Drag the slider (or press RESOLVE) to add zeta-zero waves to the smooth axis one instrument at a time (with reduced motion on, RESOLVE seats the whole orchestra at once). Each wave is 2√x·cos(γ·log x − α)/|ρ|, featureless alone. Watch the interference sharpen into vertical jumps at exactly the primes and prime powers (each jump has height log p). Staircase teeth at 2, 3, 5, 7, 11… emerge from waves that individually know nothing. Guard: the dashed staircase is the true ψ(x), summed from a live sieve, never fitted to the waves. Viewport: scroll-wheel or pinch to zoom (centred on the cursor), drag to pan, a tap sets the probe. ⌖ PROBE zooms to a window of ~4 average prime gaps (8·ln x₀) around the probe: the staircase's natural local scale. Note what happens past x ≈ 300 at N = 100: the top instrument is γ ≈ 236.5, and it resolves log-space detail no finer than 2π/γ, so the far staircase blurs. Push N toward 270 (γ₂₇₀ ≈ 500.3) and watch it sharpen: the zero-counting formula predicts ~270 instruments to articulate primes to 500, and the slider lets you verify it. More primes, more instruments, forever.
The single wave you just added: a stretching chirp in linear x. Hit LOG X and it snaps into a pure sinusoid, the primes' native coordinate system. The dashed envelope is ±2√x/|ρ|, the wave's amplitude law.
The reconstruction's y-value at one probe point x₀ (crosshair; tap anywhere on CH·1 to move it) as a function of wave count N. The amber trace is the partial sum hunting the true value (dashed); the dim tail is its future. Probe mid-gap and it settles fast; probe just next to a prime and watch it ring: each wave overshoots and corrects, the Gibbs struggle at a discontinuity rendered as a time series.
The same trace replotted as a lag-1 return map: each point is (ψN−1, ψN). The dashed diagonal is y = x; the circled dot is the fixed point (true ψ, true ψ). Convergence appears as an orbit spiralling into the fixed point: points hugging the diagonal mean consecutive sums barely changed, wide loops mean a big-amplitude wave just joined. Move the probe and watch the orbit's geometry change character.