The shape and locality of phase coupling between frequency triplets (f₁, f₂, f₁ + f₂) — orthogonal extensions of Higher-Order Statistics' peak-bicoherence channel.
The bispectrum B(f₁, f₂) = E[X(f₁) X(f₂) X*(f₁ + f₂)] is the Fourier transform of the third-order cumulant. Linear Gaussian processes have identically zero bispectrum (Liouville: the third cumulant vanishes for Gaussian distributions, and linear filtering preserves Gaussianity); quadratic nonlinearity and phase-locked triplets produce non-zero bispectral mass at the coupled frequencies. Normalized via Kim-Powers 1979 bicoherence² ∈ [0, 1] = |B|² / (E[|X(f₁)X(f₂)|²] · E[|X(f₁ + f₂)|²]), which removes amplitude dependence and isolates phase-locking. Computed via segment-averaged Hann-windowed direct estimation; requires ≥16 segments for reliable phase averaging. Not encoding-invariant — uses raw amplitudes.
Gini coefficient of the bias-corrected bicoherence² distribution over the principal triangle. The 2026-05-22 finite-segment bias correction (Kim-Powers) subtracts the per-cell noise floor 1/n_segs and clips below-floor cells to zero before computing the Gini. The empirical semantic is therefore not the naive "low spread / high concentrated" reading: white noise sits HIGH (~0.7-0.8) because chi-squared residuals leave a sparse above-floor pattern (many zeros + a few small positives = high Gini); pure tones and constants sit at 0 because all cells fall below the floor and clip to zero; designed (a,b,a+b) triplets sit at ~0.85+ from concentrated true coupling above the floor; broad-coupling chaos (Lorenz) sits ~0.69. Read it as a discriminator of above-floor bispectral mass shape, not coupling intensity — that's what HOS:bicoherence_max already provides. Replaces the earlier Shannon-entropy formulation which saturated near 1.0 for ~93% of sources because normalized-entropy of an O(K²) noisy array sits very close to max log(K²/2) for almost any input.
mean(bicoherence² off the f₁=f₂ diagonal) / mean(bicoherence² on diagonal + off). Distinguishes self-harmonic coupling (single fundamental with harmonics → mass concentrated on the diagonal) from cross-mode coupling (multiple oscillators interacting → off-diagonal mass). Rössler Hyperchaos (0.88), Weierstrass (0.85), and the 3/4/5-Torus Quasiperiodic sources (0.70–0.73) score highest — genuine multi-mode interactions. Period-3 (0.11) and Euler Totient Ratio (0.19) sit at the diagonal-dominated end (single harmonic series). Constants are 0 (no bicoherence at all).
Mass-weighted mean of (f₁ + f₂) over the principal triangle, normalized to the Nyquist bin. Captures where in frequency the phase coupling lives — orthogonal to amount-of-coupling metrics (bicoherence_concentration, off_diagonal_ratio, HOS:bicoherence_max). Logistic period-3 and period-5 windows top out at 0.71–0.72 (windowed-periodic dynamics couple at high harmonics, beating with the chaotic backbone); broadband chaos (Ikeda, Lorenz-96) sits at 0.36–0.42; Blood Pressure Waveform at 0.27 and Hodgkin-Huxley at 0.32 (low-frequency cardiac / bursty neural coupling). The uniform-on-triangle baseline is ~0.50, where Sine, White Noise, AES, and Pi Digits cluster — they have no preferred coupling-frequency location. F-by-domain is low (1.77) because the axis cuts across domains rather than along them: chaos has the widest within-domain std (0.077) in the atlas.
| Source | Domain | Value |
|---|---|---|
| Weierstrass | exotic | 0.9880 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.9786 |
| 5-Torus Quasiperiodic | chaos | 0.9748 |
| ··· | ||
| Sine Wave | waveform | 0.0000 |
| Thue-Morse | exotic | 0.0000 |
| 2-Torus Quasiperiodic | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 0.8113 |
| Logistic r=3.74 (Period-5 Window) | chaos | 0.7160 |
| Logistic r=3.9 (Near-Full Chaos) | chaos | 0.6752 |
| ··· | ||
| Critical Transition (Fold) | chaos | 0.1230 |
| Blood Pressure Waveform | medical | 0.1641 |
| Lorenz-96 N=4 F=16 | chaos | 0.2736 |
| Source | Domain | Value |
|---|---|---|
| 5-Torus Quasiperiodic | chaos | 1.0000 |
| Weierstrass | exotic | 0.9721 |
| 4-Torus Quasiperiodic | chaos | 0.9639 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0849 |
| Critical Transition (Fold) | chaos | 0.1213 |
| Euler Totient Ratio | number_theory | 0.1706 |
Bispectrum is the textbook complement to the power spectrum and a direct supplement to Higher-Order Statistics' peak-bicoherence channel. HOS:bicoherence_max captures the intensity of the strongest phase-coupled triplet; Bispectrum's three metrics capture the shape (bicoherence_concentration), locality (off_diagonal_ratio), and frequency location (coupling_frequency_centroid) of the full coupling distribution. The geometry stays atlas-orthogonal post-rebuild (2026-05-23): max |r| is 0.514 (bicoherence_concentration anti-correlates with 2-adic:valuation_spectra); the other two metrics stay at max |r| ≤ 0.36. F-stat ranks for the three metrics (3.32 / 1.30 / 0.91) understate utility — these are cross-domain discriminators (the axes cut across the domain dimension rather than along it), so F-by-domain compresses real signal; the coherence audit names coupling_frequency_centroid as a coherent-novel core member firing on phase-coupled oscillators (Lorenz / Van der Pol / Hodgkin-Huxley). Two earlier candidate metrics — peak_bicoherence (r = +0.752 with HOS:bicoherence_max) and phase_coupled_fraction (r = +0.898 intra, +0.757 cross) — were dropped at the 2026-05-17 redundancy bar; a coupling_anisotropy candidate (λ-ratio of the b² covariance) was dropped at the 2026-05-18 pilot for r = -0.77 with off_diagonal_ratio and r = -0.73 with HOS:bicoherence_max.