Whether the input value distribution looks like the eigenvalue spacings of a quantum-chaotic system, an integrable quantum system, or neither.
Treats input values as unfolded level spacings (≥ 0, mean-normalized to 1) and KS-tests the empirical distribution against three canonical Random Matrix Theory distributions: Poisson exp(-s) for integrable / uncorrelated spectra, GOE Wigner surmise (π/2)·s·exp(-π·s²/4) for time-reversal-symmetric chaotic systems with linear level repulsion, GUE Wigner surmise (32/π²)·s²·exp(-4s²/π) for systems with broken time-reversal symmetry and quadratic level repulsion. Berry's conjecture says classically chaotic quantum systems have Wigner-Dyson spacings; classically integrable ones have Poisson spacings.
The geometry calibrates exactly against three named source generators: GUE Spacings, GOE Spacings, and Poisson Spacings each rank closest to their own theoretical curve, with KS-distance ≤ 0.008.
KS-distance to the Poisson exponential distribution. Poisson Spacings itself ranks #1 (KS = 0.006). The next closest sources are all event-driven: EEG Seizure (0.030), Deep Earthquake P-wave (0.041), El Centro 1940 (0.042), IMS Bearing Failed (0.047). Heavy-tailed sources with bursts-at-random-times cluster here, consistent with the classical Poisson-process model.
KS-distance to the GUE Wigner surmise. GUE Spacings ranks #1 (KS = 0.006). Surprisingly, the next-closest sources are geophysical noise: Ambient Microseism (0.049), Kilauea Tremor (0.060), Wave Height Buoy (0.066), Seismic Noise ANMO (0.074). Likely interpretation: seismic noise is the superposition of many weakly-coupled independent oscillator modes, which is formally a random-matrix setup. This is a research lead worth following up; see project_research_backlog for the investigation plan.
Composite Wigner-Dyson-vs-Poisson axis: ks_poisson − min(ks_goe, ks_gue). Positive means closer to a Wigner-Dyson surmise than to Poisson (level repulsion present); negative means closer to Poisson (uncorrelated). MFPT Inner Unloaded bearing (-0.2100) ranks #1 most-Poisson — ahead of theoretical Poisson Spacings (-0.2099) by 0.0001. BTC Returns (-0.2054) and Geomagnetic ap Index (-0.2029) cluster nearby. The composite cancels the common-mode "distance from any RMT template" signal that otherwise couples the three raw KS distances.
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 0.9413 |
| Prime Indicator | number_theory | 0.9053 |
| von Mangoldt Function | number_theory | 0.9045 |
| ··· | ||
| GUE Spacings | quantum | 0.0059 |
| Ambient Microseism | geophysics | 0.0492 |
| Kilauea Tremor | geophysics | 0.0595 |
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 0.9149 |
| Prime Indicator | number_theory | 0.9053 |
| von Mangoldt Function | number_theory | 0.9045 |
| ··· | ||
| Poisson Spacings | quantum | 0.0056 |
| EEG Seizure | medical | 0.0302 |
| Deep Earthquake P-wave | geophysics | 0.0405 |
| Source | Domain | Value |
|---|---|---|
| GUE Spacings | quantum | 0.2746 |
| Wave Height (Buoy) | geophysics | 0.2739 |
| Zipf Distribution | exotic | 0.2731 |
| ··· | ||
| MFPT Inner Unloaded | bearing | -0.2100 |
| Poisson Spacings | quantum | -0.2099 |
| BTC Returns | financial | -0.2054 |
Level Statistics is the atlas's first RMT-grounded lens. Its named-source calibration makes it the most verifiable geometry in the framework — feed any of the three Spacings sources back through and they rank #1 on their own metric. The unexpected finding from the first probe: four independent geophysical noise sources (microseism, Kilauea tremor, ocean waves, seismic background) all match the GUE Wigner-Dyson surmise within KS < 0.075, suggesting the superposition of many oscillator modes produces eigenvalue-like value distributions. The wd_classification metric is the most cross-geometry independent of the three (max-r 0.37 with SL(2,ℝ):mean_trace), adding a genuinely new direction to the atlas.