Inter-Site Fluctuation Mechanics and the Tripartite Noise Hierarchy in Transition-Edge Sensors (TES)

Transition-Edge Sensors (TES) represent the state-of-the-art in ultra-sensitive cryogenic radiation detection for cosmic microwave background (CMB) polarimetry and sub-millimeter astronomy. Operating within deep cryogenic regimes on the razor-sharp edge of a superconducting transition, these devices exhibit excess low-frequency noise spectra that severely constrain instrumental sensitivity below 10 Hz. Conventional bolometric noise models—relying on classical thermal fluctuation noise (TFNN), phenomenological Hooge parameters, or standard Dutta-Horn thermal activation—fail to capture the anomalous spectral upturns and non-stationary behavior observed in disordered superconducting bilayers (e.g., Ti/Au or Mo/Au). In this work, we present a rigorous, first-principles derivation of inter-site fluctuation mechanics tailored exclusively to the dissipative DC transport regime of Transition-Edge Sensors. We dismantle the historical conflation of TES and kinetic inductance detector (KID) noise models, establishing instead a tripartite noise hierarchy governed by: (1) Quantum Fluctuations, which dictate zero-point phase dispersion and barrier tunneling across disordered microstructures when classical Arrhenius activation freezes out; (2) Superconductive Fluctuations, which drive order-parameter phase dynamics, Cooper-pair density modulation, and the mesoscopic "fluctuation of fluctuations" via the dynamical breathing of normal-superconducting domain boundaries; (3) Classical Thermal Fluctuations, recovered strictly as the high-temperature asymptotic limit. By formulating a coupled stochastic master equation and non-Hermitian dissipation kernels, we derive the exact non-stationary relaxation time distribution and inter-site resistance noise spectral density . We then incorporate this framework into a modified electrothermal feedback (ETF) and Noise Equivalent Power (NEP) formulation, providing an uncompromising theoretical foundation for next-generation astrophysical instrumentation.