A universal waveguide mass--energy relation for lossy one-dimensional waves in nature

Finite, lossy waveguides are ubiquitous: distributed attenuation with partial reflections produces feedback, resonance, delays and decay across electromagnetic, acoustic, photonic, quantum-transport and electrochemical interfaces. Yet standard impedance/scattering tools and weak-loss resonator approximations do not provide low-dimensional invariants that remain predictive under intrinsic asymmetry and realistic boundaries, nor do they cleanly separate total absorption from useful power delivered to a load. Here we develop a unified mass--energy framework for linear, single-mode, one-dimensional systems, in which energy-like and power-flow variables $(\mathcal{U},\mathcal{S})$ satisfy the universal invariant $\mathcal{U}^2-\mathcal{S}^2=|Γ_g|^2$, with the effective standing-wave ``mass'' $|Γ_g|$ becoming state-dependent under asymmetry. The Cai--Smith chart gives a bounded state-space map of stability, feedback proximity and operating states, and makes explicit the divergence between maximum absorption and useful delivery under loss. We derive four laws governing absorption and emission, validate the invariant in multiport optics via coherent perfect absorption at exceptional points using a basis-independent SVD criterion (reconciling quadratic vs quartic near-zero scaling), and map electrochemical polarization onto the same geometry by extracting $|Γ_g|$ from two-mode orthogonal fits to reveal a universal storage-to-transfer transition across pH. This framework provides a transferable design language for dissipative, boundary-controlled systems.