Why does the same polynomial appear twice in opposite roles?

Chern-Simons theory exhibits a curious duality in its large-representation, weak-coupling limit (Kashaev phase): the same Alexander polynomial appears both in classical A-polynomials and as the inverse of Jones polynomials in perturbative expansions. The resolution involves two distinct WKB branches with opposite action values—one producing hyperbolic volumes, the other yielding inverse Alexanders. This reveals how resurgence theory resolves apparent contradictions in non-perturbative behavior.