Abstract
We present the Bode-UTSDR Integral, a rigorous conservation law that constrains the area under the log-magnitude response of any linear time-invariant (LTI) system by its universal spectral decay rate γ. By integrating the classical Bode Sensitivity Integral with the Universal Spectral Decay Theorem (UTSDR), we demonstrate that the well-known trade-off between bandwidth, stability, and sensitivity—long regarded as a domain-specific engineering heuristic—is in fact a fundamental thermodynamic constraint governed by the UTSDR minimum principle.
We prove that for LTI systems possessing an arbitrary number of open-loop unstable poles, the Bode sensitivity integral obeys a tightened conservation law where Φ(γ,ζ) is a spectral correction term determined jointly by the system’s universal decay rate γ and its damping ratio ζ.
Our proof proceeds in four stages.
The result implies that every LTI system—regardless of domain—operates under a single, universal conservation law connecting its pole structure, spectral decay topology, and thermodynamic efficiency.
