Lean4 Yang-Mills Mass Gap Proof WIP Checkpoint

1. Progress Achieved A. Retirement of the no_normal_subgroups Axiom Before (May 12, 2026): The simplicity of the gauge group (IsSimpleLieGroup SU3) was classified as a custom AXIOM on the critical path, meaning the codebase was logically dependent on unproven assumptions. After (May 20, 2026): This axiom was fully proved and retired (commit 255285c, tagged wilsonrp-no-normal-subgroups-closed). Running #print axioms IsSimpleLieGroup SU3 now returns only standard Lean axioms. The Proof Path: The simplicity was proved by building a bridge through standard Lie theory: Proving $SU(3)$ is a connected space (ConnectedSpace SU3, commit 36e2e48). Establishing a Lie-ideal bridge for connected normal subgroups of $SU(3)$ (commit 90edf14). Verifying the simplicity of the Lie algebra model (SU3LieAlgebraModel) and closing the discrete branches (commit e618ead). B. From Open Targets to Structured Scaffolding Before: Major reconstruction fields (including the Osterwalder-Schrader transition, the Wightman limit, and the uniform spectral bounds) were classified as completely OPEN, with no code terms existing yet. After: These fields have been upgraded to PROVIDED (scaffold) status. The proof pipeline has been fully mapped out, with placeholder structures and templates established for the entire lattice-to-continuum lift. 2. Going Forward This structural progress shifts the entire nature of the project: The Gauge-Algebra Base is Completed: The pure mathematics of the gauge group $SU(3)$—its compactness and simplicity—is now fully closed under standard Lean axioms. There is no longer any mathematical vulnerability on the group-theoretic side of the proof. The Proof "Plumbing" is Syntactically Complete: By transitioning the RunSeed fields to PROVIDED (scaffold), have successfully built the logical scaffolding that connects the discrete lattice configuration directly to the continuous Wightman limit. The compiler verifies that if the analytic bounds are met, the final Clay Millennium target (YangMillsExistenceAndMassGap SU3) logically follows. The New Critical Path is Pure Analysis: The remaining open work is no longer about setting up definitions or proving Lie theory. The frontier has narrowed exclusively to real analytic estimation. To close the remaining scaffolds, must replace the template terms with true non-perturbative proofs: Proving the analytic uniform bound for the field $a_1$ under the renormalization group flow. Moving from a template Osterwalder-Schrader scaffold to a full, analytic reconstruction theorem to lift the discrete lattice theory to a continuous, relativistic Wightman QFT on Minkowski space.