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Any sort of software for voting machines really ought to be open-source. Anything less than complete transparency is a non-starter.

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Not ought but must. It's a fundamental concept in system security known as as Kerckhoffs's Principle. https://en.m.wikipedia.org/wiki/Security_through_obscurity

The rationale is to allow anybody to audit the system, so that soundness of the design can be proven against an adversary with perfect knowledge of it.

And yet Americans let companies enforce intellectual property rights over systems whose security (flaws) could change your vote...

update: Bruce Schneier on security of electronic voting: https://www.opendemocracy.net/media-voting/article_2213.jsp

And how do you verify the machines are running that code?

And how do you know someone's just not adding to the total when it's collated?

You have redundant integrity checks involving cryptography and independent, human professionals built into the compilation/deployment/testing process.

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