Ultimate Resolution of the Hairy Ball Theorem Dilemma: A Logical Proof Based on the Ontology of Cognitive Succession

The Hairy Ball Theorem, a core classical theorem in topology, has been rigorously proven in its mathematical form, yet it has long faced a fundamental ontological dilemma: it can only assert that "there must exist at least one zero-vector singularity on a two-dimensional sphere" but cannot explain the ontological root of this conclusion, nor can it clarify the essence and existential significance of singularities. Traditional explanations attribute it to "the inherent property of the spherical topological structure", falling into a purely formal circular argument; in physical applications, singularities are regarded as "breaks and anomalies in continuous fields", leading to internal logical contradictions in theories. Based on the three-layer spatiotemporal 1:1 alignment framework and geometrized axiom system of the Ontology of Cognitive Succession, taking "the essence of cognition is succession" as the only unfalsifiable first principle, this paper proves through strict linear logical deduction that the singularity of the Hairy Ball Theorem is not a formal anomaly but an inevitable anchor point for closed successive evolutionary systems to return to their origin. This paper provides the first complete ontological foundation for the Hairy Ball Theorem, resolves the traditional contradiction between "continuous fields and singularities", unifies cross-domain singularity phenomena, and simultaneously verifies the explanatory power and universality of the Ontology of Cognitive Succession for pure mathematical theorems.