We develop a functional-analytic framework for modular constraints in two-dimensional conformal field theory based on the reflection symmetry β ↦ 4π 2 /β. This symmetry isolates a natural reflection-odd subspace of L 2 (β/β) in which all deviations from modular self-duality live. For any modular-invariant partition function, the vacuum–heavy decomposition determines a uniquely defined reflection-odd modular defect D(β) whose completed Mellin transform satisfies an odd functional equation. Pairing D with compactly supported reflection-odd test functions yields a structured hierarchy of linear functionals. Under mild assumptions on the light spectrum, these pairings obey positivity conditions that lead to smeared Cardy inequalities comparing the heavy spectrum to the Cardy density through positive Laplace kernels. We also show that the vanishing of all such pairings is equivalent to averaged Cardy saturation in L 2 (dβ/β), giving a complete reflection-odd characterization of modular self-duality. This provides a simple and flexible functional basis for modular bootstrap analyses, replacing ad hoc constructions with a unified reflection-odd hierarchy suited for both analytic arguments and numerical implementation.
Douglas F. Watson (Fri,) studied this question.