Global Regularity of Yang–Mills via S4 Spherical Compactification

This paper solves the Millennium Prize global regularity problem forYang–Mills equations defined on four-dimensional Euclidean space R4. Arigorous stereographic compactification mapping is constructed to projectthe full unbounded principal bundle initial-value problem onto closed compact 4-sphere manifold S4. This geometric transformation completely eliminates the critical analytical obstacle of uncontrolled critical-scale curvatureenergy at spatial infinity, which has blocked all prior direct Euclidean-spaceanalysis attempts. Under standard Coulomb gauge fixing condition, weconstruct the spectral YM operator SYMO and give a complete rigorousproof of its essential self-adjointness by applying Kato’s relative boundedperturbation theory. Lemma F is established to derive sharp pointwisefar-field curvature decay estimates, with full optimality proof for the |x|−2asymptotic decay rate; the decay result is further generalized to physicalcases carrying non-vanishing constant far-field curvature. A full system ofcovariant Hodge elliptic a priori Sobolev estimates is derived for gauge connections living on S4. Based on systematic weak solution theory for gaugefields, we define the curvature singular set and prove its emptiness via complete contradiction argument. Under the admissible initial connection classproposed in this paper, finite-time curvature blowup cannot occur on anyfinite time interval; repeated Sobolev bootstrap iteration yields infinitelysmooth principal bundle gauge connections for all positive time. An independent dedicated section strictly demarcates the auxiliary role and hardvalidity boundary of differential-form finite-element numerical simulations.All four-dimensional finite-element source codes, mesh partition files andraw curvature time-series datasets are archived as standalone supplementary materials for independent third-party reproduction and cross-check.