Topological Arithmetic and Path-Dependent State Histories: A Homotopic Extension of Feynman Integrals to Foundational Logic

Standard arithmetic treats numbers as memoryless scalars (3+3=6 and 5+1=6 are considered absolutely identical). This paper proposes a radical epistemological and mathematical shift by introducing "Topological Arithmetic," where integers are redefined as dynamic, path-dependent state histories. Drawing conceptual parallels from Homotopy Type Theory (HoTT) and Feynman's path integrals, we model mathematical equality as a topological trajectory. A resultant state (e.g., "6") is treated as a superposition of its genealogical formation paths, each carrying a distinct ontological "memory of binding." We explore the rigorous scientific applications of this framework: Cryptography: The development of path-dependent hashes that require the correct computational history, not just the correct scalar value. Complex Signal Analysis (EEG): Distinguishing identical brainwave frequency states based on their topological history to identify pathological attractors. Physical Topology (TRFC): Modeling elementary particles as topological defects that retain the kinematic memory of their bifurcation history. This paper establishes a new formal language for systems where the process of becoming is permanently encoded into the structure of being.