Fiber Geometry of the Protocol Ratio Map: Submersion Structure, Symmetry-Induced Projected Folds, and Level-Set Design in Dynamical-Decoupling Spectroscopy

Fiber-geometric analysis of the protocol ratio map in dynamical-decoupling spectroscopy. The ratio map is shown to be a surjective submersion with smooth trivial fibers, while the previously identified fold at Rc=4/3R_c=4/3Rc=4/3 is reinterpreted as a symmetry-induced projected fold arising from tangency to level surfaces. Exchange symmetry, equivariant kernel structure, and level-set transversality are used to derive fold-free design criteria and explain symmetry-breaking fold lifting.