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arXiv · 2608.20606

Ptolemaic negative type and the values q(6) and q(7)

Eren Ercan
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Abstract

Baker, Huh, Kummer, and Lorscheid define the triangular-hyperfield threshold q(n)=q(U_{2,n}) and conjecture exact values for all n. Using their identity q(n)=P(n-1), where P(m) is the universal negative-type exponent of m-point Ptolemaic metrics, we prove q(6)=\log_2(9/4) and q(7)=1. For five points, we separate zero-sum coefficient vectors by sign pattern. A sharp two-summand inequality proves the inequality for the 1+4 pattern. For the 2+3 pattern, a sharp four-point partial-correlation bound and an exact copositivity identity prove the required inequality. For six points, we construct an involutive coefficient transport under metric inversion that preserves the negative-type quadratic form. Applying the transport at an index with positive local contribution converts a hypothetical 3+3 counterexample into a 2+4 counterexample. Complete split graph metrics attain both bounds.

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Eren Ercan. 2026-08-20. Ptolemaic negative type and the values q(6) and q(7). https://arxiv.org/abs/2608.20606

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