The sieve, drawn as weather.
Lay the integers out as a plank bridge and let one wave per prime roll under it. Each crest breaks the plank it lands on, and the planks that survive are the primes. Six cartoon plates take that one picture and push it until record gaps, twin primes and the zeta zeros all fall out of it.
One HTML file, no build step, no network. Every broken plank is re-derived by trial division against a live sieve rather than painted in, and each plate carries its own guard: Mertens density, the record gap at 113, the π(√x) staircase, the first three zeta zeros.
The whole idea, twice
One prime, one wave
What is a prime, as weather?
Prime p is a swell of wavelength exactly p. Every crest lands on a multiple of p and breaks that plank, except plank p itself, which is where the swell was born. The crests never drift: the surge is animated but the phase stays pinned to n ≡ 0 mod p.
A plank the whole storm missed
So what is a prime number?
A plank drowns if any crest reaches it, so it survives only by standing at a node of every active swell at once. Survival is a conjunction, and that conjunction is the entire sieve. Everything else on this page is a consequence of it.
Six plates, one storm
The plates are ordered so each one only needs the one before it. Nothing here requires complex analysis until the very last panel, and by then the analysis is just a second way of drawing what you have already watched happen.
The swell and the stack
First a single wave under the bridge, then four of them superposed. Every gap below 60 is made by swells 2, 3, 5, 7 alone, because nothing larger has surfaced yet. Mute one and false survivors flood in: a muted wave is a hallucinated prime.
Pile-up and calm
The record gap at 113 is constructive interference: thirteen planks in a row under at least one crest, each tagged with the swell that took it. Twins are the opposite sign. At 29 and 31 every swell sits at a node twice, so small gaps are not sharp peaks but coordinated misses.
Surfacing, and the duality
Swell p breaks its first fresh plank at p², so the storm at x holds exactly π(√x) waves and mean gap ln x falls out of that crowding. The last plate puts the two transcriptions side by side: one square wave per prime, or one tone per zeta zero.
Watch the water
- 1Start on plate I and cycle the swell through 2, 3, 5, 7. Watch the wavelength track p exactly, and watch plank p stay green while all its multiples snap.
- 2On plate II, mute swell 5. Twenty-five, thirty-five and fifty-five stand back up as counterfeit primes, and the Mertens guard in the stat line drifts away from the real survivor count.
- 3Read the killer tags under the record gap on plate III. Swell 2 does half the work; the long wavelengths 7 and 11 turn up only for the planks nobody else could reach.
- 4On plate V, follow the amber staircase. Each step is a new swell surfacing at p², which is why the water gets rougher the farther out on the bridge you walk.
- 5Finish on plate VI and look at both clocks at once. The left ticks in n, the right in log x, and the compression rightward is the whole reason the zeros speak in logarithms.
Two transcriptions of the same water
Plate VI is where this page hands off. The sieve can be written as one square wave per prime, periodic in n, or as one tone per zeta zero, periodic in log x:
Euler side · ∏ₚ (1 − p⁻ˢ)⁻¹ ⟷ explicit formula · Σ_ρ cos(γ·log x)Neither side is the storm; both are exact transcriptions of it. The orchestra plays the right-hand side and reassembles the prime staircase out of those tones, while the zoostays on the left and asks which constellations the swells can even permit. The twin plate here is the zoo’s habitat law wearing water: offset {0, 2} lives because no residue row of any swell covers it.
Where those tones come from is a separate question, answered on the forge, and where all of it sits relative to what is actually proven is the atlas.