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

Quantum field theory in curved spacetime allows time-dependent gravitational backgrounds to generate field excitations, while semiclassical gravity feeds the renormalized stress-energy expectation value back into the geometry. This suggests a logically possible feedback loop: a geometric perturbation modifies quantum matter, the resulting stress response perturbs the geometry, and the combined response may either decay or amplify. The purpose of this paper is not to assume that such a loop is unstable, nor to identify it a priori with the Big Bang, but to formulate a minimal mathematical closure in which the stability question can be answered exactly. Starting from the semiclassical Einstein equation and the linear-response form of the stress tensor, we introduce two signed perturbation variables: a coarse-grained curvature amplitude and a matter-stress response. A local Markovian closure leads to a coupled reaction–diffusion system with four linear response coefficients. Its dimensionless feedback gain is \[ G=\frac{bc}{ad}. \] For a bounded connected spatial domain with homogeneous Neumann boundary conditions, we prove from the spectral and energy formulations that the linear system is exponentially stable when \(G< 1\), has a neutral homogeneous mode at \(G=1\), and possesses a genuinely growing homogeneous mode when \(G>1\). No instability is therefore postulated: it occurs if and only if the positive cross-response exceeds the product of the two intrinsic damping rates. We then examine a nonlinear homogeneous reduction with cubic geometric saturation. We prove global existence, boundedness, global exponential attraction of the vacuum branch for \(G< 1\), and the appearance of two locally asymptotically stable nonzero branches for \(G>1\), constituting a supercritical pitchfork in the signed perturbation variables. The resulting framework distinguishes three notions that are often conflated: gravitational particle production, semiclassical backreaction, and self-amplifying curvature–matter feedback. The analysis also shows why a Big-Bang singularity does not follow from feedback alone: the minimal unstable closure produces exponential growth, while nonlinear saturation can instead generate a bounded high-curvature phase. The model is therefore proposed as a mathematically controlled metaframework for testing curvature--matter feedback mechanisms, not as a completed theory of quantum gravity.