We establish foundational results at the intersection of infinite-dimensional optimization, spectral geometry, and adaptive mesh refinement. By analyzing the interaction between saddle maps, Lie group symmetries, and curvature-driven refinement in densified sweeping nets, we present: (1) a conjectured logarithmic relationship between refinement complexity and saddle spectral gaps, (2) a symmetry-saddle correspondence lemma linking unstable manifold codimension to broken symmetries, and (3) a stochastic avoidance theorem guaranteeing almost-sure escape from spectral saddles. The introduced saddle codimension operator provides a spectral measure of instability, while numerical experiments validate the theoretical framework.