Meta-Axiom: A Mathematical-Philosophical Framework for Universal Optimization

LICENSE Apache2.0

This repository provides a formal Lean 4 formalization of the "Four Meta-Axioms" (collectively referred to as F-Theory) proposed by Takeo Yamamoto.

Overview

The framework establishes a rigorous foundation for the conceptual laws of the universe through four fundamental pillars:

• Extremum Principle (A1): $F[x] = \text{Extremum } L(x)$
• Topological Space (A2): $x \in X \subset \mathbb{R}^n$
• Logical Consistency (A3): $C[F] = 0$
• Hierarchical Structure (A4): $F_{macro} = \sum w_i \cdot F_{micro}(i)$

These axioms are formalized in Lean 4 and verified to be internally consistent and mathematically well-formed via the Curry-Howard correspondence.

Status: Fully Proven

All theorems and definitions in this repository are fully proven. The following files contain no sorry placeholders, and the proof state is verified by continuous integration (GitHub Actions).

• Metaaxiom.lean – Core axioms and consistency proofs.
• Axioms.lean – Auxiliary lemmas and mathematical properties.
• Ftheory.lean – Application to F-theory and cosmology.
• Collatz.lean – Application to the structural analysis of the Collatz conjecture.
• Dna.lean – Application to genetic systems and information theory.
• Medical.lean – Application to biological systems and homeostatic optimization.

Resolved Proofs

The following properties are integrated into the formal verification:

• shannon_entropy_nonneg: Non-negativity of Shannon entropy (proven via log-convexity).
• consistency_preserved: Invariance of logical consistency under Meta-Axioms (derived from A3).
• compact_extremum_exists: Existence of extrema on compact spaces (using IsCompact.exists_isMin/exists_isMax).
• unity_principle: Formal derivation of hierarchical convergence (derived from A4).

License

This work is licensed under the Apache2.0 license.
See the LICENSE file for details.

Author: Takeo Yamamoto

Citation

@software{yamamoto_meta_axiom_2025,
  author       = {Takeo Yamamoto},
  title        = {Meta-Axiom: A Mathematical-Philosophical Framework for Universal Optimization},
  year         = {2025},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.18603974},
  url          = {[https://doi.org/10.5281/zenodo.18603974](https://doi.org/10.5281/zenodo.18603974)}
}