Exact discrete Wright-Fisher fixation probabilities and conditional absorption times characterize selection boundaries across finite populations

Does continuous diffusion accurately capture discrete Wright-Fisher allele fixation across finite populations? Solving exact absorbing Markov chains reveals that Kimura's diffusion approximation systematically overestimates single-mutant fixation probability, with relative error scaling from 0.29% at selection coefficient 0.01 in population size 100 to 3.31% at selection coefficient 0.1 in population size 20. Using Doob $h$-transforms, conditional absorption times demonstrate trajectory time symmetry: fixation times for deleterious and beneficial alleles differ by at most % across all evaluated populations and selection coefficients up to . In the neutral limit, discrete fixation probabilities match the theoretical inverse population size with residual , and conditional absorption time for population size reaches generations, within % of the asymptotic bound of generations.