A continuous Legendre transform spanning the polar angle (θ) domain −∞<θ<∞ embodies the essence of a nonconventional integral representation derived here for source‐excited time‐harmonic pressure fields in the presence of a thin elastic spherical shell immersed in different interior and exterior fluids. Avoiding the conventional Sommerfeld–Watson transform route, the formulation identifiesdirectlythetravelingwaveswith their multiple encirclements of the shell by extending the θ domain from its conventional 0≤θ≤π range with periodicity constraints into the unbounded (multi‐sheeted) domain without these constraints. Periodicity for the closed shell is recovered by summing contributions from an infinite array of image sources located in the ‘‘nonphysical’’ portion θπ of the angular space. The rigorous solution is obtained by synthesis over a complex spectral continuum. Various alternative representations are derived from it, with special attention given to those that lend themselves to ray‐acoustic asymptotics. The rigorously derived ray acoustic constituents include incident and geometrically reflected ray fields as well as surface guided ray fields. The latter are excited by phase matching of the incident ray field to the traveling wave modes in the shell, and they reach the observer by phase‐matched detachment [see also P. L. Marston, J. Acoust. Soc. Am.83, 25–37 (1988)]. The phase matching applies to directly excited leaky waves as well as to waves that decay initially into the fluid; the latter are excited from an exterior source by evanescent tunneling. Transitional effects, where simple ray theory fails, are discussed as well, as is the special case of far‐field plane wave scattering. The ray parametrization for the canonical closed spherical shell will play an essential role in subsequent treatment of deformed and truncated shell geometries [cf. L. B. Felsen and I. T. Lu, J. Acoust. Soc. Am.86, 360–374 (1989)].