Continuous Trajectories of Exceptional Points via Dual-Boundary Confinement in Fully Passive Architectures

Exceptional points (EPs) offer new paradigms in non-Hermitian physics, but their realization has long relied on active gain-and-loss mechanisms. In our previous work, we established that clusters of innumerable EP loci can be achieved within purely real-parameter passive systems under single-fixed and semi-definite boundaries. However, extending this framework to double-ended boundaries introduces highly coupled transcendental boundary conditions, thereby increasing the complexity of the conventional exact tracking. This study addresses this challenge by considering a both-fixed two-degree-of-freedom (2DOF) passive system. Through rigorous algebraic proofs, we achieve an analytical mapping of the global coalescing trajectories, specifically encompassing the complex degenerate roots (loci of EPs) and real double roots (loci of critical damping points) within a purely real state-space via discrete boundary points. We demonstrate that the dual-boundary constraints fundamentally morph the topologies of the EP clusters, giving rise to novel localized phase transitions and critical damping loci. These exact analytical boundaries reveal that dual confinement acts not as a restriction but as an unprecedented design flexibility to precisely manipulate non-Hermitian singularities, even under purely real-parameter passive conditions. Our exact formulation provides a foundational theoretical framework that is potentially applicable to the environmentally robust passive tuning of next-generation device architectures, ranging from micro-scale electromechanical resonators to high-frequency communication components, offering a passive alternative to mitigate the instabilities inherent in conventional active non-Hermitian systems.