Ask

what this is

The Dirichlet problem is a rule of total obedience: fix a value at every point of a region's boundary, ask for the interior function that is perfectly smooth in the harmonic sense, and there is exactly one. The interior has no say. Paint the rim of this room and every point inside must settle on one tone, the average of what the walls say weighted by how visible each wall is from where the point stands. The probability behind that sentence is literal: the value at a point is the expected brightness a blind Brownian wanderer first touches when released there, and this page computes the room by releasing wanderers, tens of thousands per second, and letting each one report the single boundary value it dies on.

The wanderers do not shuffle in small steps. Each walker draws the largest circle around itself that touches no boundary, then jumps in one move to a uniformly random point of that circle, which is exactly where Brownian motion would first cross it. This is walk on spheres: the whole diffusion collapsed into a handful of leaps, about three more per walk each time the finish line is drawn ten times finer. Drag the probe and you can watch it happen, nested circles shrinking toward the rim like a held breath. Raise the wall and the room reorganizes around the shadow; there is no formula for that room, and the walkers never notice.

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 module this page draws from. All twenty-four checks held on the first blind run, nothing tuned afterward; the second clean first board in this vein.

• 1 · the exact exit law
  Held: from two interior points, the chance of landing on a fixed arc matched the Poisson kernel's prediction, 0.5686 and 0.3436, within a thousandth on both seeds.
• 2 · the center answers in one jump
  Held: from the middle of the room the first circle is the rim itself, so every walk ends in exactly one jump, uniformly. Two hundred thousand walks, not a single exception, and the mean of a cosine boundary read 0.4997 against an exact 0.5.
• 3 · the mean-value property
  Held: the field equals its own average over interior circles to within 0.0014, including with the wall up, where no closed form exists to check against and the property is the only oracle left.
• 4 · the wall orders the room
  Held: a point sitting two hundredths above the dark wall read 0.218 where pure rim light would read 0.8; the room grades strictly from the wall's value to the rim's along the perpendicular.
• 5 · a constant per decade
  Held: tightening the finish line from a hundredth to a millionth raised the average walk from 6.0 to 19.2 jumps, an even 3.3 per decade on both seeds: logarithmic cost, the algorithm's whole claim in one number.
• 6 · superposition
  Held: on shared walks, the answer for two boundary lights together equals the sum of the answers for each alone to one part in ten trillion. Asking is linear.

the picture

The interior you see is never finished, only increasingly sure: each cell shows the running average of every walker released from it, so a fresh painting comes in grainy and settles toward smoothness as evidence accumulates. When you repaint, the room's memory of its old answer fades under the new samples rather than vanishing. The number under the disk keeps the picture honest in the one place a formula exists: with no wall raised, the measured value at the probe is printed beside the exact Poisson integral of whatever you painted, and you can watch the walk close the gap. This piece follows Depart, which measured where blind walks land; Ask is what the landing is worth. Monochrome, procedural, no paid instruments.