Proof of the Cordero-Torregrosa Conjecture for Methods with Arbitrary Rules and Its Refinement under Paired Function and Jacobian Evaluations

We study the Cordero–Torregrosa conjecture on the order that can be guaranteed when solving nonlinear systems with prescribed budgets of function and Jacobian evaluations. We consider deterministic, causal, memoryless methods whose rules may be discontinuous, nonmeasurable, or noncomputable. We give an independent proof of a scalar barrier for queries of \(f\) and \(f'\) by constructing a fixed entire function, a simple root, and a single sequence of initial points that simultaneously exclude every exponent greater than \(2^{d-1}\), where \(d\) is the maximum number of queries. An exact simulation of vector-valued responses yields the Cordero–Torregrosa inequality \(p\leq 2^{N_F+N_J-1}\) for function classes containing the analytic test fields. The proof holds for \(N_F,N_J\geq 0\), \(N_F+N_J\geq 1\), without assuming \(N_J\leq N_F\); in particular, it covers \(N_J>N_F\). Under pairing at the end of the cycle, we obtain the sharper bound \(p\leq 2^{N_F-s+1}3^{s-1}\), \(s=\min\{N_F,N_J\}\geq 1\), without requiring temporal precedence. The proofs use indistinguishable perturbations, kernels of constraints, and a construction that preserves responses while producing a single entire witness. These results are upper bounds on the universal local order of a cycle; they do not establish attainability for every budget.