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

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

The framework establishes a rigorous foundation for the conceptual laws of the universe, aiming to unify pure mathematics, theoretical physics, and biological structures through minimality, consistency, topology, and hierarchical organization.

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── Theoretical Overview: The Four Meta-Axioms

The entire framework is grounded in the Extremum Principle and structured around four fundamental pillars:

1. Extremum Principle ($A_1$)
   Every systemic behavior or state is determined by the extremization of a governing functional $L(x)$: $$F[x] = \text{Extremum } L(x)$$

2. Topological Space ($A_2$)
   The underlying domain of states $x$ is embedded within a well-defined topological space $X$: $$x \in X \subset \mathbb{R}^n$$

3. Logical Consistency ($A_3$)
   The system satisfies strict, internal logical consistency, meaning the contradiction or error functional $C[F]$ vanishes identically: $$C[F] = 0$$

4. Hierarchical Structure ($A_4$)
   Macro-level laws and behaviors emerge as a weighted summation or convergence of micro-level dynamics: $$F_{\text{macro}} = \sum w_i \cdot F_{\text{micro}}(i)$$

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── Status: Fully Proven (CI-Verified)

All theorems, definitions, and applications within this repository are fully proven via the Curry-Howard correspondence. There are no sorry placeholders in the main branch. Continuous Integration (CI) via GitHub Actions ensures the integrity and axiomatic consistency of the proof state.

Key Resolved Proofs

• compact_extremum_exists: Proof of the existence of extrema on compact topological spaces using IsCompact.exists_isMin and exists_isMax ($A_2 \implies A_1$).
• consistency_preserved: Mathematical proof showing the invariance and preservation of logical consistency under the transformation of Meta-Axioms ($A_3$).
• unity_principle: Formal derivation of hierarchical convergence, proving how localized micro-dynamics scale into unified macro-structures ($A_4$).
• shannon_entropy_nonneg: Non-negativity of Shannon entropy derived rigorously via log-convexity properties, supporting the information-theoretic applications.

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── Repository Structure

├── .github/workflows/    # CI configuration (ci.yml) for automated Lean 4 verification
├── Metaaxiom.lean        # Core definitions of the 4 Meta-Axioms and consistency proofs
├── Axioms.lean           # Auxiliary lemmas, topological properties, and foundational math
├── CMA.lean              # ComputeMetaAxioms framework for algorithmic validation
├── Derivations.lean      # Minkowski trace proofs and algebraic simplifications
├── Ftheory.lean          # Application to cosmology, fundamental physics, and F-theory
├── Collatz.lean          # Structural analysis and constraints on the Collatz Conjecture
├── Dna.lean              # Application to genetic systems and information-theoretic bound
└── Medical.lean          # MetaRepair module for biological systems & homeostatic optimization