A Generalized Direction-Correction Framework for Coupled Mappings

Abstract

We develop a generalized mathematical framework for mappings that combine directionaland correction components and allow their coupled extension. The basic structure is thedirection–correction decompositionZ(x, u) = η(x, u) + ∆(x, u),where η(x, u) represents a directional component and ∆(x, u) represents a correction com-ponent. Two such mappings are then coupled through an interaction operator, producingZC = Z1 + Z2 + C(Z1, Z2).The framework separates combined direction, combined correction, and interaction into dis-tinct mathematical components. Fundamental properties, balance relations, limiting cases,and illustrative coupling structures are developed. A scalar bilinear interaction demonstrateshow non-additive terms arise. The formulation is intended as a mathematical foundation forsubsequent developments in optimization, game theory, quantum theory, physics, statistics,and related fields.