Sharp Order Barriers and Dimensional Saturation in a Frozen-Jacobian Model

The search for high-order methods for nonlinear systems requires comparing improvements in convergence order with the work needed to obtain them. We study an operational class of deterministic, causal methods without intercycle memory that construct new directions through a frozen inverse of the Jacobian. Scalar rules may be arbitrary, while vector operations are specified by their types and provenance. We prove that, with at most s < n inverse queries, the universal local order cannot exceed s + 1. The obstruction uses a fixed quadratic vector field, a sequence of accessible subspaces, and a uniformly bounded separating functional. Exact response reconstruction yields the law S_min(p,n) = min{p − 1, n} for every integer target order p ≥ 2. We study exact initial data, distinguish their rank from the cost of preparing them, and delimit the effects of additional vector resources. When direct and transpose solves are allowed, we obtain the sharp barrier for a total budget of up to three solves in the subdimensional regime. Finally, we characterize when the integer optimum for Shamanskii's method lies in that regime. Under a normalized cost model with fixed parameters, we prove that its optimal number of corrections satisfies m_*(n) ~ n/(3 log n). These relations allow us to rule out candidates that improve neither the order nor the cost of the reference implementation. They do not establish universal optimality of Shamanskii's method under unrestricted matrix processing or in execution time.