The Axioms of Quantum Field Theory as Theorems of Haar Measure

We derive the six Wightman axioms of relativistic quantum field theory as theorems of three mathematical inputs: Haar measure on the Grassmannian Gr(2,4), the Penrose twistor correspondence, and the Peter-Weyl decomposition of L^2 spaces on compact groups. The Hilbert space is L^2(Gr(2,4), dmu_Gr), the vacuum is the unique SU(4)-invariant vector, Poincare covariance arises from the conformal embedding P+ into SU(2,2), the spectrum condition from forward-tube analyticity of the Penrose transform, locality from twistor non-incidence (spacelike separation implies non-incident twistors, non-incidence implies holomorphic integrand, holomorphicity implies vanishing contour integral by Cauchy's theorem), cyclicity from irreducibility of the vacuum sector, and temperedness from elliptic regularity on the compact Grassmannian. The CPT theorem emerges from Haar measure self-duality (the map Lambda->Lambda-perp is the geometric CPT). The spin-statistics connection emerges from the topology of the Plucker line bundle. The Osterwalder-Schrader axioms follow by Wick rotation. No physical postulate enters the argument. The Wightman axioms are necessary consequences of Haar measure on the space of light rays.