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.
2D analog of the 1D Laplacian's bounded spectral fraction: low-radial-frequency energy / (low + high) on the radial 2D Laplacian power spectrum. Smooth continuous-time signals reshape into smoothly-varying 2D fields whose curvature is entirely low-frequency — Sine Wave (0.99), Lorenz (0.94), Van der Pol (0.92), Clipped Sine (0.84), Triangle Wave (0.81). Discrete chaos reshapes into noisy 2D fields with high-frequency curvature (logistic chaos near 0.0). White Noise (0.03) and Constants (0.0) sit at the floor.
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 |
| Partition Function | number_theory | 0.0169 |
| Logistic r=3.2 (Period-2) | chaos | 0.0750 |
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 7.0095 |
| Duffing Oscillator | chaos | 5.7091 |
| Chua's Circuit | exotic | 5.4864 |
| ··· | ||
| Circle Map Quasiperiodic | chaos | -1.2723 |
| Phyllotaxis | bio | -1.2723 |
| Temperature Drift | climate | -0.9237 |
| 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 |
| Primes | number_theory | 0.0007 |
| Minkowski Question Mark | exotic | 0.0010 |
| Source | Domain | Value |
|---|---|---|
| Arnold Cat Map | chaos | -0.4922 |
| RANDU | binary | -0.5085 |
| Beta Noise | noise | -0.5457 |
| ··· | ||
| Constant 0xFF | noise | -30.0000 |
| Primes | number_theory | -9.3568 |
| Partition Function | number_theory | -9.2914 |
| Source | Domain | Value |
|---|---|---|
| Partition Function | number_theory | 1.0000 |
| Primes | number_theory | 1.0000 |
| Minkowski Question Mark | exotic | 0.9941 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | -0.3288 |
| Thue-Morse | exotic | -0.3135 |
| Logistic r=3.68 (Banded Chaos) | chaos | -0.2945 |
| 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 |
| Primes | number_theory | 0.0000 |
| Partition Function | number_theory | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 0.0030 |
| ECG Ventricular | medical | 0.0029 |
| Square Wave | waveform | 0.0026 |
| ··· | ||
| Devil's Staircase | exotic | -0.0054 |
| Lorenz-96 N=36 | chaos | -0.0024 |
| Hawkes Process | exotic | -0.0020 |
| Source | Domain | Value |
|---|---|---|
| Partition Function | number_theory | 0.9943 |
| Takagi Function | exotic | 0.9807 |
| Minkowski Question Mark | exotic | 0.9747 |
| ··· | ||
| 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 | 0.6948 |
| Kolakoski Sequence | exotic | 0.6355 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.6190 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Primes | number_theory | 0.0002 |
| Partition Function | number_theory | 0.0004 |
| Source | Domain | Value |
|---|---|---|
| Hawkes Process | exotic | 541.4338 |
| Seismograph (ANMO) | geophysics | 363.0755 |
| SIR Epidemic | bio | 315.5674 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| fBm (Persistent) | noise | 0.6666 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.7029 |
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 0.9989 |
| Chua's Circuit | exotic | 0.9967 |
| Duffing Oscillator | chaos | 0.9964 |
| ··· | ||
| Temperature Drift | climate | 0.2000 |
| Circle Map Quasiperiodic | chaos | 0.2384 |
| Phyllotaxis | bio | 0.2384 |
| Source | Domain | Value |
|---|---|---|
| Pomeau-Manneville | chaos | 0.7934 |
| Lotka-Volterra | bio | 0.6731 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.6564 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Partition Function | number_theory | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0156 |
| Source | Domain | Value |
|---|---|---|
| Partition Function | number_theory | 9.5425 |
| Sine Wave | waveform | 8.5648 |
| Chua's Circuit | exotic | 7.7037 |
| ··· | ||
| Logistic r=3.5 (Period-4) | chaos | -25.3439 |
| Logistic r=3.2 (Period-2) | chaos | -23.8776 |
| Logistic Edge-of-Chaos | chaos | -15.8175 |
| Source | Domain | Value |
|---|---|---|
| Primes | number_theory | 9.6959 |
| Partition Function | number_theory | 9.4474 |
| Minkowski Question Mark | exotic | 9.0735 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -42.8803 |
| Logistic r=3.5 (Period-4) | chaos | -42.5136 |
| Logistic Edge-of-Chaos | chaos | -28.3020 |
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.