Symmetry-Controlled Observability in Dynamical-Decoupling Filter Maps

Symmetry determines observable dimensionality in dynamical-decoupling filter maps.Mirror-preserving timing deformations enforce an even filter response, leading to a rank-deficient local Jacobian and a symmetry-protected collapse of observable directions. Breaking this symmetry generically restores a finite first-order direction, with the recovered singular value varying continuously with the symmetry-breaking strength. A phase diagram in deformation amplitude and mirror-breaking strength separates symmetry-protected collapse from multi-dimensional accessibility. The results establish that observable dimensionality is controlled by symmetry class rather than parameter count. The analysis is shown to be independent of the specific feature representation by reproducing the same hierarchy, phase-diagram geometry, and scaling structure using smooth feature functionals that exclude non-differentiable peak observables.