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June 10, 2026Open Access

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

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The study aims to analyze the fiber geometry of the protocol ratio map in dynamical-decoupling spectroscopy and explore symmetry-induced projected folds.
Analyzed the protocol ratio map to identify fiber-geometric features.
Examined the nature of folds and symmetry properties in the context of level-set designs.
Developed criteria for fold-free designs based on derived structures.
Demonstrated that the ratio map serves as a surjective submersion with smooth fibers.
Reinterpreted a specific fold as a symmetry-induced projected fold at Rc=4/3.
Introduced design criteria that avoid folds by leveraging symmetry-breaking properties.

Abstract

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=4/3Rc=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.

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

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Abstract

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