Counterexamples to the Five Neighbour Density Conjecture

We construct periodic packings of unequal circular disks in which every disk touches exactly five others, the smallest-to-largest radius ratio exceeds the Fejes Tóth threshold, and the density is below the conjectured bound √3π/7. These give a negative answer to Problem 1 of Section 2.5 in Brass, Moser and Pach, attributed there to G. and L. Fejes Tóth. A two-radius example has density between 0.72030 and 0.72031. A three-radius example has exact radius ratio 0.53329642 and density between 0.67254600021 and 0.67254600022. Each uses twelve disks in a period cell. We give explicit algebraic coordinates, explain the geometric search strategy, and reduce verification to polynomial identities, rational inequalities and a short bound covering all remaining translates. A small companion package provides the exact parameters, contact lists and an independently implemented verifier.