A Continuous Family of Counterexamples to the Middle Runner Conjecture

Abstract

The 2015 AIM middle runner conjecture allows arbitrary initial positions and predicts a symmetry between complementary order statistics for runners with commensurable nonzero speeds on a circular track. For velocities (1,2,3) and initial positions (0,0,a), this manuscript determines the average median position exactly as a piecewise quadratic function of a. Modulo one, the conjectured value holds only at a=0 and a=1/2; every other offset is a counterexample, giving an uncountable continuous family. The proof is a complete piecewise integration and is accompanied by exact rational verification. Scope: The result concerns this fixed one-parameter three-runner family and the hypotheses printed in the 2015 problem list; it does not classify arbitrary three-runner systems or address a strengthened common-start version. Status: Public Beta v0.1; internally verified candidate proof; external mathematical review pending.