Ô-HAT Framework

6-Dimensional Topological Noise Framework — phase-space operators for structured-noise detection, with a candidate 7th dimension.

What it is

Ô-HAT is a family of topological operators that characterize structured noise — residual fields that are neither pure noise nor smooth signal. It was validated on plate-motion GPS residuals (NGL 8480 stations) and typhoon vorticity fields, where it separates core-condensed (aggregating) from shell-dominant (releasing) regimes.

The core vector is

Ô₆D = [dH_curl, θ₁, χ_eff, H, Ô, D₁]

The candidate 7th dimension D_fold measures the singularity-set dimension difference (negative → folding / structure formation):

D_fold = d(S_θ) − d(S_θ^null)

Install

pip install ohat-framework

Quick start

import numpy as np
from ohat import OHATEngine

rng = np.random.default_rng(0)
z1 = rng.normal(size=(20, 20))   # e.g. 850 hPa standardized vorticity
z2 = rng.normal(size=(20, 20))   # e.g. 200 hPa standardized vorticity

core = np.zeros((20, 20), bool); core[7:13, 7:13] = True
shell = ~core

engine = OHATEngine()
vec = engine.compute(z1, z2, core_mask=core, shell_mask=shell)
print(vec)   # {'dH_curl', 'theta_1', 'chi_eff', 'H', 'delta_H', 'O', 'D1'}

Plate-GPS domain

from ohat import OHATEngine

engine = OHATEngine()
result = engine.compute_plate_residual(
    dve, dvn,         # residual east/north velocity grids (Nx, Ny)
    lon2d, lat2d,     # coordinate grids (degrees)
    threshold="p90",  # D_fold singularity threshold
)
# result: 6D vector + 'D_fold' dict

Operators reference

Each operator is importable directly:

from ohat import dH_curl, theta_1, chi_eff, helicity, o_hat, d1
from ohat import d_fold, box_counting_dimension, singularity_set_dimension

Tests

python -m pytest tests/ -v

16 tests covering sign conventions, noise baselines, box-counting convergence (line → 1, grid → 2) and the D_fold random-null property.

Citation

@misc{tygtDc2026ohat,
  author = {{tygtDc, Deep Research}},
  title  = {Ô-HAT: A 6-Dimensional Topological Noise Framework},
  year   = {2026},
  doi    = {10.5281/zenodo.21760992},
  url    = {https://doi.org/10.5281/zenodo.21760992}
}

Status & caveats

• dH_curl has two estimators: continuous-field (core–shell mean difference) and discrete spatial-curl (plate GPS grid).
• D_fold is a candidate 7th dimension: validated on plate GPS (18 segments, 12/12 subduction negative) but not yet cross-domain validated (typhoon). Treat results accordingly.
• θ₁ of a random noise field is not a statistical invariant — it is a single-draw PCA angle; use distributional comparisons across ensembles.