The Atiyah-Sutcliffe determinant at five points: positivity and failure of the deletion inequality

Abstract

To investigate Atiyah's linear independence conjecture, Atiyah and Sutcliffe associated a normalized determinant to each configuration of distinct points in Euclidean 3-space and proposed a hierarchy of nonvanishing and determinant inequalities. We show that this hierarchy separates at five points. For every five-point configuration, the real part of the determinant is at least 5/768, proving Atiyah's conjecture in this cardinality. At the same time, an explicit planar configuration with integer coordinates violates the stronger deletion inequality, while its determinant and all five deletion determinants exceed one. Five is the smallest cardinality for such a failure, and adjoining sufficiently distant points gives planar integer counterexamples in every larger cardinality. The positivity proof combines a polarized determinant, signed-volume inequalities and an exact cut-variable certificate. The counterexample is certified by rational interval arithmetic. Complete verification programs and data accompany the paper.