Covering Rigidity, Forced Singularities, and Bπ-Enhanced Correspondence Singularity Theory

This record contains a preprint by Yanliang Ma entitled Covering Rigidity, Forced Singularities, and Bπ-Enhanced Correspondence Singularity Theory. The manuscript studies the relation between algebraic-topological obstructions and forced singularities in equidimensional smooth maps. Starting from the inverse function theorem and covering-space theory, it formulates a covering-rigidity principle: a proper smooth map between equidimensional manifolds with full-rank differential everywhere must be a finite covering. Consequently, any behavior of the induced fundamental group, rational cohomology, or mapping degree that cannot arise from a finite covering forces the critical set to be nonempty. The paper then argues that ordinary characteristic-class Thom polynomials depending only on the relative virtual bundle are not complete for detecting fundamental-group-type covering obstructions. To capture the missing noncommutative path information, the manuscript introduces a Bπ-enhanced framework, defines the covering-kernel defect K_pi(f), constructs Bπ-enhanced Thom cycles and their spectrum-level descendants, and embeds ordinary maps into the span/correspondence category. For stable maps between closed surfaces, the paper defines decorated discriminant data induced by an actual stable map and proves that this data reconstructs the source surface, the corresponding graph-type correspondence, and the covering-kernel defect. A minimal pure-fold map from the torus to the torus is used as an explicit example where all positive-degree ordinary Thom-polynomial classes depending only on the relative virtual bundle vanish, while the covering-kernel defect remains nontrivial. This is a mathematical preprint. It is intended for scholarly communication and has not yet undergone peer review.