Orbit: Mathematical Explorations

This repository contains computational explorations in number theory and related mathematics, implemented as a Wolfram Language paclet. The work is ongoing, unpublished, and has not been peer-reviewed.

Featured: Rational Circle Algebra

42 characters. The golden 17-pointed star from Gauss's monument in Braunschweig:

Graphics@{Hue@π,Polygon@κ@ρ[17,7Range@17]}

This uses our Rational Circle Algebra where multiplication becomes addition:

• t₁ ⊗ t₂ = t₁ + t₂ + 5/4 (stays rational!)
• ρ[n,k] = 2k/n - 5/4 (n-th root of unity, always rational)
• κ[t] converts to coordinates only when needed

All circle operations stay in ℚ until the final κ bridge. See CircFunctions.wl.

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What's Here

The repository includes:

• Orbit Paclet: Wolfram Language implementations of various mathematical explorations

  • Prime structure analysis
  • Primorial computation methods
  • Square root approximation techniques
  • Modular arithmetic utilities

• Documentation: Mathematical context and reference materials in docs/

• Scripts: Computational experiments and analysis tools in scripts/

Modules

Primorials: Methods for computing primorials using rational sums

SemiprimeFactorization: Explorations in factorization formulas

ModularFactorials: Efficient computation of factorials modulo p

SquareRootRationalizations: High-precision square root approximations using Chebyshev polynomials, Pell equations, and related methods

CircFunctions: Rational circle algebra where multiplication is addition. Includes operators ⊗ (multiply), ⊙ (power), and Greek-named bridges κ (coordinates), φ (complex), ρ (roots of unity)

See CLAUDE.md for technical details and module documentation.

Using the Code

All scripts require Wolfram Language (Mathematica or free Wolfram Engine):

# Load the paclet
wolframscript -code "<< Orbit\`"

# Or run individual scripts
wolframscript -file scripts/[script-name].wl

Documentation

Quick navigation:

• Documentation Index - Complete organized index of all documentation
• Reference Documentation - Mathematical foundations and design rationale
• Session Narratives - Discovery narratives by date
• Papers - LaTeX papers and longer-form documents

Run make preview to generate HTML previews of all documentation.

Recent Results

Latest work (December 2025):

• ✅ Rational Circle Algebra: Circle multiplication as t₁ + t₂ + 5/4 — stays in ℚ until coordinate conversion. Gauss 17-star in 42 chars!
• ✅ Multiplicative Decomposition Theorem: For composite n = md, lobe areas of n-gon Chebyshev polygon function satisfy Σ A(n, k≡r mod m) = 1/m (proven via roots of unity cancellation)
• Connection between Chebyshev composition Tₘ(Tₙ(x)) = Tₘₙ(x) and geometric lobe area structure
• Unified Chebyshev framework σ_m for square root iteration with arbitrary integer convergence order ≥ 3

See STATUS.md for current status and lobe-area-kernel.tex for the mathematical details.

A Historical Note

During literature review, we traced the history of quadratic residue pattern counting back to N. S. Aladov (1896) — a mathematician virtually unknown in anglophone literature. This attribution comes from Keith Conrad's expository notes; Russian mathematicians (Kiritchenko, Tsfasman, Vlăduț) have maintained awareness of Aladov independently.

Aladov's identity remains a minor historical mystery — see our investigation.

Status

This work represents personal mathematical explorations and computational experiments. Nothing here has been peer-reviewed. All results should be considered provisional and subject to revision.

Individual results have different epistemic statuses (✅ PROVEN, 🔬 NUMERICALLY VERIFIED, 🤔 HYPOTHESIS) - check specific documents for details.

Feedback Welcome

If you find something interesting, puzzling, or incorrect, please open a GitHub issue. Questions, corrections, and independent verification are appreciated.

License

MIT License (code) / CC-BY 4.0 (documentation)

Technical Notes

• Run make generate-index to regenerate documentation index
• Use wolframscript -file for script execution
• See CLAUDE.md for development protocols and conventions