Path Theory

Path Theory is a new mathematical framework proposing that a computational process depends not only on its initial and terminal states, but fundamentally on the specific route (path) traversed through a system of connected nodes. In conventional mathematics, intermediate steps between an initial node S and a terminal node T are often regarded as secondary if the endpoints coincide. Path Theory takes the opposite viewpoint: the intermediate route itself is treated as a primary mathematical object that dictates the final computational output. Consequently, two distinct paths P and Q with identical endpoints may yield entirely different results: Phi(P) != Phi(Q). Key Features and Concepts: Nodes and Values: Nodes represent locations in a graph or network, each assigned a mathematical object (numbers, vectors, matrices, functions, or algebraic structures). Connections and Directed Transitions: Edges represent permitted computational transitions. Directed connections ensure that forward and reverse transitions are distinct. Sieves (Filtering Maps): Connections are equipped with "sieves" (maps) that selectively remove or transform specific components of an intermediate state (e.g., discarding imaginary parts or fractional denominators) during traversal. History Dependence & Revisitation: Paths allow repeated node visits, backtracking, and cycles. Because non-invertible sieves alter computational states dynamically, revisiting the same node does not restore the previous computational state. Path Switching & Concatenation: A dynamic process where a computation can switch intermediate routes whenever admissible connections exist, forming concatenated path operators. Algebra of Path Operators: Formalizes path operators under composition, identity paths, path equivalence relations, and defines general criteria for path-independence versus path-dependence.