> The key idea is that the verifier doesn’t go looking for a parity check that fails — it just verifies the one that’s provided. So r isn’t there because the verifier couldn’t find a violation, but because it doesn’t try to; it just checks whether the one it's told about actually fails.

Then you failed. When using the certificate and verifier definition of NP, if the correct answer to the problem instance is false, there must be no certificate that makes the verifier return true.

A nondeterministic Turing machine is one that can clone itself and take two different paths without taking extra time. The copies can also make copies so it can clone as many times as it needs to. Copies that return false delete themselves, and if ANY copy returns true the overall answer is true (and we can delete all the copies). Any problem that can be solved by one of these is in NP. That's the definition of NP.

Equivalently, if we already know which one is going to return true, each time the machine wants to clone itself, we can tell the machine which copy is going to return true, and then it either continues being itself or pretends to be the clone, whichever one we told it, without actually cloning. Then we can run this on a normal Turing machine that doesn't clone itself. Of course, if we tell it wrong, it will return false even though it could have returned true if we told it right. This leads to the certificate definition of NP. The data we tell the machine about "the right way to pretend to clone" is the certificate.

Both are equivalent ways to define which problems are NP. The cloning version is the actual definition, but the certificate version is often more convenient, because, ya know, actual computers don't clone themselves.

If the answer to the problem is false, in the cloning version, all copies of the machine return false. That means in the certificate version, which only runs one copy, it will return false no matter which copy we choose, i.e. no matter which certificate we give it. Even if we give the wrong certificate, it's not possible to get a certificate machine for an NP problem to return true when the correct answer is false.

Your verifier machine is claimed to be a certificate machine that solves your problem, but you also say it sometimes returns true when the correct answer is false, if you give the wrong certificate. So there's something wrong with the setup here. I think the verifier machine is just wrong as it doesn't do what it needs to do to prove what you think it's proving.