Finite semigroup realizations for binary measure-many endpoints

Construct finite semigroup realizations that determine whether binary-alphabet measure-many automata can attain the upper endpoint of the strict-cutpoint probabilistic state-cost interval, including phase-assisted constructions when applicable.

Background

For measure-many automata, the paper proves capacity and state-cost bounds of the form M ≤ SC ≤ M+1. It also gives a scalar profile where the exact cost is M rather than M+1, showing that the dynamic upgrade to the upper endpoint is not automatic.

The available lower-bound construction uses multiple input symbols, while the paper notes an architecture-specific obstruction to compressing it directly to a binary alphabet. The unresolved issue is therefore whether another finite semigroup mechanism, potentially involving a phase-assisted construction, can realize the upper endpoint with binary input.

References

The remaining open cases now have a precise location: rank-one reversible readouts require lower-bound invariants beyond affine-complete shattering to approach full quadratic capacity, while binary channel endpoints and phase-assisted measure-many constructions require different finite semigroup realizations.

— Behavioral Memory under Symmetry in One-Way Quantum Automata  (2609.01451 - Chen, 1 Sep 2026) in Section 9, Conclusion; see also Section 7, subsection on Intermediate halting and nonhalting behavioral memory