For an icosahedral system, phason-strain fields can be classified into
two classes, transforming respectively under the GAMMA4 and GAMMA5 ir
reducible representations of the icosahedral group. Group theory predi
cts that the GAMMA4 phason-strain field, as the one observed in as-cas
t quasicrystalline AlLiCu samples, is not coupled to phonon modes. The
low temperature specific heat of AlLiCu samples was measured after 10
days of annealing which allows phason-strain elimination and leads to
almost perfect samples. No vibrational contribution evolution is obse
rved. Some consequences and implications are discussed.