A precision relation between (K ^+-)(t) and B(KL ^+-)/B(KL æ)

We find that the phase appearing in the unitarity relation between B(K-L mu(+)mu(-)) and B(K-L -> yy)is equal to the phase shift in the interference term of the time dependent K mu+mu- decay. A probe of this relation at future kaon facilities constitutes a Standard Model test with a theory precision of about 2%. The phase has further importance for sensitivity studies regarding the measurement of the time-dependent K mu(+)mu(-) decay rate to extract the CKM matrix element combination |VtnVtd sin(beta + beta n)| approximate to A(2)lambda(5)eta. We find a model-independent theoretically clean prediction, cos(2) phi(0) = 0.96 +/- 0.03. The quoted error is a combination of the theoretical and experimental errors, and both of them are expected to shrink in the future. Using input from the large -NC limit within chiral perturbation theory, we find a theory preference towards solutions with negative cos phi(0), reducing a four-fold ambiguity in the angle phi(0) to a two-fold one.