Two complementary 2D structures derived from the 1D signal — the Laplacian field of a row-major reshape, and the Hodge-Helmholtz decomposition of an empirical phase-space velocity field.
The scalar-field branch 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. The phase-space branch delay-embeds the signal into a 64×64 bin grid, averages the per-sample velocity (Δs(t), Δs(t+τ)) inside each bin, and applies the Hodge-Helmholtz decomposition v = ∇φ + ∇⊥ψ + h via Leray projection in Fourier space. The split between gradient (∇φ) and rotational (∇⊥ψ) components is a real signal axis the framework otherwise lacks.
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.
log₁₀ of the mean kinetic energy per cell in the empirical phase-space velocity field. Distinguishes high-power dynamical flows (oscillators, low-d chaos) from low-power random walks where per-bin average velocities cancel to ~0. Arnold Cat Map (-0.49), RANDU (-0.51), Beta Noise (-0.55), Gzip (-0.56), and Pi Digits (-0.56) sit at the top — their delay-embedding bins are filled densely enough that the mean velocity per bin is large. Primes (-9.36), Partition Function (-9.29), and Minkowski ?(x) (-8.78) sit at the bottom — these number-theoretic signals trace tight quasi-monotone manifolds in delay space, so almost every bin is empty and the average velocity vanishes. Constants saturate at the floor (-30.0).
Fraction of the phase-space velocity field's energy carried by the rotational component: ‖∇⊥ψ‖² / (‖∇φ‖² + ‖∇⊥ψ‖²). 0 means pure gradient flow (sources and sinks), 1 means pure rotation (closed orbits), 0.5 is balanced. Sine Wave (0.999), Chua's Circuit (0.997), Duffing (0.996), Hénon-Heiles (0.996), and Berry Random Wave (0.994) saturate at the rotational end — their phase portraits are dominated by closed-orbit structure. Temperature Drift (0.20), Circle Map QP (0.24), and Phyllotaxis (0.24) sit at the gradient end. This is a real signal axis the framework otherwise lacks: rotational vs gradient-like dynamical structure on delay-embedded phase space.
log(‖∇×v‖² / ‖∇·v‖²) on the phase-space velocity field. The same rotational/gradient axis as solenoidal_fraction, but k²-weighted so it emphasizes small-scale vorticity vs divergence. Decouples from solenoidal_fraction on scale-anisotropic flows where the dominant rotational structure lives at a different scale than the dominant divergence. Sine Wave (7.01), Duffing (5.71), Chua's Circuit (5.49), Hénon-Heiles (5.46), and Berry Random Wave (5.15) score highest. Phyllotaxis (-1.27), Circle Map QP (-1.27), and Temperature Drift (-0.92) score lowest.
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 7.0095 |
| Magnetic Pendulum (3-Magnet) | motion | 5.9458 |
| Spring Pendulum | motion | 5.9013 |
| ··· | ||
| Circle Map Quasiperiodic | chaos | -1.2723 |
| Phyllotaxis | bio | -1.2723 |
| Critical Circle Map (Bronze Mean) | chaos | -0.8954 |
| Source | Domain | Value |
|---|---|---|
| Lotka-Volterra | bio | 5.3673 |
| Seismograph (ANMO) | geophysics | 5.0545 |
| Halvorsen Attractor | chaos | 4.7706 |
| ··· | ||
| LIGO Hanford | astro | 0.0000 |
| LIGO Livingston | astro | 0.0000 |
| Critical Transition (Fold) | chaos | 0.0196 |
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 0.9989 |
| Magnetic Pendulum (3-Magnet) | motion | 0.9971 |
| Period-Doubled Wave | waveform | 0.9970 |
| ··· | ||
| Temperature Drift | climate | 0.2042 |
| Circle Map Quasiperiodic | chaos | 0.2384 |
| Phyllotaxis | bio | 0.2384 |
| Source | Domain | Value |
|---|---|---|
| Pomeau-Manneville | chaos | 0.7421 |
| Anderson 1D Localized | quantum | 0.6773 |
| Solar Flares Daily Peak | astro | 0.6735 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 0.0156 |
| Mian-Chowla | number_theory | 0.0182 |
| Devil's Staircase | exotic | 0.0988 |
Hodge-Laplacian is the 2D complement to the 1D Laplacian geometry. The scalar-field metrics capture spatial structure that 1D analysis misses: gradient coherence, source/sink balance, and directional anisotropy from row-major reshaping. The Hodge-Helmholtz metrics (solenoidal_fraction, curl_div_ratio, flow_energy) add a phase-space dynamical axis: the rotational-vs-gradient split of the empirical velocity field cleanly separates closed-orbit dynamics from gradient flows, and flow_energy separates dense-attractor systems from sparse-manifold or random-walk systems. In the atlas, biharmonic_energy is a top-5 PC1 loading and the Hodge branch contributes a unique rotational axis that no other geometry exposes.