We have this result already. Interaction with thermal noise ruins wave coherence, and causes classical physics to emerge. This comes when you in a sense replace waves with their average values.
It's not so much a question of scale, but of statistical noisiness. Quantum effects are primarily observable in the low-temperature domain (this is not necessarily "thermometer temperature", but a statistical measure)
I am not a physicist, but I thought that it is generally accepted that decoherence does not solve the measurement problem. It explains why quantum effects are not observable in a noisy environment but still does not explain the non-unitarity of measurements.
It does not solve the measurement problem, but it does explain why you don't see quantum phenomena such as superposition or tunneling at the classical limit.
This doesn't answer the question. We know how classical physics emerges from a large number of quantum events through statistics. That was not the point of Schrödinger's cat though experiment! The idea is to link a macroscopic event to a random subatomic event that is in a superposition. When or how does the wave function collapse, does it even collapse?
It's not so much a question of scale, but of statistical noisiness. Quantum effects are primarily observable in the low-temperature domain (this is not necessarily "thermometer temperature", but a statistical measure)