Twistor Theory and the Resolution of the Googly Problem via Haar Measure Self-Duality on Gr(2,4)

Penrose's twistor programme, initiated in 1967, encodes massless fields through holomorphic structures on projective twistor space PT ≅ CP³. The Penrose–Ward correspondence gives a bijection between anti-self-dual solutions of the vacuum Yang–Mills and Einstein equations and holomorphic vector bundles over PT trivial on every twistor line, but naturally produces only one helicity sector. Completing the construction to include both helicities from a single geometric object is the obstruction Penrose named the googly problem. We prove, using the antiunitary character of the time-reversal operator T, that the shadow transform Δ ↔ 2 − Δ of celestial conformal field theory is precisely T acting on massless fields at null infinity. This identification is forced by self-dual Haar measure on the Grassmannian Gr(2,4), the compact parameter space of null rays in complexified Minkowski space, and is derived through an explicit three-step chain: the orthogonal complement involution on Gr(2,4) acts as the Hodge star on Plücker coordinates, induces the antipodal map on the celestial sphere via the Penrose correspondence, and forces energy inversion ω ↦ ω⁻¹ through the spinor symplectic pairing — which under the Mellin transform sends Δ ↦ 2 − Δ. The googly problem is resolved: the self-dual sector (positive helicity, the googly modes) resides in the T-conjugate spacetime and communicates with our sector through the shadow transform at the shared celestial boundary. Both sectors are unified in a Yang–Mills twistor datum (E, Ẽ, φ) where E is the Ward bundle encoding the ASD sector, Ẽ = T(E) is the shadow bundle encoding the SD sector via conjugate transition functions, and φ is a shadow pairing satisfying the Knizhnik–Zamolodchikov equations at the shared conformal boundary. Dual twistor space PT* is not an independent structure; it is the T-image of PT. We further prove that every massive Dirac fermion samples both helicity sectors within each Compton period through zitterbewegung oscillation across the T boundary at frequency ω = 2mc²/ħ, making both helicities simultaneously accessible to forward-time observers. The oscillation arises from interference between positive-energy solutions on M⁴₊ and negative-energy (Feynman–Stueckelberg) solutions on the T-conjugate sector M⁴₋, mediated at their shared conformal boundary. The left-handed character of the Standard Model weak interaction, the 4π periodicity of spin-1/2 representations (derived topologically from the fundamental group π₁(Spin(3) ×_T Spin(3)) ≅ ℤ₄), and the Majorana condition on neutrinos all follow as corollaries of the two-sided T-boundary structure. The paper derives connections to Conformal Cyclic Cosmology (the Weyl Curvature Hypothesis is derived as a theorem, not postulated), CMB homogeneity without inflation, and the Riemann Hypothesis (the shadow symmetry Δ ↔ 2 − Δ becomes the functional equation s ↔ 1 − s under the identification Δ = 2s, with Haar self-duality on Gr(2,4) and on (ℝ⁺, ×) being two restrictions of the same identity on GL(4, 𝔸)). Three-point amplitude predictions are verified numerically to 4×10⁻¹⁵ relative error. The googly obstruction was not a deficiency in the geometry of PT; it was a failure to identify the shadow transform with time reversal.