We introduce OSIDE (Operator-Synthetic Inverse Design Engine), a two-stage algorithm for inverse design of quantum metamaterials based on behavioral equivalence classes. Rather than seeking a unique Hamiltonian realizing a target observable, OSIDE identifies the full preimage class of observationally equivalent generators and selects the optimal representative according to a resource functional encoding disorder robustness, coupling sparsity, or fabrication constraints. Three theorems establish the mathematical foundation: existence and structure of behavioral fibers (via the constant-rank theorem and implicit function theorem), existence of optimal representatives (Weierstrass), and a first-order robustness characterization showing that observable variance under Gaussian parameter disorder scales as σ²‖DΦ‖²F, so minimizing the Jacobian operator norm within the equivalence class minimizes disorder sensitivity. Numerical demonstrations use exact diagonalization on small lattice systems (N = 4 and N = 6 sites, single-excitation subspace). For excitation transport, OSIDE finds a solution with 60% sensitivity reduction and 13% variance reduction under coupling disorder (σ = 0. 08, 500 Monte Carlo samples) at matched nominal fidelity. For dual-edge-state engineering — simultaneous ground- and first-excited-state localization at chain termini — OSIDE achieves 62% variance reduction and 47% sensitivity reduction at identical nominal performance. All results are fully reproducible from the provided Python code (NumPy/SciPy, no special dependencies). The geometric structure of the design space is developed in a self-contained section: the behavioral quotient space Θ/∼ is identified as a stratified space (smooth fiber bundle on the regular locus, singular fibers at rank-drop loci), OSIDE is reinterpreted as fiberwise Morse optimization, and the pullback metric G = DΦᵀDΦ is shown to unify the robustness functional with the covariance propagation formula. Three open mathematical problems are stated honestly: derived-geometric formulation of singular fibers, path-integral interpretation of the design ensemble, and BV–BRST cohomological classification of gauge redundancy. Primary target platforms: superconducting qubit lattices, programmable spin systems, photonic quantum metamaterials.
Francis Procaccia (Sun,) studied this question.