Wait -- consider any 3 consecutive (ccw) vertices A,B,C of a convex n-gon. If the triangle ABC is cut from the n-gon and re-glued so that they now appear in ccw order C,B,A, then it's easy to see that doesn't change the circumcircle (by its symmetry about the perpendicular bisector of AC). Since all permutations of the cyclic ordeer of the edges are generated by ones of this sort,

Actually, they aren't, if the polygon has an even number of sides. Color the sides alternately red and blue, and note that your permutations always switch edges of the same color.

that shows the circumcircle and circumradius are functions purely of the multiset of edge lengths.

But the proof you posted later, implicit in the equation that relates the circumcircle radius to the sequence of edge lengths (which, if you examine it, doesn't depend on edge order), is fine.

(If true, I never realized this independence before.)

Think of cutting a circular disk into pieces; one is the polygon, and the others are each bounded by a side and an arc. Discard the polygonal piece. No matter how you rearrange the other pieces, they include a total arc length of 2piR, so they will fit together to form a full circle of the same radius no matter what order you line them up in. Andy