Numerous scenarios explain the origin of the eukaryote cell by fusion or endosymbiosis between an archaeon and a bacterium (and sometimes a third partner). We evaluate these hypotheses using the following three criteria. Can the data be explained by the null hypothesis that new features arise sequentially along a stem lineage? Second, hypotheses involving an archaeon and a bacterium should undergo standard phylogenetic tests of gene distribution. Third, accounting for past events by processes observed in modern cells is preferable to postulating unknown processes that have never been observed. For example, there are many eukaryote examples of bacteria as endosymbionts or endoparasites, but none known in archaea. Strictly post-hoc hypotheses that ignore this third criterion should be avoided. Applying these three criteria significantly narrows the number of plausible hypotheses. Given current knowledge, our conclusion is that the eukaryote lineage must have diverged from an ancestor of archaea well prior to the origin of the mitochondrion. Significantly, the absence of ancestrally amitochondriate eukaryotes (archezoa) among extant eukaryotes is neither evidence for an archaeal host for the ancestor of mitochondria, nor evidence against a eukaryotic host.