Parallel Transport on Spectral Subbundles of the Similarity Group

We construct a connection-theoretic framework for parallel transport of spectral components along parameter families of signals on the similarity group G˜=R×SO(2). Let {ft}t∈I be a signal family that evolves under a C1 group trajectory. The frequency support of the associated scale-rotation transforms produces three Hilbert subbundles over the parameter interval, and the trajectory velocity induces a covariant derivative on each subbundle. The standard spectral viewpoint treats transformation behavior at individual parameter values. Our formulation instead organizes the propagation of spectral components along the entire parameter path and provides closed-form transport operators together with error bounds on each subbundle. We derive three explicit parallel transport formulas. On the equivariant subbundle the transport is an exact isometric translation. On the coupled subbundle, the transport combines log-scale translation with a phase factor ein0Δθ. On the invariant subbundle, the transport is approximate, with the quantitative bound ∥ΠinvF−F∥≤ε|Δτ|∥F∥, where Πinv denotes the parallel transport operator on that subbundle. We introduce the notion of non-parallelism rate as a pointwise measure of deviation from parallel evolution, and we prove that cumulative deviation along the path is bounded by the path integral of this quantity. The bound separates into two parts. One part is controlled by trajectory estimation error and reflects geometric mismatch. The other part is controlled by intrinsic appearance variation and reflects non-geometric drift. We also show that regularity transfers from the signal family to the spectral sections, and we establish a discrete transport theorem whose finite-sum error bounds recover the continuous estimates in the small-step limit. The framework provides a quantitative geometric tool for multi-scale feature evolution under continuous scale-rotation transformations.