Weil's World Lab
The one place a Riemann Hypothesis is proven: curves counted in clock arithmetic. Take one curve, y² = x³ + ax + b, and count its points mod p — for every prime p. Hasse (1933) proved, and Weil (1948) extended to every curve, that each count stays within 2√p of p + 1. That bound is the Riemann Hypothesis for this curve over each prime's field: the two zeros of its zeta function lie exactly on a circle. Every panel below checks it, prime by prime.
Every prime's count, measured against the proven bound — a_p / 2√p, where a_p = p + 1 − (points) · the band is ±1
The zeros, one pair per prime — scaled by √p, they must lie on the unit circle
How the angles spread — Sato–Tate
Reading it. For each prime p, the curve's count mod p gives one number a_p = p + 1 − (number of points). The curve's zeta function over that prime's field is a small fraction whose top is 1 − a_p·u + p·u²; its two zeros α and β multiply to p. They lie on the circle of radius √p — equivalently, in s-coordinates (u = p−s), on the line Re s = ½ — exactly when |a_p| ≤ 2√p. Hasse and Weil proved that bound for every prime and every curve; the band and the circle show it holding. Writing a_p = 2√p·cos θ_p gives each prime an angle θ_p; for an ordinary curve the angles spread like (2/π)·sin²θ — the Sato–Tate law, proven in 2008–2011 (Taylor, Clozel, Harris, Shepherd-Barron, Barnet-Lamb, Geraghty). The two "special" curves have extra symmetry (complex multiplication): for them half the primes give a_p = 0 exactly, and the angles pile up at 90°.
One curve over one prime's field — every point drawn, and the zeros predicting the next field
In clock arithmetic mod p there are only p × p possible points (x, y), and the curve is the handful of them where y² = x³ + ax + b holds — plus one extra "point at infinity". Count them, and you have the curve's size over that field. The two zeros α and β, found from that single count, then predict the count over the bigger field with p² elements: N₂ = p² + 1 − α² − β². The lab builds that bigger field (numbers a + b√d with d a non-square mod p), counts the points there directly, and checks the prediction. It matches exactly, for every prime — the zeros are not a summary of the counts, they generate them, over every extension of the field. That is the finite, closed system Weil's proof rests on: a small matrix (the "Frobenius") whose two eigenvalues are α and β.