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.
| Source | Domain | Value |
|---|---|---|
| Devil's Staircase | exotic | 3.0211 |
| Classical MIDI | binary | 2.4903 |
| ECG Supraventr. | medical | 2.2679 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Rainfall (ORD Hourly) | climate | 0.6625 |
| Source | Domain | Value |
|---|---|---|
| VLF Radio (Eclipse) | geophysics | 0.6682 |
| Van der Pol Oscillator | exotic | 0.6682 |
| Triangle Wave | waveform | 0.6681 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Rainfall (ORD Hourly) | climate | 0.0895 |
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.