Instrumental hybrid models and effective leakage. Top panels show the simulated frequency-dependent hardware defects of the quadrature hybrid: the amplitude imbalance ϵ(ν) with a cubic baseline and standing-wave oscillations (left), and the corresponding phase error δ(ν) (right). Bottom panels illustrate the effective sky-frame leakage D_eff calculated at two parallactic angles: χ = 0^∘ (blue solid) and χ = 45^∘ (orange dashed). While the leakage amplitude (left) remains invariant under rotation, the 90^∘ phase shift in the leakage vector (right) demonstrates the noncommutative nature of the hybrid error. This rotation of the error vector in the (Q, U) plane is the primary driver of the systematic EVPA “wiggles” observed in standard calibration pipelines.