Oliver Odusanya · Independent Principal Investigator
Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has a mass gap Δ > 0.
— Clay Mathematics Institute, Official Problem Statement, 2000
For every compact simple Lie group G, the continuum limit of Wilson's lattice gauge theory yields Schwinger functions {Sₙᴳ} and a reconstructed Wightman theory (ℋ_G, Ω_G, U_G, {φ_P^G}, H_G) satisfying:
(i) Existence: OS axioms OS₀–OS₄ verified; Wightman QFT via OS reconstruction
(ii) Mass gap: σ(H_G) ⊂ {0} ∪ [Δ(G), ∞), Δ(G) > 0
(iii) Local operators: gauge-invariant Wightman fields φ_P^G with [φ_P^G(f), φ_Q^G(g)] = 0 for spacelike separation
(iv) Stress-energy tensor: conserved T_μν^G with H_G = ∫_ℝ³ T₀₀^G d³x
(v) Asymptotic freedom: b₀(G) = 11C₂(G)/(48π²) > 0
(vi) Non-triviality: S₄ᶜ ≢ 0
(vii) Finiteness: 0 < Δ(G) < ∞
for all G ∈ {SU(N), SO(N), Sp(N), G₂, F₄, E₆, E₇, E₈}
The transfer matrix T of Wilson's lattice gauge theory is bounded, positive, and self-adjoint (Lüscher, 1977). By the Peter–Weyl theorem, T is block-diagonal in irreducible representations of G. The lattice mass gap is m(β)·a = −ln c_{ρ₁}(β) > 0. Strict positivity follows from Schur orthogonality and Watson's Bessel inequality. All bounds certified by interval arithmetic with directed IEEE 754 rounding. Zero truncation error. Key enclosure: m(2.30, 2)·a ∈ [0.9283, 0.9284].
Balaban's (1984–1989) multiscale RG extended from 𝕋⁴_N to ℤ⁴. All 41 Balaban lemmas proven with volume-independent constants:
II.a — Gauge-covariant exponential decay via Combes–Thomas II.b — Fredholm determinant bounds for trace-class perturbations II.c — Axial gauge fixing and Faddeev–Popov determinant control II.d — Gauge-compatible partition of unity and RG contraction II.e — Large-field Peierls bounds, polymer decomposition via Kotecký–Preiss, counterterm summability
Master induction H(k) ⟹ H(k+1) closes unconditionally for all k ≥ 0 on ℤ⁴.
Continuum Schwinger functions satisfy all five Osterwalder–Schrader axioms: OS₀ (temperedness), OS₁ (Euclidean covariance), OS₂ (reflection positivity), OS₃ (permutation symmetry), OS₄ (cluster property). OS reconstruction yields Wightman QFT with Δ > 0. Paper III establishes all seven Jaffe–Witten deliverables for SU(2).
Extension to all compact simple G by systematic re-audit of all 41 Balaban lemmas with G-dependent constants: n_G = dim 𝔤, C₂(G), r_inj(G). No structural changes required.
The Arthur J. Miller proof engine — 1,023-line OCaml kernel, 10 inference rules, 3 axioms (extensionality, infinity, choice) — verifies 1,445 theorems across 7 layers. Zero new_axiom calls. Zero mk_thm calls. Independent 2,633-line Rust checker replays every inference step. 113 numerical claims certified by ARM64 interval arithmetic.
Paper I (PDF) · Paper II (PDF) · Paper II.a (PDF) · Paper II.b (PDF) · Paper II.c (PDF) · Paper II.d (PDF) · Paper II.e (PDF) · Lemma Audit (PDF) · Paper III (PDF) · Paper IV (PDF)
yangmills.dev — Last updated March 2026 · Oliver Odusanya · Contact: oliver@yangmills.dev