How your signal behaves when fed into the Mandelbrot iteration as a starting condition -- does it escape, get trapped, or linger at the boundary?
Takes consecutive byte pairs, maps them to the complex plane as the parameter c, and iterates z -> z^2 + c starting from z = 0. For each c, it records how many iterations before the orbit escapes (or whether it stays trapped). The distribution of escape times, the fraction of trapped orbits, and the entropy of the escape-time histogram characterize how the signal's byte-pair distribution intersects the Mandelbrot set's boundary structure.
Shannon entropy of the escape-time histogram. Human Proteome (4.21) has the richest distribution -- its byte pairs map to c-values that scatter across the boundary region, producing many different escape speeds. Critical Circle Map (3.27) and Classical MIDI (3.23) are close behind. Fibonacci Word and Logistic Period-2 score near zero: their byte pairs map to a narrow region where everything escapes at the same speed or not at all.
How spread out are the escape times? Devil's Staircase (953.8) and ETH/BTC Ratio (951.8) have enormous variance -- their byte pairs hit both deep interior and far exterior of the Mandelbrot set, creating a bimodal distribution. Weierstrass (910.1) follows. Constants and Fibonacci Word score 0.0 -- all their byte pairs land in the same escape-time bin.
Fraction of byte pairs whose orbits never escape (trapped inside the Mandelbrot set). Earthquake P-wave (0.97) and Bearing Inner (0.97) trap almost everything -- their byte-pair distributions map overwhelmingly to the Mandelbrot interior. Divisor Count (0.0) and Prime Gaps (0.0) trap nothing -- their byte pairs all map to the exterior, where orbits escape quickly.
Standard deviation of the external-potential field log|z_N|/2^N over non-escaping points, measuring how jagged the Mandelbrot potential is across the byte-pair cloud. Logistic r=3.5 Period-4 (4.00) and Edge-of-Chaos (3.54) score highest — their tightly-clustered pairs sample the filament-rich potential near the set's boundary. Circle Map Quasiperiodic (3.11) and Fibonacci Quasicrystal (2.58) follow. Smooth physical signals saturate at the low end: PID Controller (0.008), ETH/BTC (0.008), Devil's Staircase (0.008). Correlates +0.88 with Lorentzian:spacelike_fraction — both proxy the "acausal jump rate" of the byte sequence. Chaos (1.96) and binary (1.95) domain means lead; climate (0.28) and astro (0.36) trail.
| Source | Origin | Value |
|---|---|---|
| Noisy Period-2 | iterated-maps | 3.4422 |
| GUE Spacings | random-matrix-quantum | 3.4236 |
| GOE Spacings | random-matrix-quantum | 3.3505 |
| ··· | ||
| Thue-Morse | symbolic-dynamics | -0.0000 |
| Pell Word | symbolic-dynamics | -0.0000 |
| Collatz Cycle Word | number-theory | -0.0000 |
| Source | Origin | Value |
|---|---|---|
| Spectral Form Factor | random-matrix-quantum | 925.5001 |
| Halvorsen Attractor | continuous-flows | 914.4082 |
| Ruelle-Takens Cascade | continuous-flows | 911.8885 |
| ··· | ||
| Pell Word | symbolic-dynamics | 0.0000 |
| Penrose Substitution | symbolic-dynamics | 0.0000 |
| Gap-Word β-Shift | symbolic-dynamics | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Stable Noise | stochastic-process | 0.9980 |
| Student-t Noise | stochastic-process | 0.9937 |
| MFPT Inner Loaded | machine-vibration | 0.9816 |
| ··· | ||
| DNA Thermus | genomic | 0.0000 |
| DNA Centromere | genomic | 0.0000 |
| DNA Plasmodium | genomic | 0.0000 |
Fractal (Mandelbrot) is most discriminative for signals whose byte-pair distributions straddle the Mandelbrot set boundary. The escape_entropy metric uniquely identifies Human Proteome at the top -- its amino-acid-derived byte distribution intersects the cardioid boundary in a way no other natural signal does. The interior_fraction metric cleanly separates physical sensor data (seismic, bearing vibration -- high interior fraction) from number-theoretic sequences (prime gaps, divisor counts -- zero interior), reflecting how their value distributions cluster in the complex plane.