How many holes and connected components survive as you zoom out -- the topological skeleton that persists across scales.
Delay-embeds the signal into 2D, deduplicates the point cloud, then builds a Vietoris-Rips filtration: imagine inflating a ball around every point and tracking when components merge (H0) and when loops form and fill in (H1). Features that persist across a wide range of scales are topologically robust. Features that blink in and out are noise. The distribution of lifetimes, their entropy, and the total persistence mass characterize the signal's topological complexity.
Count of H0 features with lifetime above 0.1. Dice Rolls (34.1) produces the most significant components -- its random jumps create many well-separated clusters that merge at different scales. Intermittent Silence (24.9) and Collatz Stopping Times (20.8) follow. Rössler (1.0) has just one significant component -- its attractor is a single connected blob in the delay embedding.
Total lifetime mass of all H1 (loop) features. Dice Rolls (1.83) dominates, with loops forming at many scales as its scattered point cloud creates ring-like structures during filtration. Collatz Gap Lengths (0.56) and Codon Usage (0.54) have modest loop structure. Logistic Chaos and Tent Map score 0.0 -- their dense, space-filling attractors leave no room for persistent loops.
Lifetime of the single most persistent loop. Rule 30 (0.414), Symbolic Lorenz (0.414), and Morse Code (0.414) share the top score -- their binary-valued signals create specific point-cloud geometries in delay embedding where one dominant loop persists across scales. This metric identifies signals with a single robust circular or toroidal structure.
Shannon entropy of the lifetime distribution. High entropy means the topological features have diverse lifetimes (no single scale dominates). Low entropy means one feature dominates all others. This separates multi-scale signals from those with a single characteristic scale.
Lifetime of the single most persistent H0 feature (connected component). Dice Rolls, L-System Dragon, and Rule 110 all hit 1.414 (= sqrt(2), the maximum distance in the unit square delay embedding). Henon Near-Crisis (0.75) has the shortest maximum lifetime — its attractor is compact, so the last two components merge early. This measures the geometric diameter of the delay-embedded point cloud.
Sum of all H0 feature lifetimes. Dice Rolls (8.04) has the highest total — many well-separated clusters, each persisting for a long time. Constants and Quantum Walk score 1.0 (a single component with unit lifetime). This is an integral measure of topological complexity: it weights both the number and duration of connected components.
| Source | Domain | Value |
|---|---|---|
| Dice Rolls | exotic | 1.7809 |
| Partition Function | number_theory | 1.5740 |
| Moebius Function | number_theory | 0.8284 |
| ··· | ||
| Logistic Chaos | chaos | 0.0000 |
| Tent Map | chaos | 0.0000 |
| Sawtooth Wave | waveform | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Morse Code | waveform | 0.4142 |
| Liouville Function | number_theory | 0.4142 |
| L-System (Dragon Curve) | exotic | 0.4142 |
| ··· | ||
| Logistic Chaos | chaos | 0.0000 |
| Tent Map | chaos | 0.0000 |
| Sawtooth Wave | waveform | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Pell Word | exotic | 1.4142 |
| Morse Code | waveform | 1.4142 |
| Dice Rolls | exotic | 1.4142 |
| ··· | ||
| LIGO Hanford | astro | 0.0000 |
| LIGO Livingston | astro | 0.0000 |
| Noisy Period-2 | chaos | 0.5630 |
| Source | Domain | Value |
|---|---|---|
| Dice Rolls | exotic | 33.7000 |
| A5 Artin a_p | number_theory | 25.4000 |
| Partition Function | number_theory | 24.6000 |
| ··· | ||
| LIGO Hanford | astro | 0.0000 |
| LIGO Livingston | astro | 0.0000 |
| Sine Wave | waveform | 1.0000 |
| Source | Domain | Value |
|---|---|---|
| Poisson Counts | exotic | 5.1722 |
| Human Proteome | bio | 5.0500 |
| Catalan G Digits | number_theory | 4.9885 |
| ··· | ||
| Mian-Chowla | number_theory | 0.0000 |
| OTOC Growth | quantum | 0.0000 |
| PID Controller | exotic | 0.2738 |
| Source | Domain | Value |
|---|---|---|
| Dice Rolls | exotic | 7.9542 |
| Partition Function | number_theory | 7.3142 |
| Markov Chain (10-state) | exotic | 6.1449 |
| ··· | ||
| LIGO Hanford | astro | 0.0000 |
| LIGO Livingston | astro | 0.0000 |
| OTOC Growth | quantum | 1.0000 |
Persistent Homology is the atlas's most direct topological detector. The n_significant_features metric uniquely identifies signals with complex multi-component structure -- Dice Rolls, Intermittent Silence, and Collatz Stopping Times form a cluster that no other geometry highlights. The H1 metrics separate signals with genuine loop topology from those that are simply connected, a distinction only available through filtration-based analysis.