Modular Prime Rattling Framework
This project implements a novel mathematical model—Rattling Agents in Modular Prime Fields—which explores the emergent spatial structure of prime number distributions using entropy fields, harmonic alignments, and biased stochastic motion.
📖 Paper
The accompanying research paper outlines the full theory, formal lemmas, conjectures, and simulation results. It is formatted for LaTeX and ready for academic submission.
- Title: Structured Rattling and Harmonic Resonance as Predictors of Prime Distribution in Modular Fields
- Format: LaTeX + BibTeX
- Paper available via Repo (Update - second paper now available as well)
🧠 Theory Summary
- Modular Matrix Construction: Prime indices are mapped into a 2D modular field.
- Entropy Field: Prime scarcity is quantified via local entropy.
- Rattling Agents: Biased walkers traverse entropy gradients.
- Harmonic Alignment: Zones aligned to the golden ratio φ form resonant bands.
- Prediction Field: Combines entropy and φ-alignment to forecast probable prime zones.
🔬 Simulation
This repo includes a complete Python simulation to visualize:
- Modular structure
- Entropy fields
- Rattling path density
- Entropy curvature
- Twin prime influence
- φ-alignment
- Composite predicted prime zones
🔧 Requirements
pip install numpy matplotlib scipy
▶ Run the simulation
python modular_prime_rattling_simulation.py
This will generate modular_prime_rattling_analysis.png with 7 panels summarizing the system.
📚 References
The paper includes a formal bibliography referencing:
- Classical prime models (Cramér, Hardy-Wright)
- Entropy and resonance field theory
- Probabilistic number theory
✍ Authors
- Casey Allard — Originator of the Prime Rattling theory
- ChatGPT — Co-pilot for theory refinement, formalism, and simulation