Formal Verification of the Order of Integer Solutions (OIS) Axiomatic Framework in Lean 4

This repository contains the formal verification of the Order of Integer Solutions (OIS) framework, a unified axiomatic foundation that redefines mathematical existence through the conservation of "Order". The OIS meta-theory constraints the existence of mathematical objects based on the stability of their representations within a structural support 1. Core Axiomatic System The provided Lean 4 source code (OIS_v4_clean.lean) formalizes the four fundamental axioms of OIS4: O1 (Well-Ordering of Complexity): Establishes that complexity measures in discrete supports are well-ordered, prohibiting infinite strict descent. O2 (Lattice Closure / Non-Leakage): Asserts that an object exists in a support $S$ if and only if its minimal representation is closed within $S$ O3 (Non-Recursive Stability): Prohibits unstable recursive representations that would induce infinite descent O4 (Law of Order Conservation): Posulates the conservation of total information capacity, leading to the Universal Trichotomy of Condensation, Dispersion, and Collapse 2. Key Formalization Results The repository includes a secondary verification (Goodstein_OIS.lean) demonstrating the logical equivalence between the OIS framework and the termination logic of the Goodstein Sequence. Result: The Lean 4 proof assistant confirms the "No goals" state, verifying that OIS's descent axioms correctly capture the termination properties of transfinite-order sequences. Implication: This validates OIS as a rigorous framework capable of explaining Diophantine finiteness and other structural impossibilities (e.g., FLT, Faltings' Theorem) through machine-checked logical consistency. 3. Files Included OIS-V2.pdf: The complete theoretical paper. OIS_v4_clean.lean: Core axiomatic formalization. Goodstein_OIS.lean: Equivalence proof with Goodstein's Theorem. Verification_Screenshot.png: Screenshot showing the "Goals accomplished!" state in the Lean InfoView. 4.Mathematical Taxonomy: MSC2020: 11Dxx (Diophantine Equations), 03B35 (Automated Theorem Proving). Regimes: Condensation, Dispersion, Collapse