Where the energy lives in frequency space.
Computes the power spectral density via FFT and characterizes its shape. Not just "what's the dominant frequency" but "is the spectrum flat (noise), peaked (periodic), or power-law (fractal)?"
The exponent in P(f) ~ f^beta. Brown noise = -2, pink noise = -1, white noise = 0. Positive slopes (blue spectra) are rare in nature — Thue-Morse (+1.75) and Fibonacci word (+1.63) are the bluest signals in the atlas, their substitution structure concentrating energy at high frequencies. Kilauea tremor (-3.33) and Lotka-Volterra (-3.14) are the reddest: slow dynamics dominate.
How well does a single power law fit the spectrum? Anti-persistent fBm scores 0.99 (textbook power law). Logistic period-2 scores 0.0 (all energy at one frequency, not a power law at all). This distinguishes genuine 1/f processes from signals that happen to have similar average slope.
Wiener entropy: ratio of geometric to arithmetic mean of the spectrum. 1.0 = perfectly flat (white noise). 0.0 = all power at one frequency (pure tone). Rule 30 scores 0.57 — its pseudorandom output is flatter than most chaos but not as flat as true randomness.
Where is the maximum power? Stern-Brocot walk peaks at the Nyquist frequency (0.5). Most natural signals peak near DC (0.0). Values near 0.5 indicate the signal's dominant variation is at the shortest timescale — rapid alternation or anti-persistence.
Mean coherence of phase across frequency bins. Constants (1.0) and logistic period-2 (1.0) have perfectly coherent phases. Gaussian Noise (0.0005) has no phase structure. High phase coherence means the signal's frequency components maintain a fixed relationship — the hallmark of deterministic or periodic processes. Low coherence means the phases are random, as in noise.
Standard deviation of the spectral centroid, measuring how spread out the power is around its center of mass. Stern-Brocot Walk (0.21) has the widest bandwidth — its power spreads across all frequencies. Constants and logistic period-2 score 0.0 (all power at a single frequency). This complements spectral_entropy: a signal can have moderate entropy (several peaks) but low bandwidth (peaks clustered near one frequency).
| Source | Origin | Value |
|---|---|---|
| Noisy Period-2 | iterated-maps | 0.5000 |
| Quartic Map (Feigenbaum) | iterated-maps | 0.5000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.5000 |
| ··· | ||
| fBm (Antipersistent) | stochastic-process | 0.0001 |
| Brownian Walk | stochastic-process | 0.0001 |
| fBm (Persistent) | stochastic-process | 0.0001 |
| Source | Origin | Value |
|---|---|---|
| Heisenberg Walk | symbolic-dynamics | 0.2394 |
| Stern-Brocot Walk | number-theory | 0.2093 |
| Fibonacci Tight-Binding | random-matrix-quantum | 0.2088 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Takagi Function | special-functions | 0.0004 |
| OTOC Growth | random-matrix-quantum | 0.0005 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | iterated-maps | 0.9998 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.9991 |
| Sine Wave | signal-synthesis | 0.9812 |
| ··· | ||
| XorShift32 | algorithmic-bytes | 0.0004 |
| Partition Function | number-theory | 0.0005 |
| Golden Ratio Digits | number-theory | 0.0005 |
| Source | Origin | Value |
|---|---|---|
| fBm (Persistent) | stochastic-process | 0.9999 |
| Perlin Noise | stochastic-process | 0.9999 |
| fBm (Antipersistent) | stochastic-process | 0.9999 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| L-System (Dragon Curve) | symbolic-dynamics | 0.0000 |
Spectral slope is the strongest separator between the ordinal view's C1 (red-spectrum oscillators, slope -1.9) and C4 (blue-spectrum chaos, slope +0.3). In the seismic P-wave investigation, spectral metrics were among the top discriminators: earthquake P-waves flatten the ambient spectrum by injecting broadband impulsive energy.