Meta-Axiom: Mathematical-Philosophical Framework

This repository provides a formal Lean 4 formalization of the "Four Meta-Axioms" proposed in the paper
"Meta-Axioms as the Conceptual Foundation of the Universe" by Takeo Yamamoto.

Overview

• Extremum Principle (A1): F[x] = Extremum L(x)
• Topological Space (A2): x ∈ X ⊂ ℝⁿ
• Logical Consistency (A3): C[F] = 0
• Hierarchical Structure (A4): F_macro = Σ wᵢ · F_micro(i)

These axioms are formalized in Lean 4 and verified to be internally consistent and mathematically well-formed.

Status

All theorems and definitions in this repository are fully proven in Lean 4.
The following files contain no sorry placeholders:

• Metaaxiom.lean – Core axioms and basic theorems
• Axioms.lean – Auxiliary lemmas and properties
• Dna.lean – Application to DNA and genetic systems
• Medical.lean – Application to medical and biological systems
• Collatz.lean – Application to the Collatz conjecture
• Ftheory.lean – Application to F-theory and cosmology

Continuous integration (GitHub Actions) ensures that all proofs remain valid.

Open Problems (Resolved)

The following theorems, previously listed as open problems, are now fully proven:

• shannon_entropy_nonneg – Non-negativity of Shannon entropy (proven using convexity of the logarithm).
• consistency_preserved – Preservation 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 – Unity principle for hierarchical structures (derived from A4).

These theorems are now integrated into the respective application files (Entropy.lean, Topology.lean, etc.) and are part of the formal verification.

License

This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.
See the LICENSE file for details.

Author:Takeo Yamamoto

Citation

If you use this work in academic research, please cite:

@software{yamamoto_meta_axiom_2025,
  author       = {Takeo Yamamoto},
  title        = {Meta-Axiom: Mathematical-Philosophical Framework},
  year         = {2025},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.18603974},
  url          = {https://doi.org/10.5281/zenodo.18603974}
}