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.
Low/high frequency energy ratio of the Laplacian. Standard Map Mixed (143.3) dominates: its curvature energy is concentrated at low frequencies (broad smooth bends in the mixed regular/chaotic dynamics). Logistic Period-2 (0.0) has zero low-frequency curvature — all its energy is at the Nyquist frequency.
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 | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 15.9980 |
| Logistic r=3.5 (Period-4) | chaos | 15.8529 |
| Noisy Period-2 | chaos | 15.7787 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Bearing Outer | bearing | 1.0489 |
| Ocean Swell | geophysics | 1.2218 |
| Source | Domain | Value |
|---|---|---|
| Bearing Inner | bearing | 0.7878 |
| Ocean Swell | geophysics | 0.7715 |
| ECG Fusion | medical | 0.7458 |
| ··· | ||
| Phyllotaxis | bio | -0.5878 |
| Circle Map Quasiperiodic | chaos | -0.5877 |
| Critical Circle Map | chaos | -0.5832 |
| Source | Domain | Value |
|---|---|---|
| Devil's Staircase | exotic | 0.8013 |
| Logistic r=3.68 (Banded Chaos) | chaos | 0.7660 |
| Rössler Hyperchaos | chaos | 0.7477 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | -0.5000 |
| Critical Circle Map | chaos | -0.3448 |
| Circle Map Quasiperiodic | chaos | -0.3090 |
| Source | Domain | Value |
|---|---|---|
| Phyllotaxis | bio | 1.0000 |
| Circle Map Quasiperiodic | chaos | 1.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.9965 |
| ··· | ||
| Lotka-Volterra | bio | -0.3057 |
| Hodgkin-Huxley | bio | -0.2281 |
| Clipped Sine | waveform | -0.2260 |
| Source | Domain | Value |
|---|---|---|
| Constant 0x00 | noise | 1.0000 |
| Square Wave | waveform | 0.9919 |
| Triangle Wave | waveform | 0.9797 |
| ··· | ||
| Logistic Edge-of-Chaos | chaos | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Fibonacci Word | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Shuffled Blocks | exotic | 0.5405 |
| Coupled Map Lattice | chaos | 0.3372 |
| Henon Map | chaos | 0.3065 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | -0.4983 |
| Critical Circle Map | chaos | -0.3377 |
| Circle Map Quasiperiodic | chaos | -0.3090 |
| Source | Domain | Value |
|---|---|---|
| Standard Map K=0.5 (Mixed) | chaos | 143.2595 |
| Ocean Swell | geophysics | 20.6555 |
| Bearing Outer | bearing | 16.2628 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| GUE Spacings | quantum | 9739279.8412 |
| Fibonacci Quasicrystal | number_theory | 8845418.5383 |
| Phyllotaxis | bio | 8593151.5043 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Quantum Walk | quantum | 0.0173 |
| Random Steps | exotic | 0.1925 |
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.