Beyond affine-complete shattering for rank-one reversible readouts

Develop lower-bound invariants beyond affine-complete shattering that can certify strict-cutpoint probabilistic state costs approaching the full quadratic capacity for rank-one reversible readouts.

Background

For a reversible automaton with a prescribed rank-one readout on a multiplicity block of dimension m, the accepting-projector orbit has dimension only 2(m−1), whereas the full numerical Hankel capacity is m². The paper proves a ceiling of O(m log m) for affine-complete witnesses drawn from the readout orbit, so the existing sign-shattering method cannot establish a quadratic lower bound in this regime.

The authors therefore identify the unresolved issue as operational: a different lower-bound invariant or a different sign construction is needed to approach the full quadratic capacity for rank-one reversible readouts.

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 8, Proposition 8.1 and surrounding discussion

For balanced ranks, where \kappa=\Theta(m2), this ceiling leaves the fixed-readout question open.

— Behavioral Memory under Symmetry in One-Way Quantum Automata  (2609.01451 - Chen, 1 Sep 2026) in Section 8, immediately following Proposition 8.1