Curvature–Matter Feedback in Semiclassical Cosmology: A Reduced Mathematical Framework for Geometry-Induced Quantum Matter Production

Quantum fields on a time-dependent geometry can undergo particle production, but this does not establish a self-amplifying response of the geometry. We study a conditional two-amplitude closure for a signed curvature perturbation and a signed renormalized stress response. The physical assumptions are specified through constrained projection, a low-frequency response expansion, and a stable single-pole approximation; none is asserted to hold for every semiclassical state. A quantitative memory-remainder estimate and a conditional scalar-field example make the scope of this approximation explicit. For the resulting cooperative reaction–diffusion system, a weighted spectral argument identifies the first loss of stability with the homogeneous mode. For the homogeneous cubic completion, a coercive gradient potential proves global existence and convergence to an equilibrium, including asymptotic stability at the critical parameter. We also establish the local spatial bifurcation, distinguish quintic corrections from a degenerate bifurcation, and classify a constant symmetry-breaking perturbation. Renormalization of the cosmological constant, conservation, higher-curvature terms, and the distinction between physical and effective-theory runaway solutions are treated explicitly. The contribution is a response-level diagnostic and its mathematical limitations, rather than a new microscopic theory or an observational cosmological prediction.