Stretch Lab
The candidate object for the zeta zeros is not a surface. It is a particle on a line whose only motion is stretching: position grows as momentum shrinks, so that their product never changes. Drawn in the position–momentum plane, every path is a hyperbola — one end flaring out exponentially while the other closes in, which is the picture you described. Left: the flow. Right: the closed-off region whose area counts the zeros.
Left — the flow. A particle with position x and momentum p under the rule "energy = x · p" moves so that x multiplies by et while p divides by it. Every orbit is a hyperbola xp = constant; the dots ride outward along theirs. This is the stretching symmetry that makes every zero wave a fixed-frequency wave in log-time, and it is a saddle flow — squeeze in one direction, stretch in the other — which is why the saddle surface kept turning up. Right — the count. Drawn in log coordinates, where the stretching turns every hyperbola into a straight line and the closed-off region into a triangle. Ask how many pitches sit below energy T. In quantum mechanics that is the area of phase space with energy below T, divided by 2π, because each pitch occupies one cell of area 2π. Without cutoffs the region under the hyperbola has infinite area: no discrete pitches, an open system. Cut it off at x ≥ √2π and p ≥ √2π — one Planck cell in the corner, so the system is closed — and the area is exactly T·log(T/2π) − T + 2π, whose number of cells is (T/2π)(log(T/2π) − 1) + 1: the Riemann–von Mangoldt count of the zeros, to within a fraction of one zero at every height. That is Berry and Keating's 1999 observation. The counting works; what nobody has is the correct way to close the system — the boundary condition that turns "the right number of pitches" into "exactly these pitches." Connes' version closes it by gluing in every prime at once.