The curvature cascade of a 1D signal — how energy propagates through successive discrete derivatives.
Applies the discrete Laplacian (second difference) and its iterates to the signal. The biharmonic ratio measures how much energy survives two additional derivative orders; the Poisson recovery error quantifies non-periodic boundary content by solving Δu = Δf with periodic FFT and measuring the residual. Gradient metrics capture monotone run structure and its coupling to curvature.
Energy ratio ||f''''||²/||f''||². Logistic Period-2 (16.0) and Period-4 (15.9) score highest: their sharp alternating jumps propagate maximum energy through successive derivatives. Bearing Outer (1.05) scores lowest among non-constant sources — its smooth vibration envelope dies fast under differentiation.
Residual when solving Δu = Δf with periodic boundary conditions via FFT. GUE Spacings (9.7M) and Fibonacci Quasicrystal (8.8M) score highest: their structure lives at the boundaries, not in the bulk periodic component. Quantum Walk (0.02) scores near zero — its probability distribution is well-captured by periodic Fourier modes.
Fraction of consecutive first differences with the same sign (monotone runs). Square Wave (0.99) nearly always continues in the same direction. Logistic Edge-of-Chaos (0.0) and Fibonacci Word (0.0) reverse direction at every step.
Bounded fraction low_e / (low_e + high_e) measuring what fraction of curvature energy lives in the bottom quarter of the spectrum. (Reformulated post-Phase-2 from the unbounded ratio low_e / (high_e + ε), which produced values up to 4×10¹⁰ once normalization shrunk high_e below the absolute epsilon.) Smooth continuous-time dynamics saturate at 1.0 — Lorenz, Van der Pol, Double Pendulum, Duffing, Mackey-Glass, Sine Wave, Lotka-Volterra, Damped Pendulum, Berry Random Wave, Ocean Swell — their curvature energy is entirely low-frequency. Discrete maps with sharp jumps push energy to the Nyquist end: Standard Map K=0.5 (0.32), Logistic period-3 (0.17), Symbolic Lorenz (0.02), Logistic chaotic (~0.001), Logistic period-2/4 and Constants (0.0). Effectively a continuous-time vs discrete-jump curvature classifier.
Lag-1 autocorrelation of |f''|. Devil's Staircase (0.80) scores highest: its long constant plateaus create persistent curvature regimes. Logistic Period-3 (-0.50) is maximally anti-persistent — curvature events alternate with flat stretches.
Correlation of |Laplacian| computed at scale 1 vs scale 2 (dilated kernel). Bearing Inner (0.79) and Ocean Swell (0.77) score highest: their curvature structure is hierarchically consistent. Phyllotaxis (-0.59) and Circle Map QP (-0.59) have anti-correlated curvature across scales.
Correlation between monotone-run mask and |f''|, negated. Measures whether smooth runs coincide with low curvature. Phyllotaxis (1.0) and Circle Map QP (1.0) have perfect coupling. Lotka-Volterra (-0.31) has the opposite: its smooth runs carry high curvature (curved oscillation arcs).
Product of curvature_autocorrelation and gradient_curvature_anticorrelation. Shuffled Blocks (0.54) scores highest: its random block boundaries create curvature events that cluster AND couple to gradient structure. Logistic Period-3 (-0.50) is the most negative — strong anti-persistent curvature with inverted coupling.
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | iterated-maps | 15.9980 |
| Heisenberg Walk | symbolic-dynamics | 15.9087 |
| Logistic r=3.5 (Period-4) | iterated-maps | 15.8541 |
| ··· | ||
| Sine Wave | signal-synthesis | 0.0000 |
| Berry Random Wave | random-matrix-quantum | 0.0000 |
| Exponential Chirp | signal-synthesis | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| OTOC Growth | random-matrix-quantum | 0.9996 |
| Sine Wave | signal-synthesis | 0.9995 |
| Exponential Chirp | signal-synthesis | 0.9994 |
| ··· | ||
| Critical Circle Map (Silver Mean) | iterated-maps | -0.6081 |
| Phyllotaxis | number-theory | -0.5878 |
| Circle Map Quasiperiodic | iterated-maps | -0.5878 |
| Source | Origin | Value |
|---|---|---|
| OTOC Growth | random-matrix-quantum | 0.9996 |
| Sine Wave | signal-synthesis | 0.9995 |
| Exponential Chirp | signal-synthesis | 0.9994 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | iterated-maps | -0.5000 |
| Critical Circle Map | iterated-maps | -0.3451 |
| Weierstrass | special-functions | -0.3218 |
| Source | Origin | Value |
|---|---|---|
| Rudin-Shapiro | symbolic-dynamics | 1.0000 |
| Sawtooth Wave | signal-synthesis | 1.0000 |
| Phyllotaxis | number-theory | 1.0000 |
| ··· | ||
| μ-law Sine | signal-synthesis | -0.2046 |
| Rainfall (ORD Hourly) | atmospheric | -0.1612 |
| Devil's Staircase | special-functions | -0.1257 |
| Source | Origin | Value |
|---|---|---|
| Van der Pol Oscillator | continuous-flows | 0.9986 |
| Magnetic Pendulum (3-Magnet) | continuous-flows | 0.9976 |
| Exponential Chirp | signal-synthesis | 0.9969 |
| ··· | ||
| Kolakoski Sequence | symbolic-dynamics | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Periodic Word | symbolic-dynamics | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Pomeau-Manneville | iterated-maps | 0.5955 |
| Shuffled Blocks | stochastic-process | 0.5405 |
| Sawtooth Wave | signal-synthesis | 0.4980 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | iterated-maps | -0.4983 |
| Critical Circle Map | iterated-maps | -0.3380 |
| Circle Map Quasiperiodic | iterated-maps | -0.3090 |
| Source | Origin | Value |
|---|---|---|
| Quantum Walk | random-matrix-quantum | 15.6729 |
| Fibonacci Tight-Binding | random-matrix-quantum | 15.6161 |
| Fibonacci Word | symbolic-dynamics | 15.4566 |
| ··· | ||
| OTOC Growth | random-matrix-quantum | 0.0005 |
| Square Wave | signal-synthesis | 0.0020 |
| Sawtooth Wave | signal-synthesis | 0.0033 |
Laplacian sits in the Scale lens and captures derivative-order energy cascade — complementary to Hölder (pointwise smoothness) and p-Variation (path roughness). In the atlas, biharmonic_ratio separates periodic signals with sharp transitions (high) from smooth oscillators (low). The poisson_recovery_error is unique: it's the only metric that specifically measures non-periodic boundary content, lighting up on quasicrystal and number-theoretic sequences whose structure doesn't fit into periodic Fourier decomposition.