Quantum Geometric Tensor in the Wild: Resolving Stokes Phenomena via Floquet-Monodromy Spectroscopy

Standard topological invariants, such as the Chern number and Berry phase, form the bedrock of modern quantum matter classification. However, we demonstrate that this framework undergoes a catastrophic failure in the presence of essential singularities - ubiquitous in open, driven, and non-Hermitian systems ("Wild" regime). In these settings, the local geometric tensor diverges, rendering standard invariants ill-defined and causing perturbative predictions to deviate from reality by order unity $sim 100\%$. We resolve this crisis by introducing the Floquet-Monodromy Spectroscopy (FMS) protocol, a pulse-level control sequence, which experimentally extracts the hidden Stokes Phenomenon - the "missing" geometric data that completes the topological description. By mapping the singularity's Stokes multipliers to time-domain observables, FMS provides a rigorous experimental bridge to Resurgence Theory, allowing for the exact reconstruction of non-perturbative physics from divergent asymptotic series. We validate this framework on a superconducting qudit model, demonstrating that the "Stokes Invariant" serves as the next-generation quantum number for classifying phases of matter beyond the reach of conventional topology.