tokens: for the freedom in the low-order gravity field, because the likelihood function used to obtain our posterior distribution of interior models assumed an inflated variance on J 2 to accord with the numerical precision of our theory of figures implementation (see Section 2.1 ). Because the resulting distribution included a diversity of heavy-element and helium distributions, envelope transition locations, and core masses, the seismology suggests a tension with the Voyager rotation rate commonly assumed for Saturn’s interior that different three-layer interior models seem unlikely to resolve. This conclusion based on the ring seismology adds support to the notion that periodicities in Saturn’s magnetospheric emission (e.g., Desch Kaiser 1981 ; Gurnett et al. 2005 ; Giampieri et al. 2006 ) may not be consistently coupled to the rotation of Saturn’s interior (e.g., Gurnett et al. 2007 ; Read et al. 2009 ). The present model is potentially oversimplified in two major ways. First, the model is not suited to address the close multiplets of waves observed to have the same azimuthal wavenumber m , namely, the multiplets of waves in the C ring with m = 2 and m = 3. The bulk of these seem naturally explained by the model of Fuller ( 2014 ), wherein avoided crossings between the f -modes and deep g -modes of higher angular degree give rise to a number of strong perturbations with the same m value. However, in the wealth of new OLR and OVR wave patterns that have been measured from increasingly low signal-to-noise VIMS data since Hedman Nicholson ( 2014 ), it seems that only two waves add to the mixed-mode picture, both with m = 2: the close-in W76.44 wave, and the Maxwell ringlet wave whose frequency and m number were predicted by Fuller ( 2014 ). The f -modes of higher angular degree have less amplitude in the deep interior and so are less likely to undergo degenerate mixing with any deep g -modes strongly. Indeed, there is not yet any direct evidence for f -modes with l > 3 undergoing avoided crossings with deep g -modes, although the outlying (5, 4) OVR warrants closer scrutiny in the mixed-mode context. The second major simplification of the present model is the assumption that Saturn rotates rigidly. While upper limits can be established for the depth of shear in Jupiter or Saturn’s envelopes on magnetohydrodynamic grounds (Liu et al. 2008 ; Cao Stevenson 2017 ), evidence gathered from spacecraft indicates that zonal wind patterns do penetrate to significant depths (Smith et al. 1982 ; Kaspi et al. 2018 ). It has been proposed that the insulating molecular regions of these planets may be rotating differentially on concentric cylinders (Busse 1976 ; Ingersoll Pollard 1982 ; Ingersoll Miller 1986 ), the zonal winds being the surface manifestation of these cylinders of constant angular velocity.The mode identifications made in Section 5 and Table 1 reveal that the seismological data set now samples a variety of radial (via the angular degree l ) and
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