Symmetry-Controlled Rank Transition in Dynamical-Decoupling Identifiability

We present a geometric formulation of parameter identifiability in dynamical-decoupling noise spectroscopy based on the projected Fisher information. The key result is an exact kernel equivalence,ker⁡Iθ,eff=ker⁡(PobsKθ)\ker \mathcal{I}_{\theta,\mathrm{eff}} = \ker(P_{\mathrm{obs}}K_\theta)kerIθ,eff=ker(PobsKθ),which identifies unidentifiable parameter directions as those invisible to the observable protocol subspace. We show that symmetry-controlled protocol design induces rank transitions in the observable projector, defining an observable horizon where new identifiable directions can emerge. Numerical results demonstrate that the lifted Fisher mode corresponds uniquely to the release of a symmetry-protected null direction. The framework separates geometric identifiability (rank structure) from quantitative sensitivity, providing a model-independent link between control symmetry and estimation limits.