The Unified Wild Geometric Langlands Correspondence for GLn\mathrm{GL}_nGLn: Spectral Stabilization via the Universal Spectral Object and Filtered Monadicity

We establish a dynamical regularization framework for the wild geometric Langlands correspondence for GL2 on P^1 with one irregular singularity at infinity. The explosive wild ramification, parameterized by unipotent Stokes matrices, presents a severe analytic and combinatorial obstruction to extending known tame equivalences. We bypass static resolution approaches by treating the wild Betti character variety dynamically, embedding the moduli stacks into a noncommutative operator algebra on the Universal Spectral Object U.

By introducing the Wide-Net Seesaw Stabilizer, we couple the automorphic parahoric data with the spectral fluctuation sector (enlarged centralizers and Stokes phenomena). Applying a functorial heat-kernel filter stabilized at the exact Lambert W fixed point, theta_eff = W(1 / sqrt(4 * pi)) ≈ 0.22521, induces a strict geometric contraction kappa = 1 - theta_eff^2 ≈ 0.94928 < 1 on the nilpotent logarithms of the Stokes matrices.

This seesaw positivity (S(theta_eff) > 0) prevents the derived nilpotent cone thickening from blowing up, ensuring that the tangent complex dimensions on both the automorphic and spectral sides match precisely. In formal completions along the tame parahoric locus, iterated application drives off-diagonal wild data exponentially toward zero, producing a candidate deformation retraction onto the tame locus. By composing this "Langlands Snap" retraction with the known tame geometric Langlands equivalence, we provide a concrete functorial pathway to extending categorical equivalences to irregular formal neighborhoods.