We show that a broad class of regulated one–loop minisuperspace amplitudes leads to a lapse kernel of the universal form Formula: see text on Formula: see text. The exponent admits an exact inversion Formula: see text (with Formula: see text), which promotes the lapse integral to a normalized self–adjoint involution Formula: see text on Formula: see text. This inversion symmetry isolates a unique fixed scale Formula: see text separating ultraviolet (A/N) and infrared (BN) regimes and becomes, under Mellin transformation, a reflection Formula: see text. In minisuperspace Wheeler–DeWitt problems the self.reciprocal sector yields the decaying Bessel Formula: see text branch associated with regular Euclidean saddles. The growing Formula: see text branch commonly linked to tunneling prescriptions arises only after departing from strict Euclidean self–reciprocity (e.g. via analytic continuation or contour deformation), and is not identified with a pure eigensector of the same involution.
Douglas F. Watson (Thu,) studied this question.