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We develop conservative tests for the mean of a bounded population under stratified sampling and apply them to risk-limiting post-election audits. The tests are "anytime valid" under sequential sampling, allowing optional stopping in each stratum. Our core method expresses a global hypothesis about the population mean as a union of intersection hypotheses describing within-stratum means. It tests each intersection hypothesis using independent test supermartingales (TSMs) combined across strata by multiplication. A P-value for each intersection hypothesis is the reciprocal of that test statistic, and the largest P-value in the union is a P-value for the global hypothesis. This approach has two primary moving parts: the rule selecting which stratum to draw from next given the sample so far, and the form of the TSM within each stratum. These rules may vary over intersection hypotheses. We construct the test with the smallest expected stopping time, and present a few strategies for approximating that optimum. Approximately optimal methods are challenging to compute when there are more than two strata, while some simple rules that scale well can be inconsistent -- the resulting test will never reject for some alternatives, no matter how large the sample. We present a set of rules that leads to a computationally tractable test for arbitrarily many strata. In instances that arise in auditing and other applications, its expected sample size is nearly optimal and substantially smaller than that of previous methods. 

The U.S. state of Georgia was central to e!orts to overturn the results of the 2020 Presidential election, including a phone call from then-president Donald Trump to Georgia Secretary of State Brad Ra!ensperger asking Ra!ensperger to ‘find’ 11,780 votes. Ra!ensperger has maintained that a ‘100% full-count risk-limiting audit’ and a machine recount agreed with the initial machine-count results, which proved that the reported election results were accurate and that ‘no votes were flipped.’ While there is no evidence that the reported outcome is wrong, neither is there evidence that it is correct: the two machine counts and the manual ‘audit’ tallies disagree substantially, even about the number of ballots cast. Some ballots in Fulton County, Georgia, were included in the original count at least twice; some were included in the machine recount at least thrice. Audit handcount results for some tally batches were omitted from the reported audit totals: reported audit results do not include all the votes the auditors counted. In short, the two machine counts and the audit were not probative of who won because of poor processes and controls: a lack of secure physical chain of custody, ballot accounting, pollbook reconciliation, and accounting for other election materials such as memory cards. Moreover, most voters used demonstrably untrustworthy ballot-marking devices; as a result, even a perfect handcount or audit would not necessarily reveal who really won. True risk-limiting audits (RLAs) and rigorous recounts can limit the risk that an incorrect electoral outcome will be certified rather than being corrected. But no procedure can limit that risk without a trustworthy record of the vote. And even a properly conducted RLA of some contests in an election does not show that any other contests in that election were decided correctly. The 2020 U.S. Presidential election in Georgia illustrates unrecoverable errors that can render recounts and audits ‘security theater’ that distract from the more serious problems rather than justifying trust.