there is exponentially more room ahead than behind. No step is ever aimed; the leaving is the geometry's doing.

The disk is Poincaré's: the rim is infinitely far away, so a walk that approaches it is leaving for good, and the point where it lands is the direction it chose. Those landings are marked as ticks on the rim. Their law is the oldest harmonic measure there is, the Poisson kernel of the disk, drawn outside the rim as a thin predicted curve; the measured landings assemble themselves into it as you watch. Move the source and the prediction reshapes instantly; the walks take a moment to agree, and always do. One quiet subtlety: the landing law is the same in both worlds, because in two dimensions the exit angles cannot feel curvature. What the curvature decides is not where the walks land but whether they come home first.

the gates

Six gates were fixed in the project notes before any code existed, with formulas and tolerances in writing, and run headless against the same walk module this page draws from. All twenty-six checks held on the first blind run, the first clean board in this vein; nothing was tuned afterward.

• 1 · isotropy
  Held: twenty thousand walks from the center land uniformly on the rim, chi-square 26.9 and 15.7 on two seeds against a threshold of 49.7.
• 2 · the poisson law
  Held: from sources at two off-center distances, the landing histogram matches the Poisson kernel bin by bin, four chi-square values between 24.1 and 35.3, all far under threshold.
• 3 · the hitting law
  Held: the chance of touching a ball before escaping to infinity is log tanh(r/2) divided by log tanh(a/2), and the walk measured it to within 0.013 at worst over three geometries and two seeds.
• 4 · the exact mean, and the speed
  Held: the spherical mean of cosh r is an eigenfunction identity, so E[cosh r] after n steps is exactly cosh(ε)ⁿ; measured to 0.04% at five hundred steps. The escape speed converged to 2 log cosh(ε/2) within 0.7%.
• 5 · the direction settles
  Held: the leftover turning after radius six is smaller than after radius four by the predicted factor e⁻², measured 0.130 and 0.132 against 0.135.
• 6 · the flat control
  Held: with curvature off, the same engine obeys E|z|² = nε² to 0.5% and grows like √n (exponent 0.49) while the curved walk grows linearly (exponent 0.89): recurrence and escape, one number each.

the picture

Trails are drawn once and left to fade, so the image is always the recent past of fifty-six walks; the rim collects what does not fade, the landings. The counter under the disk keeps the two worlds honest side by side: how many walks have departed, how many ever came home, and how closely the landings agree with the kernel. This piece stands on the two before it: Pave built the geometry, Erase and Recurrence walked the flat world and found it always leads home. This is what the same walk does when the world stops folding back. Monochrome, procedural, no paid instruments.