Dimensional Boundary Geometry: Testing the Coordination Number Hypothesis

Pre-registration: DOI: 10.5281/zenodo.18287219 Fulfillment Paper: DOI: 10.5281/zenodo.18303513
Author: Nathan M. Thornhill (ORCID: 0009-0009-3161-528X)
Date: January 19, 2026

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Overview

This repository contains the complete experimental protocol, code, and data for testing whether information loss during dimensional embedding depends on coordination number k according to the hypothesis:

L(k) = 100 × k/(k+1)

where k is the number of face-adjacent neighbors in the lattice topology.

Key Finding

Hypothesis falsified: Information loss is independent of coordination number k for static patterns. All topologies (k=4, 6, 8) show identical loss (~83%) regardless of neighbor structure. Loss originates from space dilution (1/L ratio), not topological geometry.

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Quick Start

# Clone repository
git clone https://github.com/existencethreshold/dimensional-boundary-geometry.git
cd dimensional-boundary-geometry

# Install dependencies
pip install -r requirements.txt --break-system-packages

# Run Phase 1 experiment (30 patterns, ~30 seconds)
python scripts/run_phase1.py

# Analyze results
python scripts/analyze_results.py

# Generate figures
python scripts/generate_figures.py

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Results Summary

Topology	k	Mean Loss	Std Dev	Min	Max	n
Von Neumann	4	82.93%	0.69%	81.46%	83.77%	10
Hexagonal	6	83.07%	0.50%	82.08%	83.89%	10
Moore	8	83.02%	0.45%	82.30%	83.52%	10

Max difference: 0.15% (not significant, p >> 0.05)

Conclusion: Hypothesis H₁ definitively falsified. No k-dependence detected.

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Mathematical Explanation

For static patterns with middle-placement embedding (L=20, p=0.5):

• Native 2D: D = 1.0 (maximum entropy)
• Embedded 3D: D' ≈ 0.169 (diluted by 1/20 ratio)
• Loss: (1 - 0.169/1.0) × 100% = 83.1%

k does not appear in Shannon entropy formula → All topologies show identical loss.

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Repository Structure

dimensional-boundary-geometry/
├── docs/              Pre-registration PDF + figures
├── data/              Experimental results (CSV)
├── src/               Python modules
├── scripts/           Executable experiments + analysis
└── tests/             Unit tests

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Reproducibility

All experiments use documented random seeds:

import numpy as np
from src.metrics import calculate_phi_static
from src.embedding import embed_2d_to_3d

# Reproduce seed 10000
np.random.seed(10000)
pattern_2d = (np.random.random((20, 20)) < 0.5).astype(int)
phi_native = calculate_phi_static(pattern_2d)

pattern_3d = embed_2d_to_3d(pattern_2d, 20)
phi_embedded = calculate_phi_static(pattern_3d)

loss = (1 - phi_embedded / phi_native) * 100
print(f"Loss: {loss:.2f}%")  # Output: 83.13%

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Related Work

Prior Publications:

1. Thornhill, N. M. (2026). Coordination Number Independence of Dimensional Information Loss: A Pre-Registered Study.
   Zenodo. DOI: 10.5281/zenodo.18303513

2. Thornhill, N. M. (2026). Pattern Loss at Dimensional Boundaries: The 86% Scaling Law.
   Zenodo. DOI: 10.5281/zenodo.18262424

3. Thornhill, N. M. (2026). The Existence Threshold (Version 2.1).
   Zenodo. DOI: 10.5281/zenodo.18124074

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Citation

@software{thornhill2026geometry,
  author       = {Thornhill, Nathan M.},
  title        = {{Dimensional Boundary Geometry: Testing the 
                   Coordination Number Hypothesis}},
  year         = 2026,
  publisher    = {GitHub},
  version      = {1.0.0},
  doi          = {10.5281/zenodo.18287219},
  url          = {https://github.com/existencethreshold/dimensional-boundary-geometry}
}

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License

MIT License — see LICENSE file.

Contact

Nathan M. Thornhill
Email: existencethreshold@gmail.com
ORCID: 0009-0009-3161-528X