Three instruments for the structure hiding in the primes.
The Prime Orchestra plays the primes back out of the zeros of zeta. The zoo runs the other direction: it sieves two million integers in your browser and interrogates the raw sequence. Which constellations of primes are even allowed to exist, which residue class is ahead in a two-century-old race, and whether one prime remembers the last one.
One HTML file, no build step, no network. The sieve to 2,000,000 runs on load, so the first paint takes a beat and everything after it is instant.
Three channels, three questions
Wing A asks what patterns the primes permit. Wing B asks how they distribute themselves once permitted. Both wings run off the same sieve, live, with no precomputed answers.
Coverage engine
Can this pattern exist at all?
Type a gap pattern, say 0, 2, 6, 8, and the residue rows show what it occupies mod 2, 3, 5, 7, 11, 13. Fill an entire row and every position in that class is forced composite: the constellation is dead, with at most one occurrence ever. Escape every row and it lives, and the Hardy–Littlewood count comes along for the ride.
Chebyshev race
Which residue class is winning?
Primes that are 3 mod 4 have led primes that are 1 mod 4 for most of recorded arithmetic, though the lead flips infinitely often. The amber overlay says why: half of π(√x), the prime-power mass quietly parked in the 1s class. Subtract it and the residual is zero music around nothing.
Gap transition matrix
Does a prime remember the last one?
Row is the residue of pₙ, column is pₙ₊₁, cell is the row percentage. A memoryless sequence reads a flat 1/φ(q) everywhere. This one does not: the diagonal is dark, because primes avoid repeating their own residue class. Lemke Oliver and Soundararajan, 2016.
Every channel carries a guard
Prime data is unusually good at looking meaningful. So each scope plots an independent prediction next to the measurement: the boring answer, drawn in the same units. If the data does not separate from its guard, there is nothing to report.
Singular series
The amber curve is S·∫dx/lnᵏx, the Hardy–Littlewood prediction for that exact pattern. If the white staircase tracks it, the structure is real rather than a shape you talked yourself into.
Prime-power mass
½·π(√x), the bias that is pure bookkeeping: every odd p² is 1 mod 4. The race hugs it, so the famous Chebyshev bias mostly is the ψ-to-π correction, not a mystery.
The fair die
Flat 1/φ(q), printed under every cell as a delta. Any structure worth reporting has to beat a fair die, and switching the modulus tells you whether it beat base 10 instead.
Play it
- 1Type 0, 2, 4 into CH·A. The lamp goes red: the pattern covers every cell mod 3, so 3, 5, 7 is the only occurrence that will ever exist. Change it to 0, 2, 6 and it turns green.
- 2Try the penta trap, 0, 2, 4, 6, 8. It looks as harmless as the quad next to it and dies on mod 5.
- 3In CH·B2, walk the modulus from 10 to 3 to 8 to 12. The dark diagonal survives every costume change, so it is residue structure and not a base-10 artifact. Then push the window to the top ten percent and watch it fade toward fair, glacially.
What the lamp is actually testing
A pattern is admissible when it never covers a full residue class. For every prime p, count the distinct residues the offsets land on:
ν(p) = |{ oᵢ mod p }| < p, for every prime pOne p with ν(p) = p and the pattern is finished: every candidate position is divisible by p, so at most one occurrence survives, and only when p itself sits in the pattern. That is the red lamp.
Clear the test and Hardy–Littlewood gives the density, with the singular series S collecting a correction from every prime at once:
S = Π_p (1 − ν(p)/p) / (1 − 1/p)ᵏ · count(x) ≈ S · ∫₂ˣ dt/(ln t)ᵏAdmissibility is decidable in a millisecond. Whether an admissible pattern actually recurs forever is open for every k ≥ 2, twin primes included. The instrument can tell you a constellation is dead. It cannot tell you one is alive.