Whether the data was generated by a substitution rule — a process that replaces symbols with fixed patterns at every scale.
A substitution rule maps each symbol to a word: Fibonacci does a→ab, b→a; Thue-Morse does a→ab, b→ba. Iterating the rule produces an infinite aperiodic sequence whose statistical properties repeat at geometrically spaced scales (powers of the inflation factor λ). This geometry binarizes the data and tests five signatures that substitution sequences must satisfy and random or periodic sequences cannot.
How linear is the growth of distinct subwords? For a substitution sequence, the number of distinct n-grams p(n) grows as c·n + d (exactly linear). For random sequences, p(n) grows exponentially (2ⁿ). For periodic sequences, p(n) flatlines. Fibonacci word and Thue-Morse both score near 1.0 (textbook linear). White noise scores 0.76 (exponential growth, poor linear fit). Logistic period-5 scores 0.46 — its near-periodicity confuses the linearity test.
Topological entropy: does the subword count grow? Zero for substitution and periodic, positive for random and chaos. White noise and PRNG outputs score ~1.0 (exponential growth = maximal entropy). Fibonacci word scores 0.19 (barely above zero — its linear complexity growth produces a small but nonzero growth ratio). Periodic orbits score exactly 0.0.
How evenly are the symbols distributed? Measures max|D(n)|/√N, where D(n) is the deviation of cumulative symbol counts from expected. Substitution sequences have bounded discrepancy (D*/√N → 0). Random sequences have D*/√N ≈ 0.5 (random walk). Thue-Morse (0.006) and Fibonacci word (0.006) have the lowest nonzero discrepancy in the atlas — their symbols are distributed with almost crystalline uniformity. Devil's staircase (28.1) has the highest: its long constant plateaus create massive cumulative imbalance.
How regular are the gaps between repeated subwords? Substitution sequences have few distinct return times because the hierarchical tiling constrains where each pattern can appear. Logistic period-2 and edge-of-chaos score 1.0 (perfectly regular returns). Fibonacci word scores 0.82. fBm scores 0.08 (widely scattered, irregular returns).
Are autocorrelation peaks at geometric (λ^k) rather than arithmetic (nT) spacings? This is the substitution fingerprint: Fibonacci has ACF peaks at Fibonacci numbers (φ^k spacings), Thue-Morse at powers of 2. Quantum walk scores highest (0.91) — its interference pattern creates geometric-ratio ACF peaks through a completely different mechanism. Fibonacci word (0.88), phyllotaxis (0.89), and circle map quasiperiodic (0.89) cluster together. White noise and logistic chaos score 0.0 (no ACF peaks at all).
| Source | Origin | Value |
|---|---|---|
| Padovan Word | symbolic-dynamics | 0.9618 |
| Riemann Zeta Zeros | number-theory | 0.9071 |
| Circle Map Quasiperiodic | iterated-maps | 0.8888 |
| ··· | ||
| Logistic Chaos | iterated-maps | 0.0000 |
| Tent Map | iterated-maps | 0.0000 |
| Brownian Walk | stochastic-process | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Sawtooth Wave | signal-synthesis | 1.0000 |
| Sine Wave | signal-synthesis | 1.0000 |
| Period-Doubled Wave | signal-synthesis | 1.0000 |
| ··· | ||
| Logistic r=3.74 (Period-5 Window) | iterated-maps | 0.6166 |
| Nikkei Returns | financial | 0.6382 |
| MINSTD (Park-Miller) | algorithmic-bytes | 0.6382 |
| Source | Origin | Value |
|---|---|---|
| Critical Transition (Fold) | iterated-maps | 30.1687 |
| Hilbert Walk | symbolic-dynamics | 27.2802 |
| Levy Flight | stochastic-process | 25.3316 |
| ··· | ||
| Mian-Chowla | number-theory | 0.0000 |
| Forest Fire | self-organized-criticality | 0.0000 |
| Logistic r=3.65 (Banded Chaos) | iterated-maps | 0.0039 |
| Source | Origin | Value |
|---|---|---|
| Nikkei Returns | financial | 1.0000 |
| MT19937 (Mersenne Twister) | algorithmic-bytes | 1.0000 |
| ln(2) Digits | number-theory | 1.0000 |
| ··· | ||
| Sine Map (Feigenbaum) | iterated-maps | 0.0000 |
| Logistic r=3.65 (Banded Chaos) | iterated-maps | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
Inflation is the framework's substitution-rule detector. The combination of linear complexity + near-zero discrepancy + geometric ACF peaks is essentially unique to substitution sequences. In the ordinal view, it separates quasicrystalline sources (Fibonacci, Thue-Morse, L-System) from both periodic and random, which the other ordinal geometries struggle to do — Ordinal Partition sees Fibonacci as "moderately constrained" without distinguishing it from other aperiodic sequences. Inflation's acf_geometric metric also catches quantum walk and ocean/wave signals, suggesting that geometric self-similarity in autocorrelation structure is a broader phenomenon than pure substitution dynamics.