nomogeo

Exact observer-geometry kernel for visible precision, hidden-load calculus, information conservation, and observer steering.

Current release: nomogeo 0.4.0.

What nomogeo does

Given a symmetric positive-definite matrix H (the ambient precision / Fisher information / Hessian) and a rank-m observer C, nomogeo computes exact split-frame geometry:

• Visible precision Phi = (C H^{-1} C^T)^{-1} and canonical lift L
• Information budget: vis_rate + hid_rate = amb_rate (exact conservation)
• Source law: A_cpl on support-stable strata, with hidden-defect decomposition
• Observer steering: adapted observer (B = 0), capture curves, Grassmannian diagnostics
• Geometry extraction: from supervised data (X, y) or covariance matrices directly to (H, Hdot)

Install

pip install nomogeo

Runtime dependencies: numpy >= 2.0, scipy >= 1.15. No other requirements.

Quick start

import numpy as np
from nomogeo import visible_precision, information_budget, steer

# Exact visible precision
H = np.array([[3.0, 1.0], [1.0, 2.0]])
C = np.array([[1.0, 0.0]])
phi = visible_precision(H, C)

# Information conservation: vis + hid = amb
Hdot = np.array([[0.1, -0.2], [-0.2, 0.3]])
budget = information_budget(H, C, Hdot)
print(f"vis={budget.visible_rate:.4f}, hid={budget.hidden_rate:.4f}, "
      f"amb={budget.ambient_rate:.4f}")
print(f"conservation residual: {budget.conservation_residual:.1e}")

Observer steering from data

from sklearn.datasets import load_iris
from nomogeo import steer

X, y = load_iris(return_X_y=True)
result = steer(X=X, y=y, rank=2, task="fisher")
print(f"vis_frac={result.visible_fraction:.3f}, "
      f"advantage over PCA: {result.advantage_over_pca:.3f}")
print(f"exact sector: {result.exact_sector}")

Core API

Geometry

Information conservation and source law

Extraction and steering

Observation fields and regime classification

0.4.0 Technical Note

The accompanying technical note contains 30+ formal statements:

• Conservation laws: vis_rate + hid_rate = amb_rate (first and second order)
• Source law: A_cpl = A_direct + hidden_defect, with completed-square decomposition
• Positivity theorem: A_cpl >= 0 on linear paths (hidden sector stabilises)
• Equivalence principle: A_cpl can always be locally cancelled
• Curvature-Gram identity: ||F_alpha||^2 = 2(Tr(G1 G2) - Tr(C^2))
• Mixed factorisation: F_alpha(ds, dt) = -beta_t R^{-1} B_s^T
• Gauge invariance: vis_rate is a gauge-invariant Noether current
• Spectral collapse lemma: no scalar constraint makes vis_rate maximisation well-posed
• KL field equation: S - Xi = (gamma/2)(H - H0) under KL-divergence regularisation
• Joint optimum theorem: closed-form solution with frozen hidden sector

Validated on 118,539 molecular pairs from HessianQM9 and 108/108 field equation checks.

Companion packages

• nomocomp (0.1.0): geometric model comparison replacing AIC/BIC with exact fibre-volume correction
• nomoselect (0.1.0): task-aware subspace selection replacing PCA with certified observer design

Both depend on nomogeo >= 0.4.0.

Architecture

nomogeo/
  src/nomogeo/
    core.py          -- visible precision, lift, projector
    connection.py    -- fixed-observer charts, transitions, currents
    source.py        -- conservation, source law, diagnostics, capture curves
    extract.py       -- data -> geometry extraction
    steer.py         -- observer steering engine
    field.py         -- observation fields, birth/death, support transport
    regime.py        -- quadratic regime classification
    singular.py      -- singularity analysis
    kernel_reduction.py -- affine-hidden Gaussian-fibre reduction
    frontier.py      -- weighted-family frontier evaluation
    ensemble.py      -- local quadratic ensemble diagnostics
    adapted.py       -- closure-adapted observer synthesis
    affine.py        -- thin affine layers
  tests/             -- 340+ tests
    audit/           -- complete research audit trail
  examples/          -- 6+ runnable demonstrations
  papers/            -- LaTeX source for technical notes

Verification

PYTHONPATH=src python -m pytest tests/ -q

340+ tests, all passing. 2 skipped (require external datasets).

Exact domain

nomogeo is exact for:

• Linear observers on finite-dimensional SPD matrices
• Gaussian/quadratic visible objects
• Matrix identities where theorems apply
• Explicitly supplied special-law sectors

It is not a generic non-Gaussian engine. Outside the exact Gaussian sector, it provides local quadratic Hessian/Fisher geometry plus separately proved special sectors.

License

BSD 3-Clause. See LICENSE