PrimeMind · Structural atlas · the comb

Comb Lab

One string. The primes are not runners on it — they are its medium: at a position given by a rule, a bump whose height is given by a rule. Strike the string and it rings at pitches none of the bumps had; those pitches are computed exactly and scored against the zeta zeros, with the same string rebuilt from fake primes as the control. Nothing about the zeros enters the construction. If the cyan ladder lands on the violet ladder and the grey ladder does not, you have found a medium that rings at the zeros — the thing the search is for.

The medium

presets:

numbers used

position rule p(n) — where the bump for n sits

weight rule w(n) — how strong the bump is

position scale (multiplies p(n))

bump strength A

bump width

string half-length U (bumps beyond it are dropped)

background potential b(u) — expression in u (0 for none)

ends

pitches to compute

pitch =

custom list

fold: mirror every bump at −p(n); string runs from −U to U (unfolded: 0 to U) also build the string from fake primes (same count and density, no arithmetic) as a control

The medium, drawn — the potential along the string

The pitches — cyan: this string · grey: the fake-prime string · violet: the zeta zeros (ladders density-matched below the chosen height)

compare up to height

Spacing statistics — how the pitches are spaced, against the chaotic-drum shape (GUE) and independence (Poisson)

Standing waves — the first three, and one from the middle of the ladder

What is computed. The string obeys −ψ″ + V(u)ψ = Eψ with V the sum of the bumps plus the background; it is discretized finely and its lowest eigenvalues are found exactly (bisection on the Sturm sequence). A real potential gives a real, symmetric operator, so the pitches are real by construction — this is a lossless string, always. What is scored. The pitch ladder is rescaled so that it has the same number of pitches below the chosen height as the zeros have (this removes the arbitrary overall scale of A and the string length), and the score is the fraction of zeros with a pitch within 0.15 of a mean spacing; chance is about 0.3, and the fake-prime string tells you what chance looks like for this configuration. What would count. A cyan score far above the grey one, holding up as K grows. What has been seen so far: no medium matches the zeros beyond chance, but the prime comb with the prime weights makes the pitches repel like the zeros do, while random bumps make them cluster — the arithmetic shows in the statistics before it shows in the positions. Why the known route can't happen here. The zeros of the explicit formula appear as output of a sum over primes in log-time; nothing here sums over primes — the primes only shape the medium — so anything that lands on the zeros is a new relation, not the old one in disguise.