Beyond the Dilute Instanton Gas: Resurgence with Exact Saddles in the Double Well

The path-integral approach to the double well has long been limited by the dilute instanton gas approximation. We show that if the finite Euclidean-time structure is taken seriously by using exact saddles, the dilute gas can be sidestepped, allowing the partition function and energy levels to be computed systematically. At each instanton order, the full resurgent structure -- which saddles contribute, what asymptotic growth is expected and how ambiguities cancel -- is encoded in a finite-dimensional Picard--Lefschetz contour integral over the quasi-zero modes with a clear geometric interpretation. Working at finite $T$ is essential: the dilute instanton gas can only access the ground-state splitting, whereas the exact finite-$T$ computation systematically produces the non-perturbative energy splittings for all excited states, including their full dependence on the level number. The key ingredients -- Weierstrass elliptic functions for the saddles, Lamé operators for the fluctuations and Picard--Fuchs equations for the periods -- form a coherent mathematical framework that both overlaps and complements that of Exact WKB.