  We address the issue of whether the golden mean $\left[\psi=(\sqrt{5}+1)/2
\simeq 1.618\right]$ universality class, as predicted by several theoretical
models, can be observed in the dynamical scaling of thermal transport.
Remarkably, we show estimate with unprecedented precision, that $\psi$ appears
to be the scaling exponent of heat mode correlation in a purely quartic
anharmonic chain. This observation seems somewhat deviation from the previous
expectation and we explain it by the unusual slow decay of the
cross-correlation between heat and sound modes. Whenever the cubic
anharmonicity is included, this cross-correlation is gradually died out and
another universality class with scaling exponent $\gamma=5/3$, as commonly
predicted by theories, seems recovered. However, this recovery is accompanied
by two interesting phase transition processes characterized by a change of
symmetry of the potential and a clear variation of the dynamic structure
factor, respectively. Due to these transitions, an additional exponent close to
$\gamma \simeq 1.580$ emerges. All these evidences suggest that, to gain a full
prediction of the scaling of thermal transport, more ingredients should be
taken into account.
