Universal Information Loss at Dimensional Boundaries

~86% information loss when embedding patterns across dimensional transitions in discrete computational systems

📊 Key Finding

When embedding patterns from lower to higher dimensions, approximately 86% of measured information is lost at each dimensional boundary—consistently across:

• Dimensions tested: 1D→2D (85.8%), 2D→3D (86.1%), 3D→4D (86.1%)
• Pattern independence: N=1,500 patterns across all transitions
• Scale independence: Robust across grid sizes N ∈ {15, 17, 20, 23, 25}
• Rule independence: Conway (86.5%) vs HighLife (87.1%) - 0.6% difference
• Geometric origin: Loss occurs from embedding itself, not pattern dynamics

🎯 Significance

This finding reveals a universal geometric scaling law in discrete systems with implications for:

• Information Theory: Quantifies dimensional embedding cost
• Computational Complexity: Pattern persistence across dimensions
• Machine Learning: Theoretical bound on dimensionality reduction
• Consciousness Studies: Potential mechanism for dimensional dispersion hypothesis

🚀 Quick Start

Installation

git clone https://github.com/existencethreshold/dimensional-boundary-loss
cd dimensional-boundary-loss
pip install -r requirements.txt

Requirements: Python 3.8+, numpy, scipy, matplotlib

Run Full Validation (~2.5 hours)

python validate_dimensional_cascade_multisize.py

This tests grid size robustness across N ∈ {15, 17, 20, 23, 25}:

• 100 patterns per grid size, all 3 transitions
• Total: 1,500 patterns
• Results: Mean loss 86.0% ± 2.4% (CV = 2.8%)

Output files:

validation_results_multisize/dimensional_cascade_N100_grid15_*.json
validation_results_multisize/dimensional_cascade_N100_grid17_*.json
validation_results_multisize/dimensional_cascade_N100_grid20_*.json
validation_results_multisize/dimensional_cascade_N100_grid23_*.json
validation_results_multisize/dimensional_cascade_N100_grid25_*.json
validation_results_multisize/multisize_summary_*.json

Generate Publication Figures

python generate_publication_figures.py

Creates 7 figures in publication_figures/ (PNG and PDF formats).

Quick Example (1D→2D)

python examples/quick_start.py

Expected output: ~86% loss

📖 Documentation

• CHANGELOG.md - Version history
• METHODOLOGY.md - Detailed methods and statistical approach
• PHI_METRIC.md - The Φ metric: R·S + D explained
• REPLICATION.md - Step-by-step replication guide

📁 Repository Structure

dimensional-boundary-loss/
├── README.md
├── CHANGELOG.md                 # Version history
├── LICENSE
├── requirements.txt
├── .gitignore
│
├── validate_dimensional_cascade_multisize.py
├── generate_publication_figures.py
│
├── cleanup.bat                  # Windows cleanup
├── cleanup.sh                   # Linux/Mac cleanup
├── uninstall.bat                # Windows uninstall
├── uninstall.sh                 # Linux/Mac uninstall
├── cleanup_utility.py           # Cross-platform utility
│
├── validation_results_multisize/
│   ├── dimensional_cascade_N100_grid15_*.json
│   ├── dimensional_cascade_N100_grid17_*.json
│   ├── dimensional_cascade_N100_grid20_*.json
│   ├── dimensional_cascade_N100_grid23_*.json
│   ├── dimensional_cascade_N100_grid25_*.json
│   └── multisize_summary_*.json
│
├── publication_figures/
│   ├── Figure_1_*.png/pdf
│   ├── Figure_2_*.png/pdf
│   ├── Figure_3_*.png/pdf
│   ├── Figure_4_*.png/pdf
│   ├── Figure_5_*.png/pdf
│   ├── Figure_6_*.png/pdf
│   └── Figure_7_*.png/pdf
│
├── examples/
│   └── quick_start.py
│
├── tests/
│   ├── test_grid_size_sensitivity.py
│   ├── test_highlife_validation.py
│   ├── test_metric_sanity_check.py
│   └── validation_data/
│       ├── grid_size_validation_*.json
│       ├── highlife_validation_*.json
│       └── metric_sanity_check_*.json
│
└── docs/
    ├── METHODOLOGY.md
    ├── PHI_METRIC.md
    ├── REPLICATION.md
    └── CLEANUP.md              

🔬 The Φ (Phi) Metric

Measures pattern persistence/information:

Φ = R·S + D

Where:
  R = Processing (alive cells / total cells)
  S = Integration (spatial transitions / total edges)
  D = Disorder (Shannon entropy of state distribution)

Higher Φ = more active information processing

See PHI_METRIC.md for detailed explanation.

