Does the signal's local structure match the Hat aperiodic monotile, and does it show chirality?
Maps the data onto a hexagonal grid path and cross-correlates the turn sequence with the projected boundary of the Hat tile (Smith et al., 2023). The Hat is the first known single tile that forces aperiodic tiling — it cannot tile the plane periodically. Critically, the Hat and its mirror image are distinct (chirality), which this geometry tests separately.
Asymmetry between left and right turns on the hex path. Wigner Semicircle (0.526) is the most left-turning signal in the atlas: its symmetric distribution creates a systematic directional bias when mapped to hex directions. Logistic Edge-of-Chaos (-0.406) turns right. Circle Map QP (-0.204) also turns right. The sign matters: positive = left bias (hex turns 1,2 where sin > 0), negative = right bias (hex turns 4,5 where sin < 0). Signals near zero (most noise, most chaos) have no directional preference. This is a genuine chirality test — it detects handedness in the data's local structure.
Fraction of hex-path windows with >0.9 correlation to the Hat's boundary kernel, scaled by kernel length. Scores are universally low across the atlas: Henon-Heiles (0.019), Rossler (0.017), and Sine Wave (0.015) lead. The Hat boundary has 90-degree angles that the hexagonal grid cannot represent exactly (hex steps are 60-degree), so this metric is an approximate motif detector. The low scores indicate no source in the atlas genuinely resembles the Hat boundary in its turn structure.
| Source | Domain | Value |
|---|---|---|
| Wigner Semicircle | quantum | 0.5262 |
| Sawtooth Wave | waveform | 0.4430 |
| Phyllotaxis | bio | 0.2044 |
| ··· | ||
| Logistic Edge-of-Chaos | chaos | -0.4059 |
| Logistic r=3.83 (Period-3 Window) | chaos | -0.2887 |
| Circle Map Quasiperiodic | chaos | -0.2044 |
| Source | Domain | Value |
|---|---|---|
| Hénon-Heiles | chaos | 0.0191 |
| Rossler Attractor | chaos | 0.0168 |
| Sine Wave | waveform | 0.0152 |
| ··· | ||
| Collatz Trajectory | number_theory | 0.0000 |
| Prime Gaps | number_theory | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
Einstein Hat is the newest geometry in the quasicrystal lens, added to detect the 2023 discovery of aperiodic monotiles. Its hat_boundary_match metric is limited by the hex-grid approximation (the true Hat lives on a 12-direction kite grid), but chirality provides genuine signal: it detects left/right asymmetry that no other geometry in the framework measures. In investigations, chirality discriminated between ECG lead orientations — the heart's electrical axis creates a physical handedness that the hex-path chirality captures. The metric's range from -0.406 to +0.526 in the atlas spans zero symmetrically, confirming it's not biased.