Does a superconductor truly have 0 Ω resistance, or is it only arbitrarily close?

A true-zero resistance would, to my understanding, cause problems with implementing Ohm's Law. Additionally, I believe there is still a non-negligible degree of Joule heating, which leads me to think that the resistance of the material is some fraction above true zero.
Exactly zero.
Flux lines in a type two superconductor
There is no issue with violating Ohm's law: Ohm's law is an empirical law which applies to certain materials, but it isn't true in general. In the same vein, we have no issue applying Ohm's law on circuits with ideal conductors: superconductors are just a practical implementation of what we do in circuit analysis all the time.
Joule heating only applies to materials with finite resistance, and a superconductor has no resistance. This is trivial from the fact that no power is expended by sending current through a superconductor, so no heating can occur (violation of conservation of energy).

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