We study the emergence of indistinguishable, but structurally distinct, allocation outcomes in convex resource allocation models. Such outcomes occur when different users receive proportionally identical allocations despite differences in initial conditions, eligibility sets, or priority weights. We formalize this behavior and analyze the structural conditions under which it arises, with a focus on fairness-oriented objectives. While the remedy — an infinitesimal perturbation — is classical, we show it can be interpreted as a principled secondary criterion for breaking ties that preserves the fairness structure encoded in the original problem. We apply this framework to a previously studied vaccine distribution model, demonstrating that such outcomes persist across multiple parameter regimes and can be systematically resolved without distorting the fairness scheme.