Spinors, Doppelgängers and the Non-Perturbative Classical Double Copy

We develop a geometric and spinorial formulation of the non-perturbative classical double copy between the Corrigan–Fairlie–'t Hooft–Wilczek sector of complexified SU(2) Yang–Mills theory and four-dimensional conformally flat gravity. We identify the gauge potential with the positive-chiral spin connection of the conformally flat metric, establishing an exact relation between the Yang–Mills field strength and the chiral projection of the Riemann tensor. This provides a four-dimensional realisation of the doppelgänger viewpoint, in which gauge and gravitational fields arise from different interpretations of common connection data. Within this sector, the sourceless Yang–Mills equations require constant scalar curvature, explaining the shared nonlinear scalar equation and extending the gravitational correspondence from traceless matter to matter with constant energy–momentum trace. By representing adjoint colour indices as symmetric pairs of spinor indices, we derive exact relations mapping the gauge field-strength spinors to the scalar-curvature and trace-free Ricci components of the gravitational curvature. These relations retain nontrivial curvature information despite the vanishing Weyl tensor. We illustrate the construction using anti-de Sitter and de Sitter spacetimes, together with ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\times \mathbb{R}$ supported by a scalar field, demonstrating the extension to matter with nonzero constant trace