Thanks. Unfortunately, I noticed an error I made during some rewording: "an low-curvature-radius-but-still-near-horizon effective theory". The indefinite article should be "a", and the reciprocals are wrong -- the radius of curvature near the horizon is high and the magnitude of the curvature scalar is low (it's a function on position in spacetime and goes to infinity as one approaches r=0; that behaviour of the Kretschmann scalar is used to show there's an actual curvature singularity rather than some artifact of a choice of how one chooses the "r" coordinate).

The radius of curvature is 1/|K| where one chooses a curvature scalar -- Kretschmann, Gauss, others may apply -- and finds a matching "kissing circle" (osculation is kissing). Here's an example in 2d, \rho is the radius of curvature and we're asking about the radius of curvarure at P on the curve AB: <https://undergroundmathematics.org/glossary/curvature/images...> (Two other examples <https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...>, <https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...>). For a point on the surface of a shell, we'd use an osculating sphere, and so on in additional dimensions.