Holonomy-Based Loop Consistency Tests for Stored-Muon Spin Transport

This paper introduces a new class of loop-consistency observables for precision storage-ring muon experiments, aimed at testing the internal consistency of standard spin-precession analyses rather than extracting new parameters. The central novelty is the formulation of experimentally realizable closed operation loops whose net spin transport is captured by gauge- and coordinate-invariant holonomy observables, enabling null tests that are independent of parameter values. Muon spin transport is treated as a holonomy in SU(2), and closed sequences of operational toggles, orderings, or analysis partitions are mapped to invariant loop observables based on the trace of the spin-transport operator. We define several previously unreported analysis-level residuals—including echo, swap, mixed-difference, and commutator residuals—that are predicted to vanish under the standard additive, reversible effective description of spin dynamics. A concrete scalar example, the orientation–focusing interaction residual, is introduced to demonstrate how loop-consistency tests can be implemented directly using conventional fitted observables such as the anomalous precession frequency ωa. This residual isolates cross-dependences that are not probed by one-dimensional stability checks and has not been reported as a primary observable in published storage-ring muon analyses. The paper emphasizes a strict separation between effective descriptions and realizations: nonzero loop residuals do not invalidate the phenomenological framework used for parameter extraction, but instead constrain classes of underlying realizations such as non-commuting spin rotations, geometric contributions to spin transport, or background-field couplings. Practical guidance is provided on interleaving, drift suppression, amplitude-scaling tests, and robustness checks relevant to existing and forthcoming muon datasets. No new experimental data are analyzed. The work is intended as a portable, analysis-level framework for consistency testing that can be evaluated within current muon storage-ring experiments and related precision spin-transport measurements.