The same vocabulary structure as the 8-bit version, but at the resolution of byte pairs.
Packs each pair of consecutive bytes into a 16-bit symbol (65,536 possible values) using a sliding window, then performs the same Zipf-Mandelbrot analysis. The larger alphabet dramatically increases sensitivity to sequential structure: two signals with identical 8-bit histograms can have completely different 16-bit Zipf profiles if their byte-to-byte transitions differ.
Zipf exponent at 16-bit resolution. Poker hands (2.74) and Sensor event streams (2.56) still top the ranking, but with lower exponents than the 8-bit version — the larger vocabulary dilutes the concentration. Collatz gap lengths drops from 3.62 (8-bit) to 2.39 (16-bit), indicating its dominance is mainly a single-byte phenomenon. Signals where alpha is similar at both resolutions have concentration that extends to pair structure.
Fit quality at 16-bit resolution. Collatz gap lengths (0.99), Poker hands (0.98), and Continued fractions (0.98) score highest — their frequency decay is genuinely power-law even at pair resolution. This is more discriminating than the 8-bit version: many signals that fit Zipf at byte level fall apart at the pair level because their byte transitions are not power-law distributed.
The same plateau parameter. Solar wind IMF and Solar wind speed still score 10.0 — their distributional plateau persists at pair resolution. Devil's staircase drops from nonzero (8-bit) to 0.0 (16-bit): its plateau structure is a single-byte phenomenon that doesn't extend to pairs. Tidal gauge (10.0) joins the top at 16-bit — its slow tidal oscillations create repeated byte pairs that flatten the top of the frequency curve.
Frequency inequality at pair resolution. Rainfall (0.97), Forest fire (0.95), and Neural net pruned (0.91) remain the most unequal. The pruned network drops from 0.94 (8-bit) to 0.91 (16-bit) — its zero-dominated weight distribution is slightly less extreme when viewed as pairs. Logistic period-3 scores 0.0 (its 3 repeated values produce 3 repeated pairs, all equally common).
Fraction of 16-bit symbols appearing exactly once. Champernowne (0.95) tops the ranking — its digit-concatenation construction creates an enormous number of unique byte pairs. Middle-Square (0.93) and Intermittent silence (0.90) follow. The hapax explosion at 16-bit resolution is expected: with 65,536 possible symbols and finite data, many pairs will be seen only once. The signals at the bottom (Sawtooth wave at 0.0) are those whose pair vocabulary is so constrained that every pair repeats.
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 4.0000 |
| Constant 0xFF | noise | 4.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 4.0000 |
| ··· | ||
| Wichmann-Hill | binary | 0.0104 |
| XorShift32 | binary | 0.0105 |
| White Noise | noise | 0.0105 |
| Source | Domain | Value |
|---|---|---|
| Nikkei Returns | financial | 1.1226 |
| NASDAQ Returns | financial | 1.1039 |
| NYSE Returns | financial | 1.0961 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 0.9724 |
| Forest Fire | exotic | 0.9512 |
| Neural Net (Pruned 90%) | binary | 0.9092 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Champernowne | number_theory | 0.9435 |
| Middle-Square (von Neumann) | binary | 0.9345 |
| Intermittent Silence | exotic | 0.9033 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Sawtooth Wave | waveform | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Spike Train | exotic | 10.0000 |
| Accel Jog | motion | 10.0000 |
| Accel Walk | motion | 10.0000 |
| ··· | ||
| El Centro 1940 | geophysics | 0.0000 |
| Rainfall (ORD Hourly) | climate | 0.0000 |
| Sandpile | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Poker Hands | exotic | 2.7400 |
| Sensor Event Stream | exotic | 2.5753 |
| Categorical Sensor | exotic | 2.5011 |
| ··· | ||
| De Bruijn Sequence | number_theory | 0.0001 |
| Gray Code Counter | exotic | 0.0001 |
| Logistic r=3.74 (Period-5 Window) | chaos | 0.0002 |
| Source | Domain | Value |
|---|---|---|
| Collatz Gap Lengths | number_theory | 0.9911 |
| Poker Hands | exotic | 0.9796 |
| Continued Fractions | number_theory | 0.9769 |
| ··· | ||
| De Bruijn Sequence | number_theory | 0.0053 |
| Gray Code Counter | exotic | 0.0058 |
| Middle-Square (von Neumann) | binary | 0.3477 |
The 16-bit version's primary value over the 8-bit is detecting transition structure. A signal with a flat byte histogram (high 8-bit entropy, low 8-bit alpha) can still have extreme 16-bit concentration if certain byte transitions dominate. Comparing alpha and hapax between the two resolutions reveals how much of the signal's vocabulary structure is local (single-byte) versus sequential (pair). In the atlas, Champernowne's jump from low hapax at 8-bit (it uses all digits) to 0.95 hapax at 16-bit (its digit pairs are almost all unique) is the clearest example: its structure is entirely in the sequence of bytes, not in their individual distribution.