We derive the celestial optical theorem from the S -matrix unitarity, which provides nonperturbative bootstrap equations of conformal partial wave (CPW) coefficients. This theorem implies that the imaginary parts of CPW coefficients exhibit a positivity property. By making certain analyticity assumptions and using the celestial optical theorem, we derive nonperturbative constraints concerning the analytic structure of CPW coefficients. We discover that the CPW coefficients of four massless particles must and can only have simple poles located at specific positions. The CPW coefficients involving massive particles exhibit double-trace poles, indicating the existence of double-trace operators in nonperturbative celestial conformal field theory. It is worth noting that, in contrast to AdS/CFT, the conformal dimensions of double-trace operators do not have anomalous dimensions.