Topological Projection of QCD Path Integrals into the Imaginary Sector

Quantum Chromodynamics (QCD) path integrals in real space are plagued by divergences from oscillatory, non-Borel-summable contributions, rendering non-perturbative evaluation intractable. Here we propose a formal framework in which gauge fields are complexified and projected into an imaginary sector, such that formerly divergent integrals become convergent and well-defined. Employing topological projectors, Lefschetz-thimble decompositions, and resurgence methods, we show that non-perturbative QCD path integrals can in principle be rendered mathematically manageable. The shift to the imaginary sector does not simplify the theory in terms of ease, but it transforms unsolvable divergences into controlled challenges: Stokes phenomena, intersection numbers, and determinant zeros. We demonstrate this framework with schematic models and discuss its implications for QCD and lattice simulations.