A Thermodynamic SU(1,1) Witness Framework for Double-Quantum NMR Signals in Neural Tissue

Entanglement criteria based on variances or Fisher information are well developed for compact collective spin algebras, but their extension to non-compact dynamical sectors is less straightforward. In particular, double-quantum (DQ) observables associated with effective SU(1,1) structures can lead to formally unbounded classical fluctuation estimates unless additional physical constraints are imposed. In this note, we develop a thermodynamic witness framework in which the classically accessible fluctuation sector is strictly bounded by finite-temperature detailed-balance conditions and motionally narrowed sequence-transfer limits. By analyzing the quantum dynamical semigroup of the spin-bath interaction, we demonstrate that spontaneous transient pair correlations generated by a stationary incoherent bath are contractively capped near an amplitude of ${10}^{-9}$. Furthermore, classical coherent sequence amplification is empirically bounded to $\mathcal{O}({10}^{-2})$ in motionally narrowed tissue. The resulting functional provides a concrete, theoretically derived bounding framework against which macroscopic DQ anomalies (e.g., fractional amplitudes on the order of $10\mathrm{\%}$ to $15\mathrm{\%}$) can be rigorously classified as classically inexplicable, provided macro-scale structural stability (constant $T_2^{\ast }$) is empirically verified.