How thoroughly the signal fills its available space.
Maps each pair of consecutive values to a point on a flat torus — a square where the top edge is glued to the bottom and the left edge to the right, so there are no boundaries. Bins the torus into a 16x16 grid and asks: how many cells are occupied, how uniform is the distribution, and how close are neighboring points?
What fraction of the 256 torus cells contain at least one point? Arnold cat map, MINSTD, and XorShift32 all hit 1.0 (perfect coverage — every cell occupied). Constant signals score 0.0 (one cell forever). Logistic period-2 scores 0.004 (two values produce one point, landing in a single cell). Coverage is the simplest possible test of distributional richness: does the data ever go everywhere?
Shannon entropy of the torus cell histogram. Wichmann-Hill, XorShift32, and AES encrypted cluster at 7.98 (near the maximum of 8.0 bits — uniform occupancy across all 256 cells). Constants and logistic period-2 score 0.0. High coverage with low entropy means the data visits many cells but spends most of its time in a few — the signature of a non-uniform but space-filling process.
Average distance from each point to its nearest neighbor, measured with wrap-around (torus metric, not Euclidean). Pi digits (0.023), Wichmann-Hill (0.022), and white noise (0.022) cluster together — well-spread points on the torus. Logistic period-4 scores 0.0 (all points land in the same tiny region). This catches clumping that coverage misses: a signal might occupy many cells but pile points into adjacent ones.
| Source | Domain | Value |
|---|---|---|
| Arnold Cat Map | chaos | 1.0000 |
| MINSTD (Park-Miller) | binary | 1.0000 |
| XorShift32 | binary | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0039 |
| Source | Domain | Value |
|---|---|---|
| Wichmann-Hill | binary | 7.9777 |
| XorShift32 | binary | 7.9775 |
| AES Encrypted | binary | 7.9773 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | -0.0000 |
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Pi Digits | number_theory | 0.0226 |
| Wichmann-Hill | binary | 0.0225 |
| White Noise | noise | 0.0224 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
Torus coverage is the framework's simplest binary test — it immediately separates data that explores its value space from data that doesn't. In the atlas, coverage and entropy together form the "distributional baseline" that all other distributional geometries refine. PRNGs cluster at the (1.0, 7.98) corner; periodic orbits cluster at the origin. The interesting signals are the ones in between: chaotic maps with 0.8-0.95 coverage, natural signals with high coverage but moderate entropy.