What intrinsic scale-symmetries does the signal have — at what dilation factor does it look like itself?
Self-dilation beat spectroscopy. It resamples the signal at dilated time, s(α·t), correlates that against the original s(t), and sweeps the dilation factor α: ρ(α) = corr(s(t), s(α·t)). Peaks in ρ(α) reveal scale-symmetries — periodic, Sturmian, fractal, or harmonic-rich — much as an optical moiré pattern emerges between a grid and a rescaled copy of itself.
Self-coherence specifically at α = φ (the golden ratio): a golden-mean self-similarity signature. Tops on exactly the golden quasiperiodic sources — Circle Map Quasiperiodic (0.41), Penrose Substitution (0.34), Fibonacci Quasicrystal (0.33), Phyllotaxis (0.30) — and goes negative on fractal signals (fBm −0.14, Takagi −0.12).
The strongest self-dilation peak. Near 1 for periodic signals that match a rescaled copy of themselves perfectly (Logistic Period-2 and Period-4 1.00, Sine Map 0.99); near 0 for aperiodic ones (Rule 30 0.02, Free Group F₂ Walk 0.02).
The dilation factor α at which that strongest peak occurs (the dominant self-scale).
Spread (entropy) of the coherence curve across α. High when self-similarity is broadband (Critical Transition 4.22, GOES X-Ray 4.20); low when it concentrates at a single α (periodic windows).
Extra coherence at integer dilation factors versus irrational ones; harmonic-rich, period-doubled structure reads positive (Logistic Period-4 0.65).
Width of the near-invariant α band around the dominant peak (Chua's Circuit 0.045, Magnetic Pendulum 0.038).
| Source | Origin | Value |
|---|---|---|
| Critical Transition (Fold) | iterated-maps | 4.2169 |
| GOES X-Ray Flux | astrophysical | 4.1951 |
| Spectral Form Factor | random-matrix-quantum | 4.1843 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.9496 |
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 1.1005 |
| Logistic r=3.5 (Period-4) | iterated-maps | 1.3905 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.5 (Period-4) | iterated-maps | 0.6457 |
| Logistic Edge-of-Chaos | iterated-maps | 0.6238 |
| Sine Map (Feigenbaum) | iterated-maps | 0.6233 |
| ··· | ||
| OTOC Growth | random-matrix-quantum | -0.4471 |
| Spectral Form Factor | random-matrix-quantum | -0.4318 |
| SIR Epidemic | continuous-flows | -0.3921 |
| Source | Origin | Value |
|---|---|---|
| Chua's Circuit | continuous-flows | 0.0445 |
| Magnetic Pendulum (3-Magnet) | continuous-flows | 0.0380 |
| Barometric Pressure (Buoy) | atmospheric | 0.0376 |
| ··· | ||
| 2-Torus Quasiperiodic | coupled-oscillators | 0.0027 |
| Rule 30 | symbolic-dynamics | 0.0028 |
| Gap-Word β-Shift | symbolic-dynamics | 0.0031 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.5 (Period-4) | iterated-maps | 1.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 1.0000 |
| OTOC Growth | random-matrix-quantum | 0.9934 |
| ··· | ||
| Gap-Word β-Shift | symbolic-dynamics | 0.0207 |
| Rule 30 | symbolic-dynamics | 0.0222 |
| MT19937 (Mersenne Twister) | algorithmic-bytes | 0.0226 |
| Source | Origin | Value |
|---|---|---|
| OTOC Growth | random-matrix-quantum | 5.0000 |
| GOES X-Ray Flux | astrophysical | 3.5169 |
| Temperature | atmospheric | 3.4965 |
| ··· | ||
| Spectral Form Factor | random-matrix-quantum | 0.5000 |
| Rule 30 | symbolic-dynamics | 0.5623 |
| Seismic b-value (SoCal) | geophysical | 0.8096 |
| Source | Origin | Value |
|---|---|---|
| Circle Map Quasiperiodic | iterated-maps | 0.4122 |
| Penrose Substitution | symbolic-dynamics | 0.3403 |
| Fibonacci Quasicrystal | symbolic-dynamics | 0.3316 |
| ··· | ||
| fBm (Persistent) | stochastic-process | -0.1351 |
| Takagi Function | special-functions | -0.1184 |
| Perlin Noise | stochastic-process | -0.1181 |
A scale-symmetry fingerprint. The standout is moire_phi_response, a clean golden-ratio quasiperiodicity detector — the Fibonacci Quasicrystal's peak dilation lands at α = φ across seeds — distinguishing golden-mean order from both periodic and fractal self-similarity.