Orthogonal components, prime by prime
Click a row to select a prime. already dead (smaller prime), killed first by this prime (that is Ck), still alive at stage k.
Sieve stack — position space (“particle” picture)
One row per prime: faint = its full periodic field Mp, bright = its orthogonal contribution Ck. Bottom row = survivors of the stack.
Observation validation — orthogonal sequences
For each prime pk, the orthogonal new-kill set is observed as Ck = pk × {1, q prime : pk ≤ q < Lk−1}. Rows compare the computed Ck prefix against that formula. The full period is expanded only when it is small enough to display; otherwise the first terms are validated.
Prime spectrum — frequency space (“wave” picture)
Every prime owns the comb of frequencies m/p, m = 1 … p−1 in ℚ/ℤ, each with amplitude exactly 1/p. Distinct primes have disjoint nonzero support; the DC term (white, at frequency 0) is shared by all masks and equals ρk for the product.
Interference field — waves collapsing into composites
[p | n] = (1/p) Σm cos(2πmn/p) is literally a sum of cosines. Summing the truncated expansions over the basis gives UH(n); as H grows the ripples sharpen into spikes at composites and the survivors emerge as exact zeros. Prime gaps = constructive interference.
Flows: multiplicative density, additive entropy
ρk = ∏(1−1/pi) (Mertens, log axis) vs. Hjoint(k) = Σ H(pi) and the per-prime increment ΔH = H(pk) ~ log pk/pk.
CRT product structure
Two coordinates of ℤ/Lkℤ ≅ ∏ ℤ/piℤ.
Survivor gaps
Gap histogram of the wheel survivors in the window.
Reading the numbers
- ×(1−1/p) — the attenuation factor of this filter stage.
- ρk — cumulative survivor density (exact, not asymptotic).
- ρk−1/p — natural density of the orthogonal kill-set Ck.
- ΔH — marginal (Shannon) entropy injected by this prime's alive/dead indicator, in bits.
- hk = Hjoint/Lk — joint entropy per integer of period; collapses super-exponentially.
- comb — (p−1) nonzero harmonics at amplitude 1/p; nonzero spectral power (p−1)/p².
Caveat kept from the paper: all masks share the DC frequency, so only the
nonzero supports are disjoint. The frequencies here are rational points of
ℚ/ℤ — categorically unlike zeta zeros. Serve this directory over HTTP (ES
modules won't load from file://).