This manuscript establishes a rigorously closed interface between continuous operator theory and quantum-inspired randomized linear algebra. By bridging the physics of spectral collapse with the computational framework of Sampling-Query (SQ) access models, the paper provides a computationally tractable pathway for detecting instability in large-scale Omega-Sigma operator systems. The framework formalizes systemic collapse through the Birman-Schwinger principle, defining instability as the exact point where a regularized Fredholm determinant vanishes. Because computing this determinant is traditionally intractable for massive systems, the manuscript introduces a quantum-inspired randomized estimator. Utilizing stochastic trace estimation and Rademacher vectors, the algorithm achieves sublinear estimation of the determinant in time polynomial to the system's effective rank. Alongside the algorithmic upper bounds, the paper proves an information-theoretic lower bound, demonstrating that sublinear collapse detection is strictly impossible without specific structural constraints, such as low effective rank or visible instability mass. Finally, the manuscript establishes the infinite-dimensional closure of the model using Zeta-regularized spectral action. This work successfully reduces the continuous, abstract phenomenon of operator-theoretic collapse into a concrete problem of randomized low-rank matrix estimation, completely anchoring the theoretical physics of spectral instability to the rigorous bounds of modern computational complexity.
Andrew Kim (Mon,) studied this question.