Topological Numbers: Path-Based Arithmetic via Homotopy

We develop a formal theory of topological numbers, where each element is a continuous path in a topological space and equality is defined by homotopy equivalence. The resulting structure forms a group under path composition, with the identity element being the constant path and inverses given by path reversal. We prove nine theorems establishing fundamental properties of this system, including winding number additivity, length bounds under composition, and a Gauss-Bonnet-type constraint relating curvature, length, and winding. The key insight is that winding number forms a group homomorphism from the fundamental group to the additive reals. The proofs are formalized in the Platonic proof system (Lean 4 exportable) with 26 verified declarations, 10 hypotheses encoding the axiomatic properties of path length, winding, and curvature, and 7 foundational axioms defining the type and operations. Applications include gauge theory holonomy, topological quantum computing via braid groups, vortex linking in fluid mechanics, and configuration space path planning in robotics.

Keywords: fundamental group, homotopy, winding number, topological invariants, path composition, Gauss-Bonnet

MSC 2020: 57M05 (Fundamental group, homotopy), 55Q05 (Homotopy groups), 57R19 (Algebraic invariants of manifolds)