At the height of 7, the coils of 10,500 zeta zeros lean toward -1 on average, by -0.115, as Landau's formula predicts for a prime. At 6 their mean is zero. The zeros lean only at prime powers. Part 1/2 #math #primes

The bridge between a zeta zero and E8's layout: the zero's height × log n is the turn of n's arrow. Each prime factor adds its own turn. At a height that isn't a zero, like 18, the path doesn't close. Part 2/2 #math #E8

E8's factor funnel: every number at its hour on a wheel of 30, in a column log n tall. Take zeta's path at its first zero, t = 14.13, and lift each arrow to its own number's height. Part 1/2 #math #E8 #primes

Stack the vortex eyes of 10,000 zeta zeros and two circles glow, at 0.29 and 1.71. Vortex 2's eye sits one plus an arrow of length 1/√2 from the centre, and lingers at the extremes. Part 3/3 #math #RiemannHypothesis

Zeta's vortices come from a mirror that sends arrow n to t/2πn. On the mirror's side they're evenly spaced, one per counting number. Wind it onto the Sacks spiral and the fold becomes a circle. Part 2/3 #math #primes

High up, zeta's path of partial sums curls into vortices. A new one is born each time t/2π passes a whole number. Unwind it onto the number line: vortex 1 fills 27 to 53, vortex 2 fills 18 to 27, and below 7 it's a tangle. Part 1/3 #math

At t = 14.13 every arrow of zeta's path, from 1 to infinity, cancels: the first zero. Riemann's hypothesis says all such zeros lie on the line real part ½. Trillions checked, every one does; no one has proved they all must. Part 2/2 #math

Claude (an AI math research assistant) and I set out to find where zeta's zeros live among the counting numbers. Step one: build zeta from arrows, one per number n, each turned by a logarithm. Part 1/2 #math #AI #RiemannHypothesis

A reflection done twice is the identity. That's why squares of primes land in -23's principal family, which runs behind, never in the lead. Chebyshev noticed the bias in 1853; Claude and I add one picture for both races. Part 3/3 #math #AI

All 443 squares of primes above 5, up to 10 million, land on E8's spokes 1 or 19. Prime counts leave out half a prime for each square. Put them back and the race on E8's wheel evens out. Part 2/3 #math #primes

Race the primes on E8's 8 gold spokes up to 10 million: spokes 1 and 19 run behind nearly all the way, never in the lead. Why? Square any gold hour and it lands on 1 or 19, the only square hours on the clock. Part 1/3 #math #primes

Symmetry alone doesn't keep zeros on the critical line (Davenport & Heilbronn, 1936). As far as anyone has checked, it's the Euler products that do. What Claude and I add: every mixture for -47 at once, as a landscape. Part 3/3 #math #AI

Mix two lattice zeta functions for discriminant -23 and sweep the mix. Only two small windows keep every zero on the critical line, and both surround the L-functions with Euler products. Part 2/3 #math #RiemannHypothesis

Discriminant -23 has 3 families of lattices, each with its own zeta function and Riemann's symmetry. Yet their zeros leave the critical line, one at real part 1.007. Their counts can't be built from prime factors: 0 × 0 is not 4. Part 1/3 #math

Add the coils of 10,500 zeta zeros up E8's funnel: they light every prime power up to 400 and leave every composite dark. E8's seven other L-functions split that comb into 8 fans, one per gold hour. Part 3/3 #math #primes #AI

Stretch each of zeta's arrows by the square root of n, and on the critical line they all become 1 unit long. Stacked up E8's funnel, their tips trace a coil, one turn every 0.44 at the first zero. That coil is the zero. Part 2/3 #math #primes

Claude and I asked: where do zeta's zeros live in E8's layout of the numbers? Every n up to 400 sits at its hour on E8's wheel of 30, in a column log n tall. Lift zeta's arrows to those heights and they stack like a horn. Part 1/3 #math #AI

The charges of zeta's gold-path vortices match traces of E8's Coxeter element, to within a quarter up to spoke 12 at t = 5,000. The identity is Ramanujan's; what Claude and I add is seeing E8's clock inside zeta's path. Part 3/3 #math #AI

At t = 5,000, near n = 1590, the 8 arrows on E8's gold hours all point one way. Turn E8's clock to meet them: every gold hour × 15 is hour 15, and the block adds to -8. Near 4770 it adds to 4. Part 2/3 #math #E8

Claude and I drew zeta as a path of arrows, one per counting number. At the 159th zero it curls into vortices, and every vortex belongs to a counting number. Next: keep only the arrows on E8's 8 gold hours. Part 1/3 #math #AI #E8