Topological Unification Between the Regular Continued Fraction of π and the Proton

This paper establishes a rigorous topological isomorphism between the regular continued fraction of π and the internal structure of the proton. By introducing closed compact recursive topological spaces, topological charge conservation and recursive folding maps, we rigorously prove that the regular continued fraction of π is not merely a number-theoretic construction, but the geometric invariant of the minimally stable three-node topological system in three-dimensional Euclidean space. The integer term 3 corresponds to the minimal vertex number of the topological system, physically realized as the three valence quarks of the proton. The infinitely recursive tail of the continued fraction corresponds to the recursively folded field between nodes, where each denominator corresponds to the winding strength and folding layer of the gluon field between quarks, physically realized as the gluon field and the strong interaction region. This theory naturally explains the necessity of the three-quark composition, quark confinement and the extreme stability of the proton. It also distinguishes the topological closure difference between the proton and neutron, explaining the physical origin of their distinct stabilities, and is fully compatible with quantum chromodynamics (QCD). Several experimentally testable paths are proposed, the advantages and limitations of the theory are clarified, and future research directions are prospected. Keywords: π; regular continued fraction; topological stability; recursive folding; proton structure; quark confinement; topological conservation; neutron topology