Do the signal's high-value positions form a Sidon-like set — one whose pairwise sums rarely collide?
Take A = the positions of samples above the 95th percentile, form the sumset A + A, and measure its isolated-point fraction (sums with no neighbor at ±1). Inspired by Erdős problem #152. A dense or chaotic top saturates the sumset (isolation ≈ 0); a smooth, contiguous top forms one block (≈ 0); a sparse, modular, or periodic top scatters into isolated points (≈ 1). It reads the additive combinatorial structure of where the peaks fall — a number-theory-flavored lens.
Isolated-point fraction of the peak-set sumset. Saturates at 1 on sparse, regularly-spaced peaks — periodic-window chaos (Logistic Edge-of-Chaos, Period-2, Period-4, Period-3 Window all 1.00) and modular number-theory signals (Goldbach r(2n), von Mangoldt, Euler Totient Ratio); near 0 on dense or smooth tops (Gray Code Counter, Tank Drain Cascade). Noise-blind (White Noise, AES, π digits ≈ 0.009) and 24% non-reconstructable from the rest of the atlas (R²_oof = 0.76).
| Source | Origin | Value |
|---|---|---|
| Logistic Edge-of-Chaos | iterated-maps | 1.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 1.0000 |
| Logistic r=3.65 (Banded Chaos) | iterated-maps | 1.0000 |
| ··· | ||
| Sawtooth Wave | signal-synthesis | 0.0000 |
| Period-Doubled Wave | signal-synthesis | 0.0000 |
| Sine Wave | signal-synthesis | 0.0000 |
Lights up where high-value samples sit at sparse, modularly-structured positions — the additive-structure signature of Goldbach counts, von Mangoldt, and periodic windows of chaos. It stays dark for dense, smooth, or random peak sets.