Signature structure of quadratic response under Zeno-Schur coarse graining in open quantum systems

Quadratic response tensors arise naturally in quantum kinetic descriptions, such as the quantum linear Boltzmann equation (QLBE), where they encode the coupled structure of drift and fluctuations beyond simple positive-definite forms. Motivated by this class of systems, we investigate how such response structures are modified under monitoring-induced coarse graining. Within the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) framework and under time-scale separation, Zeno elimination of fast degrees of freedom generates a subtractive renormalization with Schur-complement structure. As a result, positive definiteness of the response tensor is not preserved: coupling between slow and rapidly damped sectors can induce negative directions even when the microscopic tensor is strictly positive. We formulate a minimal effective flow capturing this mechanism and show that the competition between Schur-induced compression and anisotropic perturbations organizes the dynamics into distinct signature sectors. The resulting structure appears to be robust within the class of models considered and, in appropriate regimes, may be experimentally accessible. Our results establish a general framework for how quadratic response structures, as encountered in QLBE-type dynamics, are dynamically reorganized under Zeno-induced coarse graining.