Here is the line of argument
It is based on a "fixed point" mathematical argument.
The laws of nature are the same regardless of how we assemble subsystems into a larger one, or how we partition a system into subsystems. This is the core idea. There are three possible solutions:
- Hyperbolic (not present in nature)
- Parabolic (classical mechanics)
- Elliptic (quantum mechanics)
The result is based on the algebraic formulation of quantum mechanics which is a better formulation over the Hilbert space formulation in view of Araki-Haag-Kastler axioms and Haag's theorem.