📈 Results Summary

Overall Finding (Across All Grid Sizes)

Metric	Value
Mean Loss	86.0% ± 2.4%
Range	82.5% - 88.6%
Grid Sizes Tested	15, 17, 20, 23, 25
Total Patterns	1,500
Coefficient of Variation	2.8%

By Transition (Mean Across Grid Sizes)

Transition	Mean Loss	Range	CV
1D→2D	85.8%	82.5%-88.5%	2.9%
2D→3D	86.1%	83.0%-88.6%	2.7%
3D→4D	86.1%	83.0%-88.6%	2.7%

Interpretation: Universal ~86% loss across all grid sizes and dimensional transitions, with expected finite-size variation

Rule Independence

Rule	Mean Loss	Configuration
Conway	86.5%	B3/S23 (Birth/Survival)
HighLife	87.1%	B36/S23
Difference	0.6%	Effect is geometric, not rule-dependent

🔁 Replication

Full Validation (Recommended)

python validate_dimensional_cascade_multisize.py

• Runtime: ~2.5 hours (tests 5 grid sizes)
• Output: validation_results_multisize/ directory with 6 JSON files
• Verification: Compare statistics with published results
  • Mean loss: ~86.0%
  • CV across sizes: ~2.8%

Grid Size Robustness Details

The validation tests demonstrate:

• Scale-independence: ~86% loss holds from 15×15 to 25×25 grids
• Realistic variation: CV = 2.8% shows expected finite-size effects
• Consistency: All three transitions cluster around 86%

Quick Verification

# Test single transition
python examples/quick_start.py

# Expected: 80-92% loss (pattern-dependent)
# Mean across many patterns: ~86%

Validate Robustness

cd tests

# Grid size sensitivity (15-25)
python test_grid_size_sensitivity.py

# Rule independence (Conway vs HighLife)
python test_highlife_validation.py

# Metric validation (edge cases)
python test_metric_sanity_check.py

See REPLICATION.md for detailed instructions.

📊 Figures

Run python generate_publication_figures.py to generate:

1. Figure 1: Conceptual overview (1D→2D→3D cascade)
2. Figure 2: Loss distribution histogram (N=1,500)
3. Figure 3: Rule independence - Conway (86.5%) vs HighLife (87.1%)
4. Figure 4: Grid size robustness (N ∈ {15, 17, 20, 23, 25})
5. Figure 5: Φ metric components (R, S decomposition)
6. Figure 6: Visual embedding example (1D→2D)
7. Figure 7: Reverse Prism hypothesis

🧹 Repository Maintenance

Cleanup Generated Files

After running validation or generating figures, you can clean up:

# Remove generated files (keeps validation_results_multisize/)
cleanup.bat     # Windows
./cleanup.sh    # Linux/Mac
python cleanup_utility.py cleanup  # Cross-platform

Removes: publication_figures/, pycache, temp files
Keeps: validation_results_multisize/, code, documentation

Uninstall (Remove Virtual Environment)

Remove everything except validation_results_multisize/:

# Remove venv and generated files (keeps data)
uninstall.bat   # Windows
./uninstall.sh  # Linux/Mac
python cleanup_utility.py uninstall  # Cross-platform

Removes: Virtual environments, generated files, cache
Keeps: validation_results_multisize/, code, documentation

Reset to Fresh State (DELETES DATA!)

⚠️ Warning: This deletes validation_results_multisize/

# Complete reset (use with caution!)
python cleanup_utility.py reset

Use when: Starting completely fresh, re-running full validation

See CLEANUP.md for detailed documentation.

---

📝 Citation

@article{thornhill2026dimensional,
  title={Pattern Loss at Dimensional Boundaries: The 86% Scaling Law},
  author={Thornhill, Nathan M.},
  journal={PLOS Complex Systems},
  year={2026},
  note={In review},
  doi={10.5281/zenodo.18238486}
}

🤝 Contributing

This repository contains published research code. To contribute:

1. Report issues: Open GitHub issue for bugs/questions
2. Suggest improvements: Submit pull request with clear description
3. Extend research: Fork and cite if building on this work

📄 License

MIT License - see LICENSE file

Summary: Free to use, modify, and distribute with attribution

👤 Author

Nathan M. Thornhill
Independent Researcher
Fort Wayne, Indiana, USA

• Email: existencethreshold@gmail.com
• ORCID: 0009-0009-3161-528X
• GitHub: https://github.com/existencethreshold

Recent Publications

Pattern Loss at Dimensional Boundaries: The 86% Scaling Law (2026)

• Zenodo: https://doi.org/10.5281/zenodo.18238485
• Code: https://github.com/existencethreshold/dimensional-boundary-loss
• Status: Peer review (PLOS Complex Systems)

The Existence Threshold v2.1 (2026)

• DOI: https://doi.org/10.5281/zenodo.18124074

🙏 Acknowledgments

• Anthropic Claude (computational research assistance)
• Open source cellular automata community
• Peer reviewers and community feedback

---

Last Updated: January 15, 2026
Status: Peer review (PLOS Complex Systems)
DOI: 10.5281/zenodo.18238486
Version: 1.1.0