When Does Origin–Fixation Approximate Finite-Population Evolution?: A Finite-Horizon Target-Probability Benchmark with Exact Diagnostics, Stress Tests, and Long-Term Evolution Evidence

Simplified evolutionary models are useful only when the probability they assign to an event is close enough to the corresponding finite-population probability for the question being asked. We study this model-adequacy problem for finite-horizon fixation of target genotypes. A sequential origin–fixation continuous-time Markov chain is matched to an explicit haploid Wright–Fisher reference, and approximation error is treated as the primary object: its sign distinguishes under- from over-estimation, while a prespecified tolerance defines an endpoint-specific adequacy set. Exact one-locus calculations isolate a hidden finite-time cost of segregation and sweep. Across 54 independently specified hold-out cells, the sweep-time fraction φ = t_sweep/T strongly predicts absolute log-error (ρ = 0.923, p = 3.67 × 10⁻²³), and all 18 matched series contract monotonically as the horizon lengthens. A two-locus benchmark shows that sweep time remains informative but is not sufficient once mutation supply and multistep dynamics enter. In 36 predeclared biological stress conditions, every cell remains within a factor-two tolerance, yet route multiplicity, epistasis, and mutation-process heterogeneity shift signed error, with several paired perturbations producing non-additive responses. Long-term Escherichia coli data independently demonstrate strong mutation-state heterogeneity, mutation-spectrum differences, and multiple structural routes to citrate use. Taken together, these combined results replace a binary “valid/invalid” view with a reproducible, finite-horizon, event-specific map of approximation adequacy.