HVRTAnalysis

Experimental code and LaTeX source for:

Cooperative Structure in High-Dimensional Data: The Geometric Foundations of HVRT and GeoXGB Jake Peace — February 2026

Overview

This repository contains the seven controlled experiments, shared analysis utilities, numerical results, and full LaTeX source that underpin the paper above.

The central finding: HVRT's target function optimises T = S² − Q, an indefinite quadratic form whose level sets are hyperboloids. T is exactly orthogonal to Mahalanobis distance Q for isotropic data (Cov(T,Q) = 0), its expected value equals the sum of all pairwise covariances (E[T] = Σᵢ≠ⱼ Σᵢⱼ), and it is exactly preserved under isotropic additive noise while Q inflates proportionally.

Repository Structure

experiments/     Seven standalone experiment scripts (run independently)
analysis/        Shared dataset generators, metrics, and plotting utilities
results/         Auto-generated JSON results and PNG figures
latex/           LaTeX source for the arXiv preprint
  main.tex
  bibliography.bib
  figures/       Figures copied from results/

Requirements

• Python 3.13
• HVRT v2.7.0 (see note below)
• numpy, scipy, scikit-learn, xgboost, matplotlib

pip install numpy scipy scikit-learn xgboost matplotlib

Note on HVRT: The experiments depend on the HVRT library (v2.7.0), which is not yet publicly distributed. All experiment designs, numerical results (JSON), and figures are included here for full transparency. The LaTeX source and compiled paper are fully self-contained.

Running Experiments

Each script is standalone and writes its results to results/ as JSON + PNG.

python experiments/01_target_decomposition.py    # ~20s  — P≈T/2, level sets
python experiments/02_TQ_orthogonality.py        # ~30s  — Cov(T,Q)=0 proof
python experiments/03_cooperative_geometry.py    # ~30s  — T discriminates; Q cannot
python experiments/04_cone_scaling.py            # ~60s  — cone fraction and heatmap
python experiments/05_correlation_to_cooperation.py  # ~30s  — E[T]=off-diag sum
python experiments/06_noise_robustness.py        # ~90s  — E[T] preserved under noise
python experiments/07_geoxgb_noise.py            # ~8min — GeoXGB vs XGBoost

Paper

The LaTeX source is in latex/main.tex. Compile with:

cd latex
pdflatex main.tex
bibtex main
pdflatex main.tex
pdflatex main.tex

Key Results

Claim	Evidence
P ≈ T/2 (rank-equivalent)	Spearman ρ ≥ 0.998, slope = 0.500 across all d
Level sets of T are hyperboloids	Algebraic + visualised (Exp 1)
Cov(T,Q) = 0 for isotropic z	Algebraic proof + r < 0.03 empirically (Exp 2)
E[T] = Σᵢ≠ⱼ Σᵢⱼ exactly	Pearson r = 0.9998 (Exp 5)
E[T] preserved under noise	99%+ preservation at all σ tested (Exp 6)
HVRT augmentation reduces MSE 22%	0.566 vs 0.729 vs plain XGBoost (Exp 7)
GeoXGB improves under low noise	−2.9% at σ=0.25 (Exp 7)

Contact

Jake Peace — mail@jakepeace.me