PrimeMind · Structural atlas · the playground

Playground Lab

The back-and-forth substrate with every dial exposed. A set of runners (primes, integers, or a list), and for each one a speed rule, a weight rule, a phase rule and a space rule — all expressions in n you can type. A clock rule reshapes time itself. A fold mirrors the system, and an end rule sets what happens at the walls. Every configuration is rung: its waveform, its spectrum with the zeros marked, a score against the zeros, and the same score for fake primes so a coincidence can't pass as a signal. And the whole standing-wave field is drawn as a 3-D surface.

Runners

source

custom list

Rules — expressions in n (allowed: + − * / ^ ( ) log sqrt exp sin cos abs pi, and Lambda(n) = log p if n is a prime power, else 0)

presets:

speed rule f(n) — cycles per unit time

weight rule w(n)

phase rule φ(n), radians

space rule m(n) — how many half-waves runner n has across the line

clock rule τ(t) — the runner's phase is 2π·f(n)·τ(t)

end rule (how the walls reflect)

ring time span (units of t)

listen at position x (fraction of the line)

fold: mirror every runner (its wave at −t as well as +t) also ring fake primes (same count and density, no arithmetic) as a control

The ring — the waveform at the listening point (violet ticks: the zeros on the time axis, after the time scale)

time scale — multiply t by this before comparing the waveform with the zeros (1 for the log-clock presets)

compare the waveform's deepest dips with the zeros

Its spectrum — what pitches the ring contains, with the zeros marked (violet). Cyan: your rules. Grey: fake primes under the same rules.

spectrum axis scale — multiply frequency by this before comparing with the zeros

show frequencies from 0 to

The surface — the standing-wave field over position (across) and time (into the screen), rung and drawn in 3-D

turn

tilt

height

time span drawn (units of t)

How to read it. The ring is the sum over runners of w(n)·sin(m(n)·π·x)·cos(2π·f(n)·τ(t) + φ(n)) (cosine modes if the ends are free), so the speed rule sets pitch, the weight rule sets loudness, the phase rule sets timing, the space rule sets shape, and the clock rule bends time. The spectrum is the ring's frequency content; a pitch at f is what the system rings at. If a configuration's cyan peaks land on the violet lines and the grey fake-prime peaks don't, you have found something. With the plain bounce rules the peaks are at the runners themselves — the game rings at its own numbers. With the log-time rules (speed log n / 2π, weight log p / √n, listening at the end so every runner counts fully) the zeros appear in the waveform, not the spectrum: the time trace is the Euler product read along the critical line, and its largest excursions sit on the violet ticks — that is the explicit formula, the one configuration known to produce the zeros, and it produces them as output in time, not as the system's own pitches (the spectrum still shows the runners' pitches, log n / 2π). Read both scores: a configuration matters only if it beats the fake-prime control. Everything else is open. The Gamma end rule puts a frequency-dependent delay at the walls that grows like the log of the pitch — the moving far wall from the string lab — so the allowed pitches under that rule are no longer evenly spaced; the spectrum shows which pitches the walls let through.