Finite-Horizon Classification of Set Selection in the Symmetric Two-Site Hopper: Principal Classical Partitions and Preclusive Consistent Sets

The consistent-histories formulation generally admits many incompatible consistent sets, motivating additional principles for selecting physically admissible families. Wallden proposed restricting consistent sets to coarse-grainings of the principal classical partition ΛC associated with primitive preclusive multiplicative co-events, and left the criterion unanalyzed. We evaluate this proposal exactly in the finite symmetric two-site hopper at arbitrary horizon T, a cancellation-rich solvable model, and compare it with the preclusive-consistent-set (PCS) criterion. We characterize all preclusive events and primitive supports, obtain a closed form for Λ(T) C , and classify all of its consistent coarse-grainings. For odd T, only the trivial consistent set and the final-position partition survive. For even T, the dynamics itself makes finalposition certain, and every admissible positive-measure cell must contain that entire certain final sector; all remaining distinctions are zero-measure refinements. Consequently, every intermediate single-time position question is excluded by the ΛC criterion for T ≥ 3. PCS is more permissive at T = 3, 4, but from T = 5 onward every intermediate-time position cell is contained in a zero-measure event, so no such partition is PCS. These conclusions are specific to the symmetric two-site hopper and do not constitute a general no-go theorem. They show exactly how cancellation and exact preclusion control the two selection rules in this solvable model and provide a finite-horizon comparison with previously identified global-in-time obstructions for multiplicative co-events.