ATK Formal Proof Archive: Branch-Level Kernel Verification Certificates, Theorem Provenance, and Reproducible Evidence Across Mathematical Research Repositories

ATK Formal Proof Archive: Branch-Level Kernel Verification Certificates, Theorem Provenance, and Reproducible Evidence Across Mathematical Research Repositories Abstract The ATK Formal Proof Archive is a proposed research dataset for preserving and examining branch-level formal verification evidence across multiple mathematical research repositories. The archive is designed to record verified Lean declarations, source revisions, compiler configurations, axiom dependencies, and reproducible execution results. According to the preliminary project report, 132 branches containing 4,842 successfully checked theorem declarations had been published with associated certificates. These figures are provisional and require independent reconciliation against the original repository commits, source files, and verification records. The archive distinguishes kernel-checked declarations from conditional theorems, incomplete formalizations, unresolved mathematical obligations, and claims not established by the available formal evidence. 1. Branch-Level Verification Each repository branch is treated as a versioned collection of formal declarations. Verification records identify the exact commit and compilation environment, allowing results to be associated with immutable source revisions. The archive avoids treating branch names or theorem counts as substitutes for proof evidence. 2. Certificate Structure A verification certificate should identify the formal declaration, source file, source checksum, compiler version, dependency versions, and execution result. Where applicable, the certificate records the axioms on which the declaration depends. The archive distinguishes verification of a Lean theorem statement from verification that the statement faithfully expresses an intended mathematical conjecture. 3. Mathematical Research Coverage The proposed dataset organizes formal work associated with several major areas of mathematics, including: - Analytic number theory and the Riemann zeta function. - Computational complexity and P versus NP. - Partial differential equations and Navier–Stokes. - Gauge theory and Yang–Mills. - Arithmetic geometry and the Birch and Swinnerton-Dyer conjecture. - Algebraic geometry and the Hodge conjecture. - Exponential Diophantine equations and the Beal conjecture. Inclusion of a repository in the archive does not imply that its associated conjecture has been resolved. 4. Theorem Classification Each declaration is assigned a verification category. The archive distinguishes fully checked theorems from statements depending on additional mathematical hypotheses, declarations containing axioms, and incomplete proof obligations. Compiler or infrastructure failures are recorded separately from mathematical incompleteness. 5. Deduplication and Source Provenance The archive includes a methodology for distinguishing unique declarations from copies, generated variants, and declarations repeated across branches. Deduplication must use a documented equivalence criterion. Source hashes and verification manifests provide traceability between archived records and their original repository states. 6. Evidence Integrity Large verification datasets may exceed repository hosting limits. The archive therefore supports partitioned records, deterministic manifest generation, and checksum-based integrity validation. Every partition must remain traceable to a complete logical dataset. 7. Scope and Limitations The archive is a collection of formal verification evidence. It does not claim unconditional solutions to the Millennium Prize Problems. A verified Lean declaration establishes the validity of its formal statement relative to the underlying logical framework and declared assumptions. Additional mathematical analysis is necessary to establish whether that statement resolves an external open problem. Preliminary Reported Results - Branches reported with certificates: 132. - Reported checked theorem declarations: 4,842. - Additional branch-only declarations reported: 43,366. - Branch-only declarations reported as checked in an interim aggregation: 5,098. - Reported unique declarations in default branches: 19,182. These values originate from preliminary execution reports and must be reconciled before publication. They should not be combined into a grand total without explicit deduplication and snapshot consistency checks. Research Significance The archive aims to provide transparent, reproducible evidence for large-scale formal mathematical research. It supports independent auditing, provenance tracking, proof-status comparison, and the long-term preservation of computational verification records.