Loop Lab
The bounce game on circles. Each number becomes a loop whose circumference is its length. Every loop passes through one shared touch point and extends the same way, so they nest like rings inside rings, all kissing at one spot. A dot runs around each loop, starting at the touch point. The clock decides how fast each dot goes. The tilt is the phase as a direction: it turns each loop onto a different great circle of its own sphere while keeping the shared touch point. Then watch who comes home, and when.
The loops
Drag to turn the whole thing in any direction, scroll to zoom, double-click to reset. The touch point is the white ring; a dot glows when it is home. What each clock does — exactly. Equal lap time: every loop is a scaled copy of every other about the touch point, so with no tilt all the dots always lie on one straight ray from the touch point, arrive home together every frame, and never interfere. Equal speed: dot n needs c(n) units of time per lap, so it is home exactly at the multiples of its loop's length; at a whole time m the dots at home are the loops whose length divides m — with c(n) = n and the primes, the prime factors of m. The whole times when nobody is home are the numbers with no prime factor among your loops: the sieve of Eratosthenes, running as a clock (third panel). Bounce rhythm: the rational meetings of the bounce game — at t = ½ dot 2 is home and every odd dot is on the far side. Log clock: dot n turns by t·log n; with weights 1/√n the home pull dips at the zeros — the known relation, as output, like the projector. The tilt. A loop tilted by θ is another great circle of the same sphere through the touch point (tick show each loop's sphere). Tilting never changes when a dot comes home — the home pull depends only on the clock — only where it is between visits, so the tilt lives in the sideways part of the resultant.
The home pull over time — Σ w(n)·cos(dot n's angle): high when many dots are home · violet: the zeros · grey: the same loops built from fake primes · click to jump there
When is a score interesting? Three checks, all required: the deepest dips land on the zeros far more often than on the midpoints between zeros (a curve that merely has many dips hits both equally); the score beats every fake twin, not just the average; and the curve is fully sampled. The known signal — log clock, primes to 3,000, weight 1/√n, T = 60 — scores 62% on zeros, 0% on midpoints, against fakes around 40%: that is what "interesting" looks like here.