How many genes have we discovered that determine your ZIP code?

Har dee har. Non-whites were forbidden from moving into many zip codes by law until, IIRC, around 1958. Real estate agents kept enforcing those laws long after they were struck down, and there are still tons of properties that have "racial covenants" which forbid* the current owner selling to a non-white buyer. And such neighborhoods change their composition very slowly.

So no, the causative effect is the reverse of your implication: there are many well documented zip codes that significantly determine your genetics

* though enforcement of such clauses wouldn't stand in court today

Zero, that's the point ;) !

How can you prove they don't?

Do you really think that there wasn't someone a century ago that made the same point ? It took a while, but here's the answer:

https://en.wikipedia.org/wiki/Twin_study

I'll let you find the outcome. It's not going to be what you want to hear: different races are different. In just about anything. That's exactly what nature is trying to achieve by having races, and therefore shouldn't come as a surprise to anyone.

Being a resident of Earth is also heritable.

But does it solve the diamond inheritance problem ?