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. It resolves the googly problem: the self-dual sector is the T-conjugate of the anti-self-dual sector, with both arising from a single Ward construction applied to a T-symmetric pair of spacetimes sharing a common conformal boundary at null infinity. The self-dual sector (positive helicity, the googly modes) was never absent from nature; it resides in the T-conjugate spacetime and communicates with our sector through the shadow transform at the shared celestial boundary. The Ward construction on PT, together with its T-image, encodes both helicity sectors; the missing element was the identification of the shadow transform with time reversal. 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 left-handed character of the Standard Model weak interaction, the 4π periodicity of spin-½ representations, and the Majorana nature of neutrinos follow as corollaries. The algebraic skeleton of these results has been formalized in Lean 4; the full geometric content is the subject of the present proof.