Researchers have developed a geometric algorithm to identify the critical points that contribute to the asymptotic evaluation of multidimensional integrals with exponential integrands of the form $e^{ikf(\boldsymbol{x})}$ over $\mathbb R^d$. This method significantly simplifies the process by eliminating the need to compute the flows of $-\text{Re} (i\nabla f)$ in $\mathbb C^d$, a computationally intensive step required in traditional Picard-Lefschetz approaches to derive such asymptotic expansions. The precise identification of these critical points is fundamental to understanding the behavior of these integrals in the asymptotic limit.

The algorithm relies on the combination of three key elements: the values of the function $f$ at all critical points plotted in the complex Borel plane, the concept of adjacency between these points derived from algebraic resurgence and hyperasymptotic approaches, and a new geometric "South-East rule." This rule allows determining which critical points are relevant for the asymptotic contribution, regardless of whether the function $f$ remains bounded or unbounded on $\mathbb R^d$. The study illustrates the applicability of the method with both pedagogical and advanced examples.

This advance represents a significant step towards a more systematic methodology for identifying instanton contributions in real-time path integrals. The ability to efficiently discern relevant critical points has profound implications for resolving issues associated with Wick rotations and their impact on the formulation of path integrals, opening new avenues for the analysis of complex systems in theoretical and quantum field physics.