An exact mathematical boundary raises a second question: how quickly do things change when we move away from it? This research studies finite collections of distances called Ptolemaic metrics. They satisfy the triangle inequality and an additional four-point inequality. A previous candidate identified a sharp negative-type threshold. At that threshold, special split metrics sit exactly on an equality boundary. The new work asks what happens when their distances change slightly. Its main result gives the smallest and largest possible first-order rates at which the negative-type gap opens. These rates are exact, not estimates from random sampling. They depend on how many points lie on each side of the split. The units matter: the mean distance between the two parts is fixed at one, and the size of a perturbation is its largest change in any distance. The proof first identifies all feasible infinitesimal directions. It then shows that each can be realised by actual Ptolemaic metrics, with a small quadratic correction. Eighteen symbolic certificate families cover arbitrarily large sizes, while thirty-six small certificates handle the exceptions. Four points behave differently. Changing the arms of a particular star opens the gap quadratically, so the same positive linear lower bound cannot apply there. This is a local stability result, not a universal robustness guarantee for all metrics. The archive includes proofs, complete certificate tables, corrected verification code and replay records. It remains an unrefereed candidate, without formal or external verification. This is Evidence Press, Ptolemaic split stability, released on the twenty-seventh of September twenty twenty-six, version zero point three point one candidate. The paper and evidence are linked on this page. This synthetic AI voice is an explanation, not additional scientific evidence.