Protocol-Dependent Legitimacy of Global Training Metrics: A Minimal Computational Stress Test

This technical note presents a minimal computational stress test examining the structural legitimacy of global training metrics in multi-objective learning systems. In many machine learning settings, global performance indicators are used to summarize training progress across multiple objectives or interfaces. However, when optimization involves multiple targets, it is not always clear whether such global quantities are structurally meaningful or whether they depend on the protocol used to traverse the objective landscape. A simple binary classification model is trained under three optimization protocols: simultaneous optimization, sequential optimization of objective A followed by B, and the reverse order. The experiment explores three regimes: compatible objectives, aligned objectives, and structurally conflicting objectives. The results illustrate a diagnostic pattern: when objectives are compatible, different training protocols converge to similar solutions and candidate global metrics remain approximately invariant. When objectives are structurally incompatible, global closure cannot be achieved and protocol dependence emerges. The experiment is intentionally minimal and is designed to illustrate structural behavior rather than provide empirical performance benchmarks. The goal is not to introduce a new optimization method but to provide a simple diagnostic framework for identifying situations in which global performance indicators may fail to offer a protocol-invariant description of training dynamics.