Layered Dome Lab
Build a half-sphere out of layers: the smallest layer set by the first prime, the largest by the last, and every layer in between one step longer and one angle-step further round. Stack them two ways — a dome, spreading out evenly from the first layer, or a fan, every layer touching the first at one point and growing from there. Then ring it, and test the ring against the zeros with the same honest checks as everywhere else in the atlas.
How it rings. Dome: the layers form a surface, and the surface is rung like a drumskin fixed along its smallest and largest layers; waves can run around each layer and up and down between them. Fan: the layers are loops all joined at one shared point, so they ring as a network — a wave reaching the shared point splits into every loop. Each loop is bent into a circle, and the bend adds a small inward pull that depends on its size. In the fan, the tilt of each loop does not change the pitches; only the lengths and the shared point matter. The twist (fan only). A wave running once around loop k comes back turned by that loop's twist — like a wire loop threaded by a magnetic field. With no twist, a wave and its mirror image (running the other way) behave alike; any twist that is not a half turn breaks that mirror, and physics predicts the pitches of a tangled network then push apart more strongly — the same way the zeros do. The twist cannot make the network thicken like the zeros, though. The twist on the dome works the same way: a wave circling a layer comes back turned by that layer's twist — as if a field were threaded through the dome's opening. In the picture the twist shows as meridians that spiral up the dome (the amber one is a seam you can follow); rotating the round layers themselves would change nothing, because a round dome turned about its own axis is the same dome. The dome still separates into independent pieces (one for each number of turns around it), so a twist shifts its pitches but cannot make them push apart. Alternating sides and the spiral arrangement change only the picture: a network feels its loop lengths, twists and joins, not where the loops hang. Your rule versus a true sphere: on a perfect sphere, equal angle steps give unequal length steps, so "one step longer and one angle-step further" makes a dome that is sphere-like but not exactly a sphere. Both are available.
The ring against the zeros — pitch number k, stretched so the last pitch lands on the last zero
Reading the match: a pitch counts as landing on a zero if it is within 15% of the gap to the next zero. Any evenly spread set of pitches does that for roughly a quarter of any targets, so the percentage alone means little — compare it with the midpoints and with the chance ladders in the readout. Only a clear lead over those means the pitches are tracking the zeros.
How fast the pitches thicken — count against height, on log scales
How the pitches are spaced — neighbours pushing apart, or not
Three tests. Matching: the pitches are stretched so their count matches the zeros' count, then each zero asks whether a pitch landed near it — and so does each midpoint between zeros. A ring that tracks the zeros hits them far more often than the midpoints. How fast they thicken: up to a height T there are about (T/2π)·log(T/2πe) zeros — the count grows a little faster than the height itself. A surface (the dome) thickens like the height squared; a network of loops (the fan) like the height itself. The number shown is the growth rate: the zeros sit near 1.2–1.4 in this range. Spacing: the zeros push their neighbours away — close pairs are rare (the amber curve). Pitches of a system that separates neatly into independent parts do not push apart (the grey curve, where close pairs are the most common).
The additive waves — ring every pitch together and listen at each length x · ticks at log n (cyan: prime · violet: prime power · amber: composite)
The zeros, rung together this way, echo loudly at every prime, more softly at repeated primes (4, 8, 9 …) and stay silent at mixed products (6, 10, 12 …) — that is the test from the network lab. A system whose pitches were the zeros would have to do the same.