Supposedly, the reason they don't show up in mirrors is that mirrors used to be backed by silver and silver is elementally "pure."

Your geometric theorem reminds me of vampires having to count grains of sand. Maybe vampires are just giant nerds.

What is a castle doctrine?

Motor vehicles? Castle doctrine is about overriding the usual limitations (around what's reasonable use of force) on the right to self defence if you're in your home. Cars don't come into it.

Some places also extend the same protections that castle law provides to your home to your car, but that's separate from castle doctrine itself.

A, this is why magic isn’t real

I don't think the question is about magic vs reality, we're talking vampires here, right?, but to know how if vampires were a thing, it would be possible (or not) for a vampire cop to enter a house without being invited by the home owner, even with a warrant.

B, the law can say whatever the fuck it wants it still can’t bend reality.

The law doesn't need to bend any reality, it never does. Law is not about scientific laws, it's about human behavior. At least modern laws as some older ones may have tried, and maybe some particularly stupid modern ones too, but their ability to bend said reality to their will is still to be demonstrated :p

The law is a contract, with a sanction of some sort when it is broken. Be it to pay a fine, or to be prevented to do certain activities in the future, or be forced to do some other ones, or to go to jail. Up to the death penalty, in some places.

The law is about making the citizens bent to its will, not the reality.

That will is, in theory at least, is the expression of the common will, also known as the agreed upon desire of all the citizens. Citizens don't define laws of physics (which would deal with 'reality') and no matter how hard they may want, the also can't alter them.

Speed limit is not about enforcing a certain speed over which the laws of physics would suddenly (and magically) crumble. It’s about punishing people not respecting that agreed upon speed limit. That’s also why it’s very possible to have different speed limits in different places. Physics doesn't change, our expectations do.

We will drive faster on a highway than, say, next to a school despite the car being the same, with the same driver and with the same laws of physics applying, why? The place is different and also how we are expected to behave in such a place which, near a school, should obviously be to slow down so we the drivers (aka old/adults enough to have our driving license and act responsibly) can compensate for kids being… kids, aka not always being attentive to what’s going on around them, or being silly.

A warrant, for example has nothing to do with giving its carrier some magical power to enter a place (say by moving through a closed door or through walls, or by teleporting there) but it has all to do with punishing the owner of the place for not letting the warrant carrier enter their house, even if they don’t want to.

So, all I was saying is that in that ‘fantasy’ world where vampire cops would be a thing, the law may as well be written so it makes it a ‘mandatory welcoming' for the home owner to let in the vampire-cop, any refusal to comply to said 'forced invitation to enter' being sanctioned by a more or less severe punishment… Which, btw, is not far from what a warrant is supposed to be doing in our (this time, real) world ;)

Invited in by who though? You might say the owner, but then that means that kids or tenants don't count. So it might be "anyone with authority to do so", which would include judges following the prescribed process...

I meant: the warrant would equal an invitation to enter one's home, an invitation decided by the judge to which, as a law abiding citizen, the place owner would be forced to comply with.