The character of 2x2 matrix dynamics — whether the accumulated transformation rotates, shears, or stretches.
Converts each byte triple to an SL(2,R) matrix via the KAK decomposition: a rotation, a hyperbolic boost, and a second rotation. These three parameters span all three conjugacy classes of 2x2 matrices with determinant 1. The geometry classifies each matrix by its trace: elliptic (|trace| < 2, rotation-dominated), parabolic (|trace| = 2, shear), or hyperbolic (|trace| > 2, exponential stretch). A running product of matrices accumulates the signal's overall dynamical character.
Fraction of matrices with |trace| > 2. L-System Dragon, pulse-width modulation, and Morse code all score 1.0 (every matrix is hyperbolic — their binary structure pushes the boost parameter to extreme values). Earthquake P-wave scores 0.014 (almost entirely elliptic — the smooth waveform creates small boosts). This separates signals with discontinuous jumps (hyperbolic) from smooth oscillations (elliptic).
Average rate of exponential growth in the running matrix product, computed over blocks of 50 matrices. Rainfall (1.98) and Collatz gap lengths (1.96) score highest — their bursty or irregular dynamics create matrices whose product grows rapidly. Earthquake P-wave (0.020) is the lowest non-degenerate value: its smooth oscillation produces near-identity matrices that barely grow. This is a genuine Lyapunov exponent of the matrix cocycle, not an ad-hoc estimate.
Average trace of all matrices. L-System Dragon, Morse code, and Rule 110 all score 7.52 (far into the hyperbolic regime). DNA sequences score around -2.6 (elliptic regime with negative trace — the base pair frequencies create matrices with trace near the elliptic boundary). Logistic period-3 scores -2.77 (deepest into the negative-trace elliptic region). The sign of mean_trace separates two flavors of rotational dynamics: positive trace means the rotations partially cancel; negative means they compound.
Fraction of matrices near the elliptic/hyperbolic boundary (|trace²-4| <= 0.2). Ambient microseism (0.40) and bearing inner fault (0.39) score highest — their narrowband oscillations produce matrices that hover near the boundary. This is the rarest conjugacy class: most data is either clearly elliptic or clearly hyperbolic, with few matrices landing in the thin parabolic strip. High parabolic fraction signals a critical or transitional dynamical regime.
Average spectral radius (largest singular value) of the individual SL(2,R) matrices. Pulse-Width Mod, Morse Code, and Square Wave all hit 7.39 (maximum — binary jumps create large boosts). BTC Returns (1.02) is near the identity matrix. This separates signals that create "gentle" matrices (smooth oscillations, radius near 1) from those that create "violent" matrices (binary jumps, radius >> 1).
Lag-1 autocorrelation of the matrix trace sequence. Logistic period-2 (1.0) and fBm Persistent (1.0) score highest. Fibonacci Word scores 0.0 (traces change unpredictably). Trace autocorrelation captures persistence in the conjugacy class — does the signal stay in the elliptic/hyperbolic regime, or does it switch rapidly?
Standard deviation of the block-wise Lyapunov exponent estimates. BTC Close (0.61) and Van der Pol (0.59) score highest — their Lyapunov exponents vary dramatically across blocks, indicating intermittent dynamics. Constants and logistic period-2 score 0.0 (perfectly uniform growth rate). High std with moderate mean Lyapunov indicates alternating laminar and turbulent phases.
| Source | Origin | Value |
|---|---|---|
| Critical Transition (Fold) | iterated-maps | 0.7121 |
| Takagi Function | special-functions | 0.7111 |
| Spectral Form Factor | random-matrix-quantum | 0.6941 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 0.0000 |
| Logistic r=3.74 (Period-5 Window) | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Aubry-André Critical | random-matrix-quantum | 1.9883 |
| Rainfall (ORD Hourly) | atmospheric | 1.9817 |
| Mian-Chowla | number-theory | 1.9815 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Stable Noise | stochastic-process | 0.0079 |
| Student-t Noise | stochastic-process | 0.0147 |
| Source | Origin | Value |
|---|---|---|
| Periodic Word | symbolic-dynamics | 1.0000 |
| Gap-Word β-Shift | symbolic-dynamics | 1.0000 |
| Fibonacci Word | symbolic-dynamics | 1.0000 |
| ··· | ||
| MFPT Inner Unloaded | machine-vibration | 0.0064 |
| Stable Noise | stochastic-process | 0.0065 |
| Intermittency Type-III | iterated-maps | 0.0071 |
| Source | Origin | Value |
|---|---|---|
| Pell Word | symbolic-dynamics | 7.3891 |
| Penrose Substitution | symbolic-dynamics | 7.3891 |
| Golden-Mean β-Shift | symbolic-dynamics | 7.3891 |
| ··· | ||
| MFPT Inner Unloaded | machine-vibration | 1.0055 |
| Intermittency Type-III | iterated-maps | 1.0125 |
| Intermittency Type-II | iterated-maps | 1.0172 |
| Source | Origin | Value |
|---|---|---|
| Thue-Morse | symbolic-dynamics | 7.5244 |
| Collatz Cycle Word | number-theory | 7.5244 |
| Fibonacci Word | symbolic-dynamics | 7.5244 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | iterated-maps | -5.1102 |
| Euler Totient Ratio | number-theory | -1.9706 |
| Random Telegraph | stochastic-process | -1.9317 |
| Source | Origin | Value |
|---|---|---|
| Stable Noise | stochastic-process | 0.9148 |
| Student-t Noise | stochastic-process | 0.7484 |
| Intermittency Type-II | iterated-maps | 0.7010 |
| ··· | ||
| Thue-Morse | symbolic-dynamics | 0.0000 |
| Periodic Word | symbolic-dynamics | 0.0000 |
| Gap-Word β-Shift | symbolic-dynamics | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Takagi Function | special-functions | 0.9998 |
| OTOC Growth | random-matrix-quantum | 0.9998 |
| Spectral Form Factor | random-matrix-quantum | 0.9993 |
| ··· | ||
| Thue-Morse | symbolic-dynamics | 0.0000 |
| Fibonacci Word | symbolic-dynamics | 0.0000 |
| Collatz Cycle Word | number-theory | 0.0000 |
SL(2,R) is the most algebraically rich Thurston geometry in the framework. Its conjugacy class decomposition (elliptic/parabolic/hyperbolic) provides a three-way classification that no other metric replicates. In the symmetry view, hyperbolic_fraction is the primary separator: it cleanly divides binary/symbolic sources (fraction near 1.0) from smooth continuous signals (fraction near 0). The lyapunov_exponent adds a growth-rate axis, and mean_trace adds a polarity axis. Together they place each source in a 3D space where binary chaos, smooth oscillation, and critical behavior occupy distinct regions.