Geometry of Protocol-to-Filter Maps: Dimensional Collapse in Dynamical Decoupling

This work develops a geometric framework for protocol-to-filter maps in dynamical decoupling noise spectroscopy. By analyzing the singular spectrum of the local Jacobian in a normalized feature space, it shows that increasing protocol-control complexity does not necessarily increase the number of experimentally accessible filter directions. CPMG exhibits moderate anisotropy with a weak but finite secondary direction, while the XY-8 deformation studied here shows symmetry-protected strong anisotropy: the deformation direction vanishes at first order at λ=0\lambda=0λ=0 and remains strongly suppressed over a finite range. A two-component toy model confirms that the framework detects genuinely two-dimensional accessibility when independent spectral directions are present.