Effective Gaps between Singular Values of Non-Stationary Matrix Products Subject to Non-Degenerate Noise

The paper proves the ε^2 n lower bounds for singular-value gaps in both expectation and almost surely via a rigorous miniflag/fibered-entropy framework, with explicit additivity and quantitative one-step entropy production, culminating in Theorem 1.1 and Theorem 6.7. By contrast, the model’s outline hinges on unproved uniform Doeblin-type minorization on Grassmannians, a telescoping simple-root cocycle bound with uncontrolled boundary terms, and a uniform second-order perturbation lower bound for α_k that fails at degeneracies; as written, these gaps prevent a complete or correct proof.