Can a table of distances be realised by points in Euclidean space? Not always. Raising the distances to a smaller power can help by compressing the ratio between long and short distances. This anonymous research candidate asks exactly how much compression guarantees success for a special class called Ptolemaic metrics. These satisfy an inequality involving every four points. The paper proposes a sharp answer depending only on the number of points. Its analytic proof covers six points onwards; cited earlier results supply the smaller cases. The first new cardinality relative to the inspected established results is seven. The exponent in the negative-type inequality is twice the exponent applied to distances for the Euclidean embedding. Keeping those two powers separate matters. The page's seven-point diagram explains sharpness. Three points form one group and four form another. Distances are one, except between pairs in the four-point group, where they are two. A balanced positive and negative weighting gives an exact test. At the proposed threshold the test is zero; above it the test becomes positive and rules out the guarantee. This example supplies the upper bound. The universal lower bound comes from the written star-packing and inverse-correlation arguments, not from numerical experiments. The result would connect a local four-point condition to a global embedding guarantee. It does not establish practical performance improvements or settle every matroid threshold. Exact replay supports particular identities and examples, not formal verification of the proof. This is the Evidence Press release of Sharp negative type of finite Ptolemaic metrics, dated twenty-sixth September twenty twenty-six, version one point zero candidate. It is unrefereed. The paper and evidence are linked on the release page. This AI-generated synthetic voice adds no mathematical evidence.