Dynamical Stability of an Operational Access Boundary Under Local Free Evolution

An earlier note derived, for two qubits coupled by a fixed controlled-NOT and an experimenterwith local capabilities on the target qubit only, that the accessible effects are exactly those of thealgebra A1 = M2 ⊗ span{I, Z}: a non-classical algebra is accessible while the control qubit’s phaseis not. That boundary was derived for a remainder with no dynamics of its own. This note askswhether it is maintained when the remainder evolves freely under a time-independent HamiltonianH = h1 X + h2 Y + h3 Z that is fixed and known, and the experimenter may insert waits ofany chosen duration, adaptively, anywhere in a protocol. The answer is an exact dichotomy. Theaccessible effect set, and its span, remain unchanged under every finite adaptive protocol if andonly if [H, Z] = 0, i.e. h1 = h2 = 0: the remainder may precess about the axis the interaction reads.For every other member of the family the failure is demonstrated operationally — an explicit pairof states that no protocol without waits can distinguish, and the existing readout protocol with asingle wait of duration π/(4|h|) whose acceptance probabilities differ by 1 − n3^4 > 0, n = h/|h|.For the control H = X the accessible span becomes all of M4, by thirteen named protocols; whetherevery effect of M4 is then implementable is left open. Invariance of a protocol’s statistics underchanges of its wait durations is defined and reported separately from preservation of the accessibleset; in this family the two coincide, for a reason that is stated. The closed form of the wait, thecommutation criterion and the span bound are standard material; the contribution is the worked,exactly-verified criterion at both the effect and the span level. Nothing about robustness or genericHamiltonians is claimed. Technical note, not peer-reviewed. The deposit contains the note and areproducibility archive (one exact-arithmetic script with 34 checks, a failure-path harness, the runrecord and a checksum manifest).