Multiscale Violation of Onsager Reciprocity: Thermomechanical Proof, Atomic Evidence, and Graphene Predictions

Onsager reciprocity $L_{ij}=L_{ji}$ is a cornerstone of near-equilibrium thermodynamics derived from microscopic time-reversal symmetry. We develop a geometric framework in which entropy-weighted reparameterization of thermodynamic response functions leads to an effective asymmetry in cross-couplings without violating the microscopic Onsager theorem. Motivated by the parallel structure of heat capacities $C_p$ and $C_v$, we introduce entropy-weighted response variables ${\lambda }_p$, ${\lambda }_v$, ${\lambda }_s$, and ${\lambda }_t$. Their ratios ${\mathrm{\Gamma }}_c={\lambda }_p\mathrm{/}{\lambda }_v=C_v\mathrm{/}{C}_p$ and ${\mathrm{\Gamma }}_m={\lambda }_s\mathrm{/}{\lambda }_t={\kappa }_T\mathrm{/}{\kappa }_S$ form thermodynamic invariants whose product equals unity in equilibrium. Within a differential-form representation of thermodynamic state space, equilibrium corresponds to exactness of the accessibility form $\omega ={\lambda }_{p}dp+{\lambda }_{v}dv$ with $d\omega =0$, while non-equilibrium processes generate curvature $\mathrm{\Omega }=d\omega$, producing an effective asymmetry in the transformed coupling matrix. A microscopic theorem shows that entropy-weighted statistical ensembles with time-reversal asymmetry $\chi (\mathrm{\Gamma })=W(\mathrm{\Theta }\mathrm{\Gamma })\mathrm{/}W(\mathrm{\Gamma })\mathrm{\ne }1$ generate an antisymmetric contribution to the Green–Kubo transport matrix. Atomic-scale analysis using the Transforma model reveals cross-derivative asymmetries across the $3d$ transition series, peaking at configuration anomalies in Cr and Cu. Temperature-dependent Raman spectroscopy of monolayer graphene exhibits statistically significant hysteresis loops (up to $30\sigma$), providing experimental evidence for thermodynamic curvature. These results unify microscopic irreversibility, atomic structure anomalies, and macroscopic hysteresis within a geometric interpretation of entropy-weighted thermodynamic coupling.