PrimeMind · Structural atlas · superposition

Superposition Lab

A magnifier for the blue wave. Choose any list of frequencies — one pitch, the zeta zeros, the whole numbers, the primes, or a list you type — give them weights and a Gaussian window, and see their sum in log-time at a resolution that never aliases. One frequency is a smooth bell with a wiggle inside and nothing else; the beats, spikes and standing patterns you see with more than one are real interference, not drawing artifacts. The lab tells you its samples-per-cycle so you always know.

Frequencies

source

t₀ (for "one frequency")

K (how many / up to)

weight of frequency f

shape

custom list — numbers separated by spaces or commas (e.g. 14.13 21.02 25.01, or 1 2 3 5 8 13)

Window, range, and drawing

Gaussian window width s (0 = no window)

u range ±

centre of the view

line width

samples per cycle of the fastest frequency (min)

draw the envelope stems at ±log n for prime powers normalize the sum to fit ignore |u| < 0.3 when scaling (hides the u = 0 alignment spike) also draw |sum| (power)

Every frequency f contributes w(f) · cos(f·u) (or sine), times the Gaussian window e^−u²/2s². What each preset shows: one frequency — a bell with a wiggle; if you see beats here, raise the samples-per-cycle. the zeros — spikes appear at u = log 2, log 3, log 4, log 5, log 7, log 8, log 9 …: the prime powers, recovered from the zeros (turn on the stems to check). the whole numbers — the Dirichlet kernel: a sharp spike every 2π and near-cancellation between; add more and the spikes sharpen. the primes as frequencies — the "shrinking" picture: spikes at simple fractions of 2π, sized by the Möbius rule. log of the primes — the primes as log-time delays, which is the arithmetic side of the explicit formula (choose the Λ-style weights with the whole numbers to get the exact prime pulse train). The lab always samples at least the chosen number of points per cycle of the fastest frequency, so any beat or standing pattern you see is in the mathematics, not the pixels.

Delays as primes → which frequencies come out the forward direction

prime powers up to

taper of the delays (Gaussian in log n; 0 = none — without it the peaks ring)

show frequencies from 0 to

mark the twin primes (amber doublets) among the delays

Here the primes are not frequencies but delays: each prime power n is a pulse at time log n with weight log p/√n, mirrored by the fold. The panel shows the strength of each frequency τ in that pulse train, |Σ w(n) e^{−iτ log n}|, with the zeros marked in violet. With a handful of primes the curve is broad and featureless. Add more and the peaks climb onto the zeros — every zero at once, none in particular. That is the honest form of "delays as primes give the zeros": collectively, in the limit, with no prime belonging to any zero. It is the same relation you saw run backwards above, where all the zeros together gave every prime power.