Conditional Lean 4 Formalization Exercises Around the Yang-Mills Mass Gap — Phases 1 & 2

This record collects Lean 4 files exploring how statements about a hypothetical Yang-Mills mass gap function could be organized formally. All physical content is assumed via explicit axiom declarations; the theorems proven are logical consequences of those assumptions.

─── WHAT THIS IS ───

A conditional formalization exercise in Lean 4. The repository contains ~442 theorems/lemmas across Phase 1 (strong coupling framework) and Phase 2 (RG flow & continuum limit preparation). Every substantive result is conditional on one or more of the 353 axiom declarations in the codebase.

─── WHAT THIS IS NOT ───

• It is not a proof of the Yang-Mills mass gap, in whole or in part.
• It does not construct the Yang-Mills measure, a Hilbert space, a Hamiltonian, or verify Wightman/Osterwalder-Schrader axioms — the actual content of the Clay problem.
• Percentages such as "~50% of the Millennium Prize Problem" from earlier versions were not defensible and are hereby retracted.
• The claim of "zero sorry statements" in earlier versions was incorrect — 21 files contain sorry statements, fully documented in AXIOM_AUDIT.md.

─── HONEST ACCOUNTING (v34, June 2026) ───

• axiom declarations: 353 (after Etapa 0 hygiene — down from 420)
• Theorems/lemmas: ~442
• Theorems unconditional (axiom-free beyond Lean/Mathlib): ~0 substantive
• sorry statements: present in 21 files — documented in AXIOM_AUDIT.md
• gemini_* axioms removed (LLM assertions, no probative value): 21 unused deleted; 75 Phase 2 load-bearing converted to explicit hypotheses
• Phase 2 Theorems 1–8: migrated from Float to ℝ (IEEE Float theorems say nothing about continuous physics)

─── METHODOLOGY ───

Coordinated by Jucelha Carvalho using multiple AI assistants (Claude Mythos Flabe 5, Claude Opus 4.5/4.6/4.7, GPT-5.2, Gemini 3 Pro, Manus AI 1.6) for drafting and auditing Lean 4 code. AI cross-validation is not equivalent to peer review or numerical simulation. Earlier claims of "numerical validation by Gemini" referred to LLM-generated assertions, not executed lattice computations, and are retracted.

─── ROADMAP ───

The realistic Phase 3 target is the formalization of the lattice strong-coupling mass gap (Osterwalder–Seiler 1978) — a known theorem, never formalized in Lean 4. See PHASE3_ROADMAP.md in the repository.

If citing, please cite as a conditional formalization exercise, not as progress on the Millennium Problem.

🏆 Awards & Recognition - 🥇 IA Global Challenge 2025 — 1st Place (440 solutions, 83 countries ) -
🌍 UN Tourism AI Challenge —