This thread is actually clarifications from a post you were discussing with Demystifier and someone called Electric to be (in two threads):What makes you think this?

Post 1: "(One point that I have not raised is that the wave function is not the only "structure" present in QM; there is also the Hamiltonian, or Lagrangian if you are doing QFT. So one possibility that we have not discussed is that the "additional structure" is in the Hamiltonian, not the wave function; that the Hamiltonian of the cat, or the cat/environment system, is what picks out the alive/dead basis as the one that gets decohered.)"

Reference https://www.physicsforums.com/threads/irreversibility-vs-reversibility.909761/page-2#post-5734686

Post 2: "In QM the quantized Hamiltonian is still used to find the total energy in these situations It's the operator corresponding to the "energy" observable, yes. But this observable only measures the energy in the "system", i.e., the part that we are modeling using QM. It can't tell you anything about the part that we are not modeling using QM, i.e., the field. So if you consider the field as part of "the total system", then you would have to say that "the total system" is not completely modeled by QM in this case, so the "energy" you get from the QM operator is not the "total energy"."

Reference https://www.physicsforums.com/threads/hamiltonian-in-quantum-vs-classical.919627/page-2

In the second post, you stated how the "field" was not part of the QM hamiltonian. So if the additional structure is in the "field". Then it is not part of the Hamiltonian you were describing in post 1? If it's the field that is the source of the additional structure, then the right language is to say the Lagrangian of QFT couldbe the source of the additional structure that gives the preferred basis of unitary only dynamics?