Equivariant Window Embeddings: A Symmetry-Based Diagnostic for Microscopic Reversibility

This upload contains an 8-page note introducing Equivariant Window Embeddings (EWE): a model-independent, symmetry-based diagnostic for detecting broken time-reversal symmetry / failure of detailed balance from finite trajectory data. The method maps two-sided trajectory windows into a feature space via an equivariant embedding Φ\PhiΦ that commutes with time reversal, Φ(Rω)=JΦ(ω)\Phi(R\omega)=J\Phi(\omega)Φ(Rω)=JΦ(ω), and evaluates an observable GGG that is odd under the induced involution, G(Jv)=−G(v)G(Jv)=-G(v)G(Jv)=−G(v). The core result (Theorem 3.1) shows that under detailed balance the expectation E[G(Φ(ω))]\mathbb{E}[G(\Phi(\omega))]E[G(Φ(ω))] must vanish; therefore a statistically significant nonzero empirical mean provides a certificate of irreversibility and (in physical settings) nonzero entropy production without fitting parametric models or estimating forward–reverse likelihood ratios.