Unconditional Proof of the Existence and Mass Gap of Four-Dimensional Yang-Mills Theory

Unconditional Proof of the Existence and Mass Gap of Four-Dimensional Yang-Mills Theory Abstract This paper presents a rigorous proof of the existence and mass gap of four-dimensional Yang-Mills theory for the compact simple Lie group SU (N) (N ≥ 2), resolving the Clay Mathematics Institute Millennium Problem (Jaffe-Witten, 2000). The proof proceeds in four stages. Stage One (Chapters 0–1): Establish the problem statement and the rigorous mathematical foundations of lattice gauge theory. Define the Wilson action and partition function on a finite lattice, prove the absolute convergence of the SU (N) character expansion, and establish the lattice duality identity. Section 1. 4 provides a complete constructive correspondence from representation assignments to string-net configurations, including the rigorous verification of the bijective mapping and weight correspondence. Stage Two (Chapters 2–3): Independently construct the magnetic monopole string-net model. Define the string-net configuration space Ω with Polish topology. Construct the Poisson process measure μ₀ as a reference measure, define the Gibbs measure μₛtring via the Radon-Nikodym derivative e^-E, and prove the existence and uniqueness of the thermodynamic limit. Through low-energy homotopy classification, prove that low-energy states are in one-to-one correspondence with integer partitions, yielding the partition function series representation Z = Σ p (n) e^-Δn. Rigorously prove the upper bound ||δH|| ≤ 3Δ/8 0 is derived. Stage Three (Chapters 4–5): Prove the mass gap. Define the Hamiltonian Hₛtring on the Hilbert space L² (Ω, μₛtring) and prove its self-adjointness. Through the low-energy effective space decomposition Hₑff = ⊕₍=₀∞ Hₙ (dim Hₙ = p (n) ), prove the uniqueness of the ground state (E₀ = 0) and the lower bound E₁ ≥ 3Δ/4 > 0, yielding the string-net mass gap μₛtring ≥ 3Δ/4 > 0. Define the Yang-Mills measure μYM via the continuum limit of the duality map, and prove the transfer of the mass gap μYM = μₛtring ≥ 3Δ/4 > 0. Stage Four (Chapters 6–7): Verify the Osterwalder-Schrader axioms and complete the Minkowski lifting. Prove that μYM satisfies OS1–OS5, analytically continue to Minkowski spacetime via the OS reconstruction theorem, and verify all Wightman axioms. This proof is completely independent of lattice gauge theory numerical simulations and perturbative expansions, relying solely on standard mathematical tools: functional analysis, probability theory (Kolmogorov extension theorem, Prokhorov theorem), representation theory (Peter-Weyl theorem, Weyl character formula, Schur-Weyl theorem), differential geometry (Frobenius theorem, Uhlenbeck compactness theorem), and number theory (Hardy-Ramanujan formula). 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