Invariant-Based Coupling of One-Dimensional Excitations to Two-Dimensional Attractor Manifolds

This paper presents a formal mathematical construction for coupling a one-dimensional excitation field to a two-dimensional invariant attractor manifold, resulting in an effective focusing of trajectories in an embedding space. The framework combines reduced state-space projections, dissipative attractor dynamics, tensorial coupling on invariant manifolds, and invariant-based potential construction. The approach is intentionally abstract and avoids ontological or physical assumptions. It is formulated as a general mathematical model applicable to nonlinear dynamical systems, geometric coupling mechanisms, invariant-driven effective forces, and trajectory focusing phenomena. All interpretations regarding physical meaning, dimensional embedding, or empirical realization are deliberately left open. This work is part of an ongoing independent research project exploring invariant-based geometric structures and their role in emergent dynamical behavior. Author contributions and tools: Conceptual development, mathematical formulation, and manuscript preparation were carried out by the authors. AI assistance (OpenAI GPT-5.2) was used as a collaborative tool for structural refinement, language optimization, and consistency checking. All final content decisions, interpretations, and responsibility for the work remain with the authors.