Unified genus-1 potential and a parametric perturbative/nonperturbative relation

We study a parametric deformation of the unified genus-1 anharmonic potential and derive a parametric form of perturbative/nonperturbative (P/NP) relation, applicable across all parameter values. We explicitly demonstrate that the perturbative expansion around the perturbative saddle is sufficient to generate all the nonperturbative information in these systems. Our results confirm the known results in the literature, where the cubic and quartic anharmonic potentials are reproduced under extreme parameter values, and go beyond these known results by developing the nonperturbative function of real and complex instantons solely from perturbative data.