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> How can the system prove that my vote counted correctly if I can’t personally verify who my vote was counted for?

In the Scantegrity sysem you can personally verify who your vote was counted for.

You should learn about off-the-record messaging, zero-knowledge proofs and blinded signatures. It's not obvious that centralized trust is required and in fact it isn't.

Zero knowledge proofs come to mind. You could prove you voted for one of a set of people without saying which one