Laplacian field structure of a 2D reshaping of the signal — energy, gradient coherence, spectral content, and directional anisotropy.
Reshapes the 1D byte stream into a square 2D field and analyzes it through the Laplacian operator and its iterates: the Laplacian Δf (source/sink density), the biharmonic Δ²f (curvature of curvature), and Poisson recovery error (non-periodic boundary content). Two anisotropy metrics exploit the vertical/horizontal asymmetry created by row-major reshaping — temporal adjacency is preserved along rows but broken across rows.
L2 energy of the biharmonic field Δ²f, normalized. Phyllotaxis (2.34) and Circle Map QP (2.34) score highest: their angular structure creates maximum biharmonic energy in the 2D field. Logistic Period-2 (0.08) is near zero — its simple alternation produces a smooth 2D pattern.
Integrated squared gradient magnitude, normalized. Logistic Period-3 (1.57) tops the list: its three-cycle creates sharp 2D boundaries in the reshaped field. Wigner Semicircle (0.002) is near zero — its smooth distribution creates a smooth gradient field.
L2 energy of the Laplacian field Δf. Logistic Period-3 (2.29) again dominates. Thue-Morse (1.93) is second — its binary substitution creates sharp source/sink density in the 2D field.
Mean of the Laplacian field. L-System Dragon Curve (0.78) has the most positive mean (net source density); Devil's Staircase (-1.39) has the most negative (net sink density from its monotone plateaus).
Standard deviation of the Laplacian. Thue-Morse (177.2) and Rule 110 (155.7) score highest — their binary structure creates extreme Laplacian variation in the 2D field. Wigner Semicircle (1.3) is nearly uniform.
Low/high frequency energy ratio of the 2D Laplacian. Sine Wave (139.9) dominates massively: its clean periodicity concentrates Laplacian energy at low 2D spatial frequencies. Logistic Period-3 (0.0) has all energy at high frequencies.
Spatial autocorrelation of the gradient field. Wigner Semicircle (0.84) and Temperature Drift (0.82) score highest: their smooth distributions create coherent gradient fields. Logistic Period-3 (-0.33) has anti-coherent gradients (sharp alternating boundaries).
2D analog: solve Δu = Δf periodically and measure residual. Hawkes Process (541.1) scores highest — its clustered spike events create strong non-periodic boundary content in the 2D field. fBm Persistent (0.67) is near zero (well-captured by periodic modes).
Fraction of pixels where the Laplacian is positive (sources vs sinks). Logistic Period-3 (0.66) is most source-heavy; Period-2 (0.02) is almost entirely sinks.
Log ratio of vertical to horizontal gradient energy. Sine Wave (6.2) has strong vertical anisotropy because row-major reshaping preserves its temporal periodicity along rows. Logistic Edge-of-Chaos (-30.4) has extreme horizontal anisotropy — its chaotic dynamics create structure within rows but not across them.
Log ratio of vertical to horizontal Laplacian spectral energy. Tidal Gauge (6.1) scores highest: its slow ocean dynamics create anisotropic curvature structure in the 2D field. Logistic Edge-of-Chaos (-52.2) is the most extreme negative — same mechanism as spatial_anisotropy but amplified by the spectral transform.
| Source | Domain | Value |
|---|---|---|
| Phyllotaxis | bio | 2.3372 |
| Circle Map Quasiperiodic | chaos | 2.3370 |
| Logistic r=3.83 (Period-3 Window) | chaos | 2.3056 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0750 |
| Tidal Gauge (SF) | geophysics | 0.1904 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 1.5703 |
| Thue-Morse | exotic | 1.4041 |
| Champernowne | number_theory | 1.3119 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Wigner Semicircle | quantum | 0.0015 |
| ETH/BTC Ratio | financial | 0.0165 |
| Source | Domain | Value |
|---|---|---|
| Wigner Semicircle | quantum | 0.8437 |
| Temperature Drift | climate | 0.8178 |
| Van der Pol Oscillator | exotic | 0.8140 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | -0.3288 |
| Thue-Morse | exotic | -0.3135 |
| Logistic r=3.68 (Banded Chaos) | chaos | -0.2918 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 2.2857 |
| Thue-Morse | exotic | 1.9311 |
| Champernowne | number_theory | 1.7915 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Wigner Semicircle | quantum | 0.0004 |
| fBm (Persistent) | noise | 0.0076 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 0.7774 |
| ECG Ventricular | medical | 0.7350 |
| Square Wave | waveform | 0.6584 |
| ··· | ||
| Devil's Staircase | exotic | -1.3863 |
| Hawkes Process | exotic | -0.5026 |
| Random Steps | exotic | -0.4527 |
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 139.8922 |
| Van der Pol Oscillator | exotic | 56.4864 |
| Clipped Sine | waveform | 44.7186 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Logistic r=3.74 (Period-5 Window) | chaos | 0.0001 |
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 177.1768 |
| Rule 110 | exotic | 155.6543 |
| Rule 30 | exotic | 144.1670 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Wigner Semicircle | quantum | 1.2902 |
| Collatz Gap Lengths | number_theory | 1.5641 |
| Source | Domain | Value |
|---|---|---|
| Hawkes Process | exotic | 541.0809 |
| Seismograph (ANMO) | geophysics | 363.0865 |
| SIR Epidemic | bio | 315.6002 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| fBm (Persistent) | noise | 0.6683 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.7029 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 0.6564 |
| Earthquake Depths | geophysics | 0.6401 |
| Multiplicative Cascade | exotic | 0.6236 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0156 |
| Forest Fire | exotic | 0.1201 |
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 6.1985 |
| Damped Pendulum | motion | 5.8916 |
| Van der Pol Oscillator | exotic | 5.7126 |
| ··· | ||
| Logistic Edge-of-Chaos | chaos | -30.4447 |
| Logistic r=3.5 (Period-4) | chaos | -30.1888 |
| Logistic r=3.2 (Period-2) | chaos | -28.1681 |
| Source | Domain | Value |
|---|---|---|
| Tidal Gauge (SF) | geophysics | 6.0576 |
| Rossler Attractor | chaos | 5.8366 |
| Seismograph (ANMO) | geophysics | 5.5949 |
| ··· | ||
| Logistic Edge-of-Chaos | chaos | -52.2014 |
| Logistic r=3.5 (Period-4) | chaos | -52.1817 |
| Logistic r=3.2 (Period-2) | chaos | -51.4612 |
Hodge-Laplacian is the 2D complement to the 1D Laplacian geometry. By reshaping the byte stream into a square field, it captures spatial structure that 1D analysis misses: gradient coherence, source/sink balance, and directional anisotropy. The anisotropy metrics are particularly diagnostic — they measure how much the signal's structure is preserved vs destroyed by the row-major 2D reshaping, which is a proxy for temporal adjacency strength. In the atlas, it contributes 11 metrics with 4 unique (36% uniqueness), and its biharmonic_energy is a top-5 PC1 loading.