The Yang-Mills Existence and Mass Gap: Non-Abelian Emergence and the Energy-Spectrum Signature of Topological Charge Locking

                                                                                                 Abstract

    Within the framework of generative mathematics, this paper provides a complete solution path for the Yang-Mills existence and mass gap problem. The core proposition: The mass gap Δ > 0 is the minimal positive energy gap of topological charge locking—i.e., the energy scale of the smallest non-trivial closed mode of the coupling network.

Complete argument chain:

1. L3 Theorem 3.1 (Non-Abelian Emergence): The self-evolution of a multi-component phase field necessarily gives rise to SU(N) gauge groups
2. Axiom 1 ρ > 0: The intrinsic cutoff forces the spectrum to be discrete; the smallest eigenvalue is strictly positive
3. Isoperimetric optimality (Lemma 3.12): Surviving configurations enforce uniform coupling, κ_min = κ_uniform > 0
4. Axiom 5 (Correlation Emergence): Enforces connectedness—any pair of pixels is linked by a path of non-zero coupling strength
5. Synthesis: Δ = 2π · κ_uniform / ρ > 0, which is the survivor signature of the external cutting of Axiom 4 on non-Abelian modes

All steps are rigorously guaranteed by the axiomatic system and the L1–L3 layer theorems. Classical original gauge field theory references are given.

    Unification with the Millennium Problems: The Yang-Mills mass gap, Riemann's Re(s)=1/2, BSD's rank = order of zero, and P vs NP's P ≠ NP share the same core mechanism—the critical signature of isoperimetric optimality on different problem domains. The mass gap Δ is the minimal positive eigenvalue of topological charge locking, Re(s)=1/2 is the self-dual midpoint of a single generator, rank = order of zero is the spectral-topological correspondence of two generators, and P ≠ NP is the landscape classification of zero versus non-zero topological charges.

    This paper belongs to the L4 Millennium Problem series of generativism, fully relying on the underlying axiomatic framework, and is mutually supported by the companion preprints on the Riemann Hypothesis and the BSD Conjecture.

Keywords: Yang-Mills; mass gap; non-Abelian emergence; topological charge; isoperimetric optimality; generativism