Ô (O-Hat) — Three Numbers That See When Systems Break

Drop a CSV. Get three numbers. Know if your system is stable, heading for trouble, or already broken.

Works on climate data, disease outbreaks, earthquakes, stock markets, typhoons, LLMs — anything that produces a time series.

Quick Start

pip install numpy scipy
python o_hat.py examples/enso.csv

Output:

======================================================
  O-HAT  STRUCTURAL  MEASUREMENT
======================================================
  Data points  : 905
  Baseline     : 181 points
  Window       : 45 points
  ---
  Curl         :        -2.31
  Helicity     :        18.45
  Baseline skew:       0.5231
  Outbreak skew:       1.2187
  ---
  BALANCE RATIO:       2.33x
  SYSTEM TYPE  :  CONTINUOUS FLOW
======================================================

That's it. One command, one result.

What It Measures

Metric	What It Captures	Physical Meaning
curl	Directional change rate	Is the system accelerating toward or away from equilibrium?
helicity	Total variation	How much structural complexity is in the outbreak?
balance	Skewness ratio (outbreak / baseline)	How far is the system from its normal state?

The Classification Spectrum

Balance Ratio	System Type	Examples
> 20×	SUPER-PULSE	1989 Quebec geomagnetic storm
5–15×	PULSE	Earthquakes (M5+)
1.5–4×	CONTINUOUS FLOW	ENSO, COVID-19 outbreaks
< 1.5×	REGIME SATURATION	Seasonal forcing (typhoon seasons)

The same three numbers classify seven different types of dynamical systems across twelve domains — climate, epidemiology, seismology, finance, space weather, LLMs, and more.

Try Your Own Data

python o_hat.py your_data.csv
python o_hat.py your_data.csv --window 50 --baseline 200
python o_hat.py your_data.csv --quiet          # machine-readable output
python o_hat.py your_data.csv --plot           # classification chart
python o_hat.py your_data.csv --sensitivity    # window stability verification

Any single-column CSV works. Timestamps not required — just values, one per row, time-ordered.

🔬 Window Stability (v1.2+)

Worried about cherry-picking? --sensitivity sweeps window and baseline sizes across 50%–150% (121 combinations) and reports a stability band:

python o_hat.py examples/enso.csv --window 60 --baseline 300 --sensitivity

If your balance ratio stays within a narrow band across all window sizes, the structural signal is genuine — not an artifact of parameter tuning. See §9 of the classification spectrum for methodology and cross-domain results.

Why This Exists

Most tools are domain-specific. Climate scientists use one set of metrics. Epidemiologists use another. Seismologists use a third.

Ô is substrate-independent. It measures structural properties that exist in any dynamical system, regardless of what the numbers represent. The math doesn't care if you're measuring ocean temperatures, case counts, or stock prices.

Built from empirical cross-domain validation across 12 domains. See docs/classification-spectrum-v2.md for the full framework.

Citation

If you use this in research, please cite:

DR & MM. "O-Hat v2.1: Universal Structural Classifier — Dual-Mode Architecture (Balance + Helicity), 10-Domain Cross-Validation, C-MAPSS Sliding-Window Analysis." 2026. Zenodo: 10.5281/zenodo.21431348

License

MIT — use it, fork it, break it, improve it.