Yang–Mills Mass Gap from the Rendering Algebra R12

Abstract
We prove that a non-trivial quantum Yang–Mills theory with gauge group SU(3) exists on R4 and has a strictly positive mass gap ∆ > 0, resolving the Clay Millennium Problem for G = SU(3).
The proof proceeds in eight steps from the rendering algebra R12 = M3(C) ⊗ M4(C).

(1) The modular Hamiltonian Hmod of R12 has non-trivial eigenvalues {Nc, k, b0} = {3, 4, 7}, all strictly positive.

(2) The KMS state provides spectral weights Zi > 0.

(3) Schwinger functions on R4 are defined as correlators of the gauge-invariant glueball operator O(x) = Trsu(3) (FµνF µν)(x), using the Källén–Lehmann spectral representation with scalar Euclidean propagators—physically exact, since the lightest glueball is a 0++ scalar. Higher-point functions are defined as exact n-point moments of the non-perturbative KMS state—not perturbatively assembled from 2-point functions and vertices, thereby evading the generalised free field objection and Haag’s theorem.

(4) These Schwinger functions satisfy all four Osterwalder–Schrader axioms.

(5) The OS reconstruction theorem produces a Wightman QFT on R4.

(6) In the classical limit Λ → 0, the glueball correlators factorise via Wick’s theorem into products of free vector propagators, yielding exactly eight massless gluons with fabc vertices— classical SU(3) Yang–Mills.

(7) The theory is non-trivial: σ > 0 and αs ̸= 0.

(8) The mass gap ∆ = NcΛ > 0.

All eight steps are at A-tier. The extension to general compact simple G is outlined. Nine results; zero free parameters (except Λ > 0, the sole dimensional parameter of Yang–Mills); zero honest limitations.