You're absolutely right that, by definition, a verifier for an NP language must return false for all certificates if the instance is not in the language. That’s a fundamental property, and I fully agree with it.
What I intended in my construction is that the verifier accepts only if the parity check specified by the certificate actually fails — it recomputes the encoded value *_n(), verifies the parity check at index , and accepts only if there's a real mismatch. If the circuit is correct, then no matter what or you give, that mismatch shouldn't appear, so the verifier should always reject.
That said, I totally see how the way it's currently written might be unclear or misleading. It’s very possible that I need to make this logic more explicit, or even re-express the verifier definition to better align with the formal NP structure. I’ll take a closer look and make sure that the construction is not only correct, but clearly communicates the intended behavior.
Thanks again for pointing this out in such a precise way — it's exactly the kind of critique that helps me make this better.