Whether the signal's multiscale statistics are specifically consistent with turbulence — not just "has a power law" but "matches the physics of energy cascades and intermittent dissipation."
Computes structure functions S_q(r) = mean(|u(x+r) - u(x)|^q) at dyadic scales and tests them against two turbulence theories: Kolmogorov 1941 (K41, which predicts simple linear scaling exponents) and She-Leveque (SL, which predicts specific nonlinear corrections from intermittency). Also measures the direction of energy transfer through scales, intermittency burst clustering, and ESS linearity improvement.
Does She-Leveque fit the observed scaling exponents better than K41? Uses Extended Self-Similarity (ESS) to normalize by the third-order exponent, removing dependence on whether a clean inertial range exists. +1 means SL fits perfectly and K41 doesn't (strong intermittency). -1 means K41 fits better (simple self-similar scaling). Sunspot Number (+0.930), Rössler Attractor (+0.925), and Temperature Drift (+0.917) score highest — all have turbulence-like intermittent bursts. Sine Wave (-1.0) and Brownian Walk (-0.985) score lowest — their scaling is too smooth for SL corrections to help.
Which direction does energy flow through scales? Computed from the skewness of velocity increments: S_3(r)/S_2(r)^{3/2}. Negative means forward cascade (large structures break into smaller ones, as in 3D turbulence). Positive means inverse cascade (small structures merge into larger ones, as in 2D turbulence). Near zero means no preferred direction. SIR Epidemic, Hodgkin-Huxley, and financial time series saturate at ±1 (strong asymmetric dynamics). White noise sits at 0.001 (symmetric by construction).
Are energy dissipation bursts clustered in time? Computes the local intermittency ε(t) = |δu(t,8)|² / <|δu|²> at scale 8, log-transforms via log(1+ε) to stabilize variance, then measures the lag-1 autocorrelation. High positive values mean intense bursts tend to follow other intense bursts (temporal clustering), which is a hallmark of turbulent cascades. Shuffling scatters bursts randomly, driving autocorrelation to ~0. This is the strongest NS metric by Cohen's d (mean |d| = 12.8). Lorenz, Rössler, Collatz, BTC Volatility, and Van der Pol all saturate at d = 20. Logistic Chaos scores 18.3 — its intermittent bursts near the chaotic threshold are strongly clustered despite looking irregular. White Noise (0.004) and AES are clean nulls. Replaces flatness_slope, which measured the same intermittency axis less effectively (mean |d| = 7.1, |r| = 0.83 correlation). Discovered by ShinkaEvolve evolutionary search (gen 4/50, never beaten).
How much does Extended Self-Similarity improve the structure function fits? ESS replaces log S_q vs log r with log S_q vs log S_3, which dramatically linearizes the scaling for turbulent cascades. The metric is the mean R² improvement across moment orders. Fibonacci Word (+1.0), Rule 30 (+0.999), and Morse Code (+0.992) score highest — their structure functions have clean power-law scaling in ESS form that raw log-log fits miss. Devil's Staircase (-0.07) scores lowest (ESS can't help when structure functions are degenerate).
| Source | Origin | Value |
|---|---|---|
| OTOC Growth | random-matrix-quantum | 1.0000 |
| Critical Transition (Fold) | iterated-maps | 1.0000 |
| Potomac River Flow | geophysical | 0.9685 |
| ··· | ||
| Stochastic Resetting Walk | stochastic-process | -1.0000 |
| Tank Drain Cascade | self-organized-criticality | -0.9998 |
| Sawtooth Wave | signal-synthesis | -0.9994 |
| Source | Origin | Value |
|---|---|---|
| Penrose Substitution | symbolic-dynamics | 0.9996 |
| Fibonacci Word | symbolic-dynamics | 0.9996 |
| Morse Code | signal-synthesis | 0.9963 |
| ··· | ||
| Collatz Stopping Times | number-theory | -0.2047 |
| Rudin-Shapiro | symbolic-dynamics | -0.0012 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Riemann-Hardy-Littlewood | number-theory | 0.9579 |
| Triple Pendulum | continuous-flows | 0.9433 |
| Rossler Attractor | continuous-flows | 0.9224 |
| ··· | ||
| Sine Wave | signal-synthesis | -1.0000 |
| 2-Torus Quasiperiodic | coupled-oscillators | -0.9986 |
| Seismograph (ANMO) | geophysical | -0.9982 |
Navier-Stokes separates "turbulence-like" signals from "merely complex" ones. MultifractalGeometry already measures generic spectrum width; SpectralGeometry already fits generic power-law slopes. What NS adds is model-specific: does the observed nonlinearity match the specific She-Leveque prediction for intermittent turbulence? Does energy flow through scales with a preferred direction? And are dissipation bursts clustered in time? In the atlas, sources that score high on sl_fit_quality (seismic, temperature drift, Rössler) have turbulence-compatible intermittency. Sources with high intermittency_autocorr (Lorenz, Logistic Chaos, BTC Volatility) have coherent energy dissipation events — the temporal clustering signature that distinguishes genuine cascade dynamics from shuffled surrogates. Sources with high ess_quality (Fibonacci, Rule 30) have cascade-compatible multiscale structure that ESS reveals.