Extremal Euclidean throat ($V(r)\sim a(r-r_h{)}^2,\int dr/\sqrt{V}\to \infty$)

At a degenerate static horizon the Euclidean proper radial distance diverges logarithmically, so the horizon is an infinite end rather than a finite polar origin. Its time circle is not contracted at a conical tip, and local cone regularity imposes no finite period. The vanishing limiting surface gravity determines zero Hawking temperature; the simple-horizon periodicity argument must not be applied unchanged at the degenerate root.