Spectral Operator Dynamics: From Fixed-Gain PID to Adaptive Meta-Control of Non-Equilibrium Steady States

Description : This paper unifies the preceding two analyses (LSP and MCN) through the spectral operator Ĝ(z) = ∂X/∂z — the Jacobian of the deterministic drift field. Under fixed gains, Ĝ reproduces the PID-FPE isomorphism locally: tr(Ĝ) is constant, with its eigenvalue spectrum separating dissipative (P/D) and rotational (I+Ω) modes. Under adaptive gains (the κ loop), Ĝ becomes time-dependent and its instantaneous spectral gap Δ(t) provides a direct control signal replacing the entropy production proxy. Real-time thermodynamics — entropy production rate, cumulative phase-space contraction, and non-equilibrium potential — follow directly.Declaration: During the preparation of this work, the author used an AI language model to assist with mathematical formalization, code implementation, and manuscript refinement. After using this tool, the author reviewed and edited the content as needed and takes full responsibility for the final content.Keywords : spectral operator, Jacobian spectrum, phase-space compressibility, adaptive PID control, real-time thermodynamics, non-equilibrium landscape