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.
2D analog of the 1D Laplacian's Poisson residual: solve Δu = Δf periodically over the reshaped field 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. The signed counterpart to laplacian_mean, retained because the fraction is bounded and well-behaved across heavy-tailed sources where the raw mean explodes.
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 | Origin | Value |
|---|---|---|
| Sine Wave | signal-synthesis | 7.0095 |
| Spring Pendulum | continuous-flows | 5.9013 |
| Magnetic Pendulum (3-Magnet) | continuous-flows | 5.8527 |
| ··· | ||
| Phyllotaxis | number-theory | -1.2723 |
| Circle Map Quasiperiodic | iterated-maps | -1.2723 |
| Temperature Drift | atmospheric | -0.9237 |
| Source | Origin | Value |
|---|---|---|
| Lotka-Volterra | continuous-flows | 5.3673 |
| Seismograph (ANMO) | geophysical | 5.0545 |
| Halvorsen Attractor | continuous-flows | 4.7706 |
| ··· | ||
| LIGO Hanford | astrophysical | 0.0000 |
| LIGO Livingston | astrophysical | 0.0000 |
| Critical Transition (Fold) | iterated-maps | 0.0196 |
| Source | Origin | Value |
|---|---|---|
| Sine Wave | signal-synthesis | 0.9989 |
| Period-Doubled Wave | signal-synthesis | 0.9970 |
| Chua's Circuit | continuous-flows | 0.9967 |
| ··· | ||
| Temperature Drift | atmospheric | 0.2000 |
| Phyllotaxis | number-theory | 0.2384 |
| Circle Map Quasiperiodic | iterated-maps | 0.2384 |
| Source | Origin | Value |
|---|---|---|
| Fibonacci Tight-Binding | random-matrix-quantum | 0.7595 |
| Pomeau-Manneville | iterated-maps | 0.7421 |
| Anderson 1D Localized | random-matrix-quantum | 0.6773 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0156 |
| Mian-Chowla | number-theory | 0.0182 |
| Devil's Staircase | special-functions | 0.0988 |
Hodge-Laplacian is the 2D complement to the 1D Laplacian geometry. The scalar-field metrics (poisson_recovery_error, source_fraction) capture spatial structure that 1D analysis misses: non-periodic boundary content and source/sink balance from row-major reshaping. The Hodge-Helmholtz metrics (solenoidal_fraction, curl_div_ratio) add a phase-space dynamical axis: the rotational-vs-gradient split of the empirical velocity field cleanly separates closed-orbit dynamics from gradient flows. In the atlas, the rotational axis is unique to this geometry — no other lens exposes phase-space vorticity directly.