How much structure the signal has in its divisibility pattern.
Samples random pairs of byte values and measures the distance between them using the 2-adic metric: two numbers are "close" if their difference is divisible by a high power of 2. The number 48 and the number 80 differ by 32 = 2^5, so they are 2-adically very close (distance 2^-5) despite being far apart on the number line. This geometry detects hierarchical modular structure — patterns in which bits the signal's values share.
Shannon entropy of the distinct 2-adic distances. High entropy means the pairwise distances span many different powers of 2 — the signal explores multiple levels of the divisibility hierarchy. Devil's staircase scores highest (3.02): its plateaus create consecutive values whose differences hit every power of 2 as the staircase descends through its fractal levels. Classical MIDI (2.49) and ECG supraventricular (2.27) are also high — both have quantized amplitude levels that create diverse divisibility patterns. Rainfall scores lowest among nontrivial signals (0.66): its near-zero values produce differences that are almost always odd (2-adic distance 1.0), collapsing the entropy.
Average 2-adic distance across sampled pairs. VLF Radio Eclipse, Van der Pol, and Triangle Wave all score 0.668 (the maximum). Rainfall scores 0.09 — its small integer values have differences divisible by high powers of 2 more often than random data would. The expectation for uniform random bytes is near 2/3 (matching the ~0.668 maxima above); values significantly below indicate non-trivial divisibility structure.
How concentrated is the power spectrum of the 2-adic valuation sequence (the sequence of trailing-zero counts)? Logistic period-3 (1.0) and constants (1.0) have perfectly concentrated valuation spectra (one dominant frequency). RANDU (0.04) and glibc LCG (0.04) have the flattest — their divisibility patterns span all frequencies equally. Evolved via ShinkaEvolve.
How predictable is the next 2-adic valuation given the current one? Logistic period-2 (1.0) has perfectly predictable transitions. x86-64 Machine Code (0.29) has the least predictable — its byte differences have chaotic divisibility patterns. Evolved via ShinkaEvolve.
Markov predictability of the valuation sequence at multiple block sizes. Logistic period-2 (1.0) and constants (1.0) are perfectly predictable. Pi Digits (0.027) and AES (0.027) are nearly unpredictable — their divisibility sequences are essentially random at all scales. Evolved via ShinkaEvolve.
| Source | Origin | Value |
|---|---|---|
| Classical MIDI | algorithmic-bytes | 2.4595 |
| Copeland-Erdős | number-theory | 2.3325 |
| Windows PE x86-64 | algorithmic-bytes | 2.2683 |
| ··· | ||
| Mian-Chowla | number-theory | 0.0613 |
| Devil's Staircase | special-functions | 0.5058 |
| Rainfall (ORD Hourly) | atmospheric | 0.6625 |
| Source | Origin | Value |
|---|---|---|
| White Noise | stochastic-process | 0.6685 |
| VLF Radio (Eclipse) | geophysical | 0.6684 |
| Triangle Wave | signal-synthesis | 0.6683 |
| ··· | ||
| Mian-Chowla | number-theory | 0.0075 |
| Devil's Staircase | special-functions | 0.0603 |
| Rainfall (ORD Hourly) | atmospheric | 0.0895 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 1.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 1.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 1.0000 |
| ··· | ||
| ln(2) Digits | number-theory | 0.0267 |
| Pi Digits | number-theory | 0.0267 |
| XorShift32 | algorithmic-bytes | 0.0268 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 1.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 1.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.9015 |
| ··· | ||
| Tohoku Aftershock Intervals | geophysical | 0.0465 |
| Coupled Map Lattice | coupled-oscillators | 0.0465 |
| Rule 30 | symbolic-dynamics | 0.0465 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.5 (Period-4) | iterated-maps | 1.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 1.0000 |
| Square Wave | signal-synthesis | 0.9882 |
| ··· | ||
| x86-64 Machine Code | algorithmic-bytes | 0.2945 |
| Human Proteome | genomic | 0.3305 |
| Earthquake Depths | geophysical | 0.3408 |
The 2-adic geometry detects structure invisible to every other distributional metric: the pattern of trailing bits. Two signals with identical histograms and identical Torus coverage can have very different 2-adic profiles if one tends to produce differences divisible by 4 and the other doesn't. Devil's staircase topping distance_entropy is diagnostic: its Cantor-function structure creates a precise hierarchy of jump sizes that maps perfectly onto the 2-adic distance hierarchy. This makes 2-adic geometry the framework's most direct detector of fractal staircase dynamics.