Spin-resolved double-trace thermal coefficients in holography

It was previously shown that the stress-tensor sector of the OPE, together with the KMS condition, fixes holographic thermal two-point functions at vanishing spatial separation. We extend this construction to nonzero spatial separation, where the KMS condition leaves a residual ambiguity that depends only on the spatial separation. We show that this ambiguity is fixed by the zero-frequency bulk wave equation, which can be solved analytically in terms of Heun functions. This gives an efficient method for computing thermal coefficients of double-trace operators resolved by spin. We also study the Lorentzian analytic structure of the resulting correlator and show that complex bulk-cone singularities which appear at spacelike separation in the stress-tensor sector do not persist in the full two-point function; they are resolved by the double-trace contribution.