F-Theory: Cosmological Physics

This repository provides a formal Lean 4 verification of a unified cosmological framework based on the extremal principle of F-theory.

The framework establishes a rigorous structural correspondence between the two fundamental facets of reality: the Obverse (the material/physical manifestation) and the Reverse (the underlying mathematical/logical structure). By formalizing this duality, the theory derives cosmic expansion, dark matter, and dark energy as necessary logical consequences of systemic consistency.

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── Core Framework & Axioms

The mathematical foundation is formalized within Cosmology.lean and revolves around three primary axioms:

1. The Extremum Principle
   Systems and cosmic fields evolve to extremize (minimize) the governing action functional. The variation of the action $A$ vanishes identically: $$\delta A = 0$$

2. Obverse-Reverse Correspondence
   Material reality (Obverse) and logical/mathematical structures (Reverse) are linked via a continuous, structure-preserving mapping: $$\Phi: \text{Obverse} \longleftrightarrow \text{Reverse}$$

3. Logical Consistency
   The universe functions as a self-consistent formal system free of internal contradictions. The global logical error functional $C$ is strictly zero: $$C = 0$$

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── Key Derivations

Using the foundational axioms, the proofs contained in Derivations.lean mathematically derive several key cosmological phenomena:

• Cosmic Expansion: Emerges naturally from the dynamic tension and alignment between the Obverse and Reverse spaces.
• Dark Matter & Dark Energy: Formulated not as arbitrary physical particles, but as geometric and logical boundary conditions required to preserve global consistency.
• Mass-Energy Equivalence ($E=mc^2$): Derived rigorously as a direct consequence of hierarchical consistency and conservation laws within the F-theory framework.

Relationship to the Four Meta-Axioms

This cosmological system serves as a concrete physical application of the broader Four Meta-Axioms. The Extremum Principle instantiates Meta-Axiom 1, while the Obverse-Reverse correspondence serves as a direct manifestation of Meta-Axioms 2 (Topological Space) and 4 (Hierarchical Structure).

• See also: Meta-Axiom Repository

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── Repository Structure

├── .github/workflows/    # CI configuration for automated Lean 4 verification
├── Cosmology.lean        # Core definitions, axioms, and the Obverse-Reverse mapping
├── Derivations.lean      # Formal proofs of cosmic expansion, dark energy, and equivalence
├── LICENSE               # Apache License 2.0
└── NOTICE                # Copyright and attribution details