Ωε-Calculus

Reference implementation of the Ωε-calculus, a total term-rewriting system with distinguished constants governing symbolic resolution.

Overview

The calculus introduces two constants, Ω and ε, governed by the rewrite rule ε · Ω → 1. All operations are total: every ground term reduces to a unique normal form. Expressions traditionally regarded as undefined (such as 1/0 or 0/0) are treated as well-formed terms with specified rewrite behavior.

Requirements

• Python 3.x

Installation

No external dependencies required. Clone the repository:

git clone https://github.com/archudzik/OmegaEpsilonCalculus.git
cd OmegaEpsilonCalculus

Usage

Run the demonstration:

python calculus.py

Output

=== Ωε-CALCULUS ===

Primitive rules:
         1 / 0  →  Ω
         1 / Ω  →  ε
         ε · Ω  →  1
         Ω · ε  →  1
         0 · Ω  →  0

Power and scaling rules:
         Ω · Ω  →  Ω²
         Ω / ε  →  Ω²
        ε · Ω³  →  Ω²
   (3/2·Ω) · ε  →  3/2

Closure (no undefined forms):
         0 / 0  →  0
         Ω / Ω  →  1
         ε / ε  →  1

Rewrite Rules

Primitive Rules

Rule	Description
1 / 0 → Ω	Division by zero yields Ω
1 / Ω → ε	Reciprocal of Ω is ε
ε · Ω → 1	Core identity
0 · t → 0	Zero absorption
0 / 0 → 0	Closure
t / t → 1	Self-division (t ∈ {Ω, ε})

Power and Scaling Rules

Rule	Description
Ω · Ω → Ω²	Power formation
Ωᵐ · Ωⁿ → Ωᵐ⁺ⁿ	Power combination
ε · Ωⁿ → Ωⁿ⁻¹	Power reduction (n ≥ 2)
Ω / ε → Ω²	Division by ε
(c · Ω) · ε → c	Scaled collapse (c ∈ ℚ)

External Interpretation

The calculus admits external interpretations. Given a resolution unit δ > 0:

⟦ε⟧ = δ,  ⟦Ω⟧ = 1/δ

For example, taking δ as the Planck length (≈ 1.616 × 10⁻³⁵ m) yields Ω as the number of Planck lengths per meter.

License

This project is licensed under the MIT License.

Citation

If you use this work, please cite:

@misc{chudzik2025omegaepsilon,
  author = {Chudzik, Artur},
  title = {The Ωε-Calculus},
  year = {2025},
  url = {https://github.com/archudzik/OmegaEpsilonCalculus}
}