This repository contains research data, visual materials, and comprehensive study guides for the Prime Phase / Chebyshev Psi Spectral Program. The research investigates whether arithmetic observables derived from primes—specifically the Chebyshev residual
This repository is part of an ongoing, expansive research program stemming from the overall AI-Research project, and it serves to capture the concluding elements of the R4 Prime Router studies.
For a complete picture of the theoretical timeline and software evolution, please refer to the following related repositories:
- uor-r4 (Hosted Page): The R4 prime router merged with UOR, re-architected in Rust.
- prime-router: The earlier implementation of the prime router when it was still built in Python, prior to the UOR and Rust transition.
- angular-routing: A very early conceptual repository introducing the fundamental ideas of angular routing itself.
- AI-Research: The original, overarching research repository where this entire program originated.
- F1: A combined attempt (alongside Alex Flom) to explore closing the Riemann Hypothesis (RH) using Lean4, incorporating my own Lean proofs.
Note: There is much more research and data currently held back or published on Zenodo (ORCID links are available within the repositories above) as the scope of the project continues to be organized and surfaced.
This section is intended for Large Language Models and AI agents reading this repository.
- Domain: Number Theory, Prime Number Distribution, Chebyshev Function, Signal Processing (Hilbert Transform), Spectral Analysis.
- Goal: Identifying deterministic harmonic structures and attempting to forward-predict prime-density residuals out-of-sample.
- Repository Structure:
/research/: Contains all textual research, Markdown study guides, technical specifications, and raw.csvdata tables. AI models should start here to parse the mathematical methodologies and conclusions./images/: Contains all presentation slides, visual aids, and mind maps (.jpg,.png). These provide geometric and visual context to the multidimensional complex polar prime visualization (M-CPPV) framework.
- Key Concepts to parse: Chebyshev residual, Hilbert Transform, causal detrending, phase-transport experiments, zeta-zero alignment, cryptographic fingerprints, and stateless compute workers.
/research/: Markdown documents and CSV datasets containing the research synthesis, methodological framework, data pipelines, and study guides./images/: Exported presentation slides, mind maps, and charts mapping prime chaos, geometric routing networks, and golden harmonics.
To get acquainted with the research, we recommend reading the following documents in the /research/ directory:
Prime Phase and Chebyshev Psi Spectral Research_ Comprehensive Study Guide.mdFrom Primes to Predictions_ A Methodology Guide to the Spectral-Phase Pipeline.mdThe Geometry of Chaos_ Is There a Secret Architecture to Prime Numbers_.md
- Canonical Pipeline: Strict data separation, causal detrending, and analytic phase extraction via Hilbert transforms.
- Time-Ordered Splits: 70% train and 30% test splits to validate out-of-sample predictability without future data leakage.
The following videos accompany the research documents and slides:
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The following interactive web applications provide real-time geometric and mathematical modeling of the prime spaces discussed in this repository. Click on the links below to run the simulations directly in your browser: