Randy Leonardo

We establish the foundations of a theory of genuinely vectorial iterative methods for coupled nonlinear systems. Within classical local convergence analysis, we define the GV_p property through a bound guaranteeing local order at least p for every problem in a declared admissible class, under suitable smoothness and simple-root assumptions. We propose the Leonardo–Ureña certificate as a sufficient criterion for establishing this property. By ensuring class-wide cancellation of all vectorial error terms of degree below p and controlling the local remainder, the certificate guarantees a minimum local convergence order that remains valid when particular problems exhibit higher order. We organize convergence claims by distinguishing scalar, separable, trajectory-dependent, directional, and class-guaranteed orders, and introduce the structural order profile and coupling-order defect. Rooted composition trees preserve multilinear nesting and stage substitutions. We prove that polynomial scalarization is injective through degree three and loses injectivity from degree four onward, while norm-dependent weights require additional directional analysis. Finite-jet realizations provide explicit counterexamples and separate sharpness witnesses. We generalize the established CM4 scheme into a parameterized two-step family in which CM4 is recovered for α=0. We apply the proposed certification framework to establish the GV₄ property of this family through a composition-preserving proof and obtain a class-wide eighth-order lower bound for the Jacobian-defect method JD8. Explicit coupled realizations demonstrate the sharpness of both bounds. A complementary coupled counterexample excludes universal eighth-order convergence for WFM1–WFM5. A numerical assessment of 19 published methods also reveals cases in which the stated orders are not reproduced on the coupled problems considered, highlighting the relevance of the structural assumptions supporting convergence claims. Finally, we introduce EDOL as a directional diagnostic and two intra-iterative EOLI formulations: an error-based estimator requiring a reference root and a residual-based estimator evaluable along a trajectory without knowing the root. We establish their consistency under stated conditions and use numerical comparisons as evidence complementary to the analytical guarantees.