Emergence of Boolean Facts from Markovian Coarse-Graining in Relational Quantum Causal Processes

We formulate an operator-algebraic mechanism by which exact Boolean records can arise from local completely positive quantum operations without being imposed as microscopic structure.

The kinematic input is an algebraic process functional assigning probabilities to local normal completely positive operations in a finite operational context.

From the predual response of a target algebra to source interventions, relative to a background strategy class, we define an influence algebra; exact events are then, by definition, the projections in its center.

The dynamical question is whether nontrivial centers can be generated by coarse-graining rather than inserted through split-record laboratories.

We address this question using state-preserving normal unital completely positive coarse-graining channels.

If the Cesaro means of such a channel converge to a Choi-Effros infrared range and the range is asymptotically abelian in the GNS seminorm, then the represented infrared algebra is a commutative von Neumann algebra.

Its projection lattice is therefore a complete Boolean algebra.

We also give a finite-sector block-primitive criterion, motivated by locality and scrambling, which implies this asymptotic abelianness with exponential suppression of off-sector coherences and intra-sector fluctuations.

The result is a conservative mathematical statement: classical facts are not identified with arbitrary projections of a Type-III local algebra, but with central projections selected by an asymptotically abelian completely positive infrared limit.