Annihilator analogue of colon connectivity

Establish an annihilator analogue of Corollary 1? [Corollary~\ref{cor:colonconn}] that would determine the annihilator-component distance $d_{\mathcal A}$ directly, without requiring the full census of minimum-weight logical operators.

Background

The paper proves a connectivity property for minimum-weight logical operators in the colon component: every nonempty proper subset of the support has nonzero syndrome, enabling the anchored cluster search to compute $d_{\mathcal C}$ exactly. The corresponding argument does not apply to the annihilator component, because a zero-syndrome split can produce lighter colon logicals without contradicting minimality within the annihilator component.

Consequently, the annihilator-component distance $d_{\mathcal A}$ is obtained indirectly from the complete census of logical operators at relevant weights or from an irreducible-range argument combined with a one-sided witness. The authors explicitly leave open the problem of finding an analogous structural or connectivity result that would compute $d_{\mathcal A}$ directly.