Is it possible that only the state vector exists, and the 3D-space, a preferred basis, a preferred factorization of the Hilbert space, and everything else, emerge uniquely from the Hamiltonian and the state vector?

In this article no-go theorems are given, showing that whenever such a candidate preferred structure exists and can distinguish among physically distinct states, physically distinct structures of the same kind exist. The idea of the proof is very simple: it is always possible to make a unitary transformation of the candidate structure into another one of the same kind, but with respect to which the state of the system at a given time appears identical to a physically distinct state (which may be the state at any other time, or even a state from an "alternative reality"). Therefore, such minimalist approaches lead to strange consequences like "passive" travel in time and in alternative realities, realized simply by passive transformations of the Hilbert space.

These theorems affect all minimalist theories in which the only fundamental structures are the state vector and the Hamiltonian (so-called "Hilbert space fundamentalism"), whether they assume branching or state vector reduction, in particular, the version of Everett's Interpretation coined by Carroll and Singh "Mad-dog Everettianism", various proposals based on decoherence, proposals that aim to describe everything by the quantum structure, and proposals that spacetime emerges from a purely quantum theory of gravity.