Determine the underlying Hamiltonian and dynamical law of the unified matter–geometry theory

Determine the explicit form of the underlying Hamiltonian operator H_u and the corresponding dynamical law of the fundamental theory that unifies matter and geometry, as assumed in the model’s product-state framework, so that the total Hamiltonian H_tot = (1/2)(H_u ⊗ I + I ⊗ U H_u U†) is grounded in a specified microscopic dynamics and its implications for the product-state constraint and residual-based action can be analyzed.

Background

The paper assumes a fundamental quantum theory in which matter and geometry are the same state, leading to a reduced Hilbert space of product states and a total Hamiltonian expressed in terms of an "unknown underlying Hamiltonian" H_u. This assumption underpins the proposed local collapse model and the residual-based action principle used to reconcile the product-state constraint with canonical quantum gravity evolution.

The author explicitly notes that the underlying theory’s dynamical law is not known, positioning the model as a phenomenological approach pending identification of the actual microscopic dynamics. Determining H_u and the associated dynamical law would clarify how the product-state constraint emerges and how the action principle connects to a fundamental description. It is central for establishing the foundations and testable predictions of the proposed local collapse mechanism.