When you compute some nice and elegant result, dissipated heat is an undesired side effect. But let's face it: we are speaking about proof of work. Proof of work means that a computed has run during some "required" time. In other words, you have to prove that enough heat has been dissipated. Waste of energy actually is "by design" here.

I'm not sure if you're trolling. Of course that's nonsense. The work is entirely the (artificially complex) computations necessary to get to the result. If someone were to invent a 100% efficient computer, based on superconductors, which produces no heat at all, the proof of work (the final hash value) would still be equally valid. As I said, heat is an undesired, unavoidable side-effect. You don't show anybody the heat you produced, to convince them that you did the work, you show them the hash value, otherwise you could just burn some wood.