Two Casts Lab

What the chain is. Performer n carries an arrow of length 1/n^σ, turned by the angle −t·log n, scaled by its card. Lay all the arrows tip to tail, add one dashed arrow for the smoothed rest of the cast, and the chain ends exactly at the wave's value (checked against mpmath to 10⁻⁸). A zero is the chain coming home to the origin.

Same loops, same angles. Both chains use the same lengths and the same turning; only the cards differ. ζ's cards are all 1. The look-alike's repeat every five and flip some arrows, shrink others, and skip every fifth performer.

The gamma clock. There is a second rotation in the system that no one has to add: the gamma factor's phase. On the ½ shell the tip is forced to sit on a line through the origin whose angle is set by that clock, so a zero is the tip sliding through the origin to the other side. Each half-turn of the clock calls for about one more zero (the count matches to ±1 for both casts); the primes decide exactly where. Off the ½ shell the tip is free to leave the line, and silence needs a dead-centre hit.

Try it. At ½ + 14.13i, ζ's chain closes and the look-alike's doesn't. At 0.81 + 85.70i (the shell slider at 0.81), the look-alike's closes and ζ's doesn't. Off the ½ shell ζ's chain has never been seen to close; the look-alike's does, in mirror pairs.