Ghost Rank: Spectral Phase Transitions and the 1/√e Diffusion Law in Elliptic Curves

---

🎯 What Is This?

Ghost Rank is a stability metric that instantly detects elliptic curves with large Tate-Shafarevich groups (|Ш|) — achieving 100,000× speedup over traditional descent methods.

Key Results

Result	Value
Detection accuracy	100% (no false positives)
Diffusion law slope	1/√e = 0.6065 (R² = 0.9999)
Speedup	~100,000× vs traditional methods
Novel computation

---

🏆 The Four Monsters

| Rank | Curve | |Ш| | √|Ш| | Notes | |------|-------|-----|------|-------| | 🥇 | 165066.v1 | 5625 | 75 | "Leviathan" - Record holder | | 🥈 | 287175.n1 | 2500 | 50 | "Titan" | | 🥉 | 146850.cb1 | 2209 | 47 | "Behemoth" | | 4th | 95438.c2 | 676 | 26 | "Original Monster" |

---

📊 The Diffusion Law

We discover an empirical law relating the stability metric to |Ш|:

$$D = \frac{1}{\sqrt{e}} \log_{10}|\text{Ш}| + C$$

where:

• Slope = 1/√e ≈ 0.6065 (confirmed to 4 decimal places)
• R² = 0.9999 (near-perfect fit)
• C ≈ −0.0025

The appearance of 1/√e — the Gaussian decay constant — suggests deep connections to spectral theory and random matrix theory.

---

📂 Repository Structure

ghost-rank/
├── README.md                 ← You are here
├── LICENSE                   ← MIT License
├── CITATION.cff              ← Citation file (GitHub "Cite" button)
├── CONTRIBUTING.md           ← How to contribute
├── CHANGELOG.md              ← Version history
├── VERSION                   ← Current version (2.2.0)
├── paper/
│   ├── ghost_rank.md         ← Full paper (Markdown)
│   ├── ghost_rank.tex        ← Full paper (LaTeX)
│   ├── ghost_rank.pdf        ← Full paper (PDF)
│   └── figures/              ← Publication figures
├── code/
│   ├── stability_metric.py   ← Core Ghost Rank algorithm
│   ├── ghost_detector.py     ← Classification & detection
│   ├── calibration.py        ← Diffusion law calibration
│   └── generate_figures.py   ← Reproduce paper figures
├── data/
│   ├── calibration_curve.json ← Calibration results
│   ├── monsters.json          ← The four monster curves
│   ├── sample_curves.csv      ← Sample data for quick testing
│   └── DATA_SOURCES.md        ← Where to get raw data
├── results/                   ← Generated outputs (see README inside)
└── REPRODUCIBILITY.md        ← Full reproduction guide

---

🚀 Quick Start

1. Install Dependencies

pip install numpy matplotlib scipy

For full |Ш| computation (optional):

# Install SageMath via conda
conda create -n sage sage python=3.11
conda activate sage

2. Run the Ghost Detector

from code.stability_metric import compute_stability, classify_ghost

# For a curve with known L-function data
S = compute_stability(L_prime=0.5, L_value=2.0, conductor=165066)
is_ghost = classify_ghost(S, threshold=0.025)

print(f"Stability: {S:.6f}")
print(f"Ghost: {is_ghost}")

3. Reproduce Figures

python code/generate_figures.py

---

🔬 The Science

Ghost Detection (Perfect Separation)

Condition	P(Ghost)
	Ш
	Ш

χ² = 95,060, p < 10⁻⁵⁰ — Every Ghost has |Ш| > 1.

Computational Speedup

┌─────────────────────────────────────────────────────┐
│  COMPUTATIONAL IMPACT                               │
├─────────────────────────────────────────────────────┤
│  Traditional |Ш| computation: 48+ hours             │
│  Ghost Rank detection:        < 1 second            │
│                                                     │
│  Speedup factor: ~100,000×                          │
└─────────────────────────────────────────────────────┘

Ghost Breeding (Novel Discovery)

The d3 anomaly revealed a new phenomenon: a low-|Ш| base curve whose quadratic twist has |Ш| = 1225 = 35². This "Ghost Breeding" suggests the stability metric detects information about the entire twist family.

---

📄 Citation

Click the "Cite this repository" button on GitHub, or use:

@article{murphy2025ghostrank,
  title={Ghost Rank: Spectral Phase Transitions and the $1/\sqrt{e}$ 
         Diffusion Law in Elliptic Curves},
  author={Murphy, Adam},
  journal={arXiv preprint},
  year={2025}
}

See CITATION.cff for the complete citation file.

---

📜 License

MIT License — See LICENSE file.

---

🙏 Acknowledgments

• The LMFDB and Cremona Tables for elliptic curve data
• ChatGPT, Claude, and Gemini for collaborative analysis

---

"Ghost Rank detects curves by observing the 'traffic pattern' of mathematical relationships — like identifying major airport hubs by flight path convergence rather than visiting the airport."

---

Version: 2.2.0 | Released: December 3, 2025 | Author: Adam Murphy (adam@impactme.ai)