The quantum-classical bipartite universe and the logical emergence of $\hbar$ from classical temporality

Abstract

We present a foundational reinterpretation of quantum dynamics based on the standard von Neumann formalism, positing a structural logical bipartition between a classical observational environment and a quantum observed system. Against universalist frameworks, we show that theoretical consistency requires distinct propositional logics: Boolean logic governs the environment, while non-Boolean logic applies to the observed system. Within this framework, an ideal classical clock is represented as an open system whose energy spectrum is uniformly discretized. The dynamics follow from the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation under the constraint that the Hamiltonian remains observable. Although the clock's temporal evolution is driven by dissipative coupling to the measurement apparatus, it admits a stable, deterministic solution corresponding to information-preserving cyclic dynamics. This deterministic regime emerges as a logical attractor, stabilized by the Boolean structure inherent to the measurement interaction. Ultimately, $\hbar$ is derived as a fundamental scaling parameter from the classical clock's action-angle commutation relation, revealing that the quantum of action reflects the discrete logical structure underlying classical temporality. Furthermore, this framework establishes an exact mapping with 't Hooft's cellular automaton formalism, offering an alternative ontological interpretation that not require hidden variables and superdeterministic assumptions. The resulting distinction between quantum dynamics (as potential recontextualization) and classical dynamics (as temporal actualization) provides a novel logical approach to the measurement problem.