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A processing unit evaluates a function at a box center, computes a Jacobian norm, and applies a conservative bound on the Hessian spectral norm to form a Taylor-based deviation bound. When the bound exceeds the function value, the method certifies the entire box contains no solution and prunes it. Boxes that cannot be excluded undergo adaptive subdivision and interval-arithmetic evaluation; an operator-based fixed-point criterion provides inclusion and, when applicable, uniqueness certification. The system emits traceable certificates of non-existence, existence, or uniqueness, operates deterministically without reliance on initialization, and handles clustered roots, high curvature, and boundary effects. The approach improves reliability and efficiency in control, optimization, simulation, robotics, and verification workflows by replacing heuristic pruning with mathematically guaranteed decisions.

System and method that outputs (i) a scalar invariant measuring intrinsic coupling, (ii) a localization map identifying where in state space the irreducible interaction concentrates, and (iii) a proof-carrying certificate that can be verified in polynomial time from the model without exhibiting any reparameterization. The diagnostic is constructed from operator-normalized couplings and their holonomy and curvature invariants, yielding: (a) a value exactly zero if and only if the system is decomposable under admissible reparameterizations, (b) stability under small perturbations and noise, and (c) localization over state-space regions and/or network substructures.