Scale-invariant relationships between byte triples.
Lifts each triple of consecutive bytes to projective 2-space — the space of lines through the origin in 3D, where (x,y,z) and (kx,ky,kz) represent the same point. Distances are measured by the Fubini-Study metric (arccos of the absolute inner product). The geometry then samples cross-ratios — the fundamental projective invariant of four collinear points — and checks for near-collinearity in ℙ².
Standard deviation of the cross-ratio across sampled quadruples of points. Logistic edge-of-chaos (40.3) has the widest cross-ratio distribution: its dynamics create point configurations spanning the full range of the Mobius invariant. Euler totient (38.3) and critical circle map (34.4) are close behind. Logistic period-3 scores 0.0 (degenerate — too few distinct projective points). High cross_ratio_std means the data's projective geometry is wild and varied; low means it's constrained to a narrow family of configurations.
Fraction of sampled triples that are nearly collinear in ℙ² (parallelepiped volume < 0.1). Solar wind speed, speech "nine", and tidal gauge all score 1.0 (every triple is nearly collinear — their smooth waveforms keep consecutive triples in a low-dimensional projective subspace). Poisson spacings score 0.30 (most triples span the full ℙ²). High collinearity means the signal's projective image is nearly one-dimensional — it traces a curve rather than filling the plane.
Average Fubini-Study distance between nearby projective points. Collatz parity (0.937) scores highest: its binary sequence creates maximally separated projective points. L-System Dragon (0.914) is close behind. Logistic period-3 scores 0.0 (all triples project to the same point). High mean distance means successive triples jump across projective space; low means they cluster.
Standard deviation of Fubini-Study distances. Collatz Parity (0.60) and Thue-Morse (0.53) score highest — their binary dynamics create a bimodal distance distribution (some triples are close, others are maximally far). Constants and logistic period-3 score 0.0. High std with moderate mean indicates heterogeneous projective structure.
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 1.0000 |
| Mian-Chowla | number_theory | 1.0000 |
| ··· | ||
| Penrose Substitution | exotic | 0.9289 |
| Pell Word | exotic | 0.9297 |
| Kolakoski Sequence | exotic | 0.9324 |
| Source | Domain | Value |
|---|---|---|
| Logistic Edge-of-Chaos | chaos | 45.4487 |
| Sine Map (Feigenbaum) | chaos | 44.7827 |
| Quartic Map (Feigenbaum) | chaos | 41.0959 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Mian-Chowla | number_theory | 0.0000 |
| Square Wave | waveform | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 0.3073 |
| Penrose Substitution | exotic | 0.2489 |
| Pell Word | exotic | 0.2466 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Spectral Form Factor | quantum | 0.0000 |
| Exponential Chirp | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Penrose Substitution | exotic | 0.4113 |
| Pell Word | exotic | 0.3755 |
| L-System (Dragon Curve) | exotic | 0.3645 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Spectral Form Factor | quantum | 0.0000 |
| OTOC Growth | quantum | 0.0000 |
Projective ℙ² is the framework's scale-invariance detector. The cross-ratio is invariant under all projective transformations (scaling, rotation, shear, perspective) — so cross_ratio_std measures how much the data's structure varies beyond what any single projective transformation could explain. In the symmetry view, the collinearity metric separates smooth waveforms (collinearity near 1.0, data traces a projective curve) from complex or binary sources (low collinearity, data fills ℙ²). This is complementary to the Fubini-Study distance metrics in Spherical S², which operate on the same underlying point-on-sphere representation but measure different invariants.