Scale-Shift and Fractional Fourier Transform as Rotations in Representation Space over Finite Fields

The fractional Fourier transform and the reversible scale-shift, or zoom, are exhibited as two facets of one finite cyclic rotation in representation space, exact over the arithmetic symmetry shells defined over finite fields. Three results share the shell's meridian cycle. A native fractional Fourier family is additive in the meridian index, takes the Fourier quarter-turn, parity, and inverse quarter-turn as its cardinal values, and is faithful on the full cycle. A finite-field Weil realization matches this family on the cardinal Fourier skeleton. A reversible meridian-step scale-shift recasts framed-rational refinement as a coordinate-side rotation. Under a declared cyclotomic observer readout the spatial and spectral bases form a mutually unbiased pair, the finite entropic uncertainty relation is saturated by basis-localized states, and the Shannon-entropy interpolation along the cycle takes an exact closed form. The shell rotation conserves entropy; entropy is produced only by the observer readout, locating irreversibility at coarse-graining. The transform construction is finite and algebraic, with no continuum angle, trigonometric kernel, or limit invoked; the continuum enters only through the declared observer readout.