How a signal's trajectory organizes itself relative to the binary icosahedral group 2I — the 120 unit quaternions that tile the 3-sphere — and whether that organization reveals topological-sector structure, spinorial asymmetry, or phi-vertex preference.
Takes a 1D signal, delay-embeds it into ℝ⁴, projects each point onto S³, and snaps every point to the nearest of 120 precomputed "2I vertices" that form the 600-cell (the regular 4-polytope whose vertices are the unit quaternions of 2I). The resulting discrete trajectory can be decomposed in several geometrically motivated ways: into three topological sectors (inspired by the three isolated flat SU(2) connections on the Poincaré homology sphere S³/2I), into Hopf fibers (S¹ fibers over S²), into the phi/non-phi partition (96 vertices carry golden-ratio coordinates, 24 don't), and into a comparison between the full 2I and its antipodal quotient I (the icosahedral group of order 60). The metrics sit in different corners of this decomposition. The whole geometry was built from the topological ansatz in Mode Identity Theory (S¹ boundary of a Möbius strip embedded in S³ with the 2I quotient structure).
Normalized Shannon entropy of transitions between the three topological sectors (partitioned by quaternion real part relative to sin(π/5) thresholds). Low = predictable sector transitions, high = random. Accel Jog (0.64), Accel Stairs (0.62), ARMA(2,1) (0.62), Ikeda Map (0.61), and Ambient Microseism (0.61) score highest — high-complexity signals whose sector traversal looks Markov-random. The only Möbius-S³ metric confirmed to track intra-regime chaos intensity on multiple systems (ρ=+0.578 logistic, +0.801 Hénon with Lyapunov exponent), so it functions as a genuine chaos fingerprint, not just a complexity proxy.
Autocorrelation of phase increments along the S¹ fiber direction of the Hopf fibration. High = smooth fiber-tracking (coherent phase rotation), low = random hopping between fibers. Bearing signals score highest as a domain — their rotational structure produces coherent Hopf-fiber tracking in the delay embedding. Correlates strongly with Laplacian:laplacian_spectral_ratio (+0.805).
Visit-fraction on the 96 phi-coordinate vertices of the 600-cell minus the geometric expectation 96/120 = 0.80. Range [-0.80, +0.20]. Positive values mean the trajectory prefers phi-coordinate vertices, not the inverse — Pell Word (silver Sturmian, +0.20), Circle Map Quasiperiodic (+0.17), and Phyllotaxis (+0.16) ceiling the metric because their delay-window factor sets cluster near Group 3 (φ/2 and 1/(2φ)-entry) vertices. Negative values mean the trajectory concentrates on Group 1+2 vertices ({0, ±1, ±½} entries) — Fibonacci Word, Penrose Substitution, and smooth Sine Wave all floor at -0.80 because their delay-embedded points sit at simple corner-of-cube positions. Not a measure of "phi-content" of the input: smooth periodic signals and golden-Sturmian substitution words are both on the negative pole, while silver-Sturmian and quasi-rotation signals share the positive pole. The metric tracks which Voronoi subset of the 600-cell the delay embedding falls nearest, set by the joint structure of the delay window and symbol alphabet. Correlates strongly with G2:normalized_entropy (+0.855).
Coefficient of variation of return times to phi-vertices. Low = regular returns to the phi scaffold, high = bursty returns. Devil's Staircase (5.32) and similar clustered signals sit at the top; smooth waveforms have CV near 0.3. Borderline redundant with Chladni:nodal_clustering (r=+0.817) but retained because it specifically measures burstiness of phi-vertex visitation, not zero-crossing burstiness in general.
Kolmogorov-Smirnov distance of Θ-occupancy from the sin²(πΘ) distribution predicted by Mode Identity Theory's anti-periodic ground-state profile. Structured signals whose Θ-occupancy matches the sin² profile score low; signals that deviate score high. Anticorrelates with phi_vertex_excess (r=-0.824) — the two metrics are measuring different facets of the same underlying non-uniform vertex distribution.
|entropy(2I assignment) − entropy(I assignment)|: compares the full 120-vertex assignment with the 60-vertex antipodal-identified quotient. A signal that cares about the double cover (distinguishes +q from −q) will produce different entropies under the two assignments. The single metric highest-CV of the geometry (1.19) but F=1.74 — high variance without strong domain separation. Mostly noise-level for this corner of the atlas; only bearing sources show a consistent elevated signal (+1.10 z).
| Source | Domain | Value |
|---|---|---|
| Rössler Hyperchaos | chaos | 0.8264 |
| μ-law Sine | waveform | 0.8122 |
| OTOC Growth | quantum | 0.8101 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Heisenberg Walk | exotic | 0.0111 |
| Intermittency Type-III | chaos | 0.0712 |
| Source | Domain | Value |
|---|---|---|
| Mian-Chowla | number_theory | 0.7858 |
| Devil's Staircase | exotic | 0.7406 |
| Forest Fire | exotic | 0.7161 |
| ··· | ||
| Quantum Walk | quantum | 0.1072 |
| Wigner Semicircle | quantum | 0.1087 |
| Ramanujan Tau | number_theory | 0.1103 |
| Source | Domain | Value |
|---|---|---|
| Spectral Form Factor | quantum | 10.7464 |
| Devil's Staircase | exotic | 7.3937 |
| ETH/BTC Ratio | financial | 6.0617 |
| ··· | ||
| Thue-Morse | exotic | 0.0000 |
| Fibonacci Word | exotic | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Critical Circle Map (Bronze Mean) | chaos | 0.2000 |
| Critical Circle Map | chaos | 0.2000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.2000 |
| ··· | ||
| L-System (Dragon Curve) | exotic | -0.8000 |
| Fibonacci Word | exotic | -0.8000 |
| Thue-Morse | exotic | -0.8000 |
| Source | Domain | Value |
|---|---|---|
| Accel Jog | motion | 0.6444 |
| Accel Stairs | motion | 0.6225 |
| Ocean Swell | geophysics | 0.6203 |
| ··· | ||
| Penrose Substitution | exotic | 0.0000 |
| Thue-Morse | exotic | 0.0000 |
| Fibonacci Word | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Henon Map | chaos | 0.3254 |
| μ-law Sine | waveform | 0.2220 |
| Gaussian Collatz Orbit | number_theory | 0.1883 |
| ··· | ||
| Thue-Morse | exotic | 0.0000 |
| Penrose Substitution | exotic | 0.0000 |
| Tribonacci Word | exotic | 0.0000 |
Möbius-S³ tests whether a signal's trajectory on the 3-sphere carries structure visible only through the binary icosahedral quotient. Most signals don't — their trajectories distribute over 2I vertices in roughly the geometrically expected proportions, producing mid-range scores on every metric. The ones that do light up fall into two groups: (1) signals with coordinate-axis preferences that produce phi-vertex avoidance (chaotic maps with rational rotation numbers, phyllotaxis), and (2) signals whose cross-domain complexity drives sector_transition_entropy, which acts as a cheap chaos-intensity probe when you already know the system class. A careful audit (Thread 1, 2026-04-09) found that only sector_transition_entropy carries cross-system Lyapunov-tracking signal within the chaotic regime — the other five metrics either fail to track λ within chaos or are dominated by value-distribution effects. Treat the geometry as one genuine chaos-detector (sector_transition_entropy) plus five structural-asymmetry probes useful mainly for flagging signals that deviate from the predicted 2I statistics.