Stratified Geometry of Inverse Quantum Design: Singular Fibers, Robustness Bounds, and Cotangent Complexes

We develop a rigorous geometric framework for inverse quantum design problems centered on the forward map Φ: Θ → ℝ, where Θ ⊂ ℝᵈ is a compact parameter space. The central result is a Stratified Robustness Control Law: under Gaussian fabrication disorder of amplitude ε, the variance of the observable satisfies ℛ(θ) = ε²‖∇Φ(θ)‖² + O(ε⁴‖H_Φ‖²), with the leading coefficient identified as the Fisher-information pullback metric and the remainder bounded uniformly on each Whitney stratum. We prove that the loss landscape Hessian on the behavioral fiber factors algebraically as 2 DΦᵀ DΦ, yielding exactly d − rank(DΦ) flat fiber-tangential directions. Applied to the N = 4 quantum transport problem, Monte Carlo validation (200,000 samples per noise level) recovers the predicted leading coefficient to 3.4% accuracy and confirms the ε⁴ residual scaling. The framework is positioned as the second-order Taylor refinement of the first-order Symmetry-Forced Fiber Stability Theorem.