  We study, by means of numerical methods, new $SU(N)$ self-dual instanton
solutions on $\mathbf{R}\times \mathbf{T}^3$ with fractional topological charge
$Q=1/N$. They are obtained on a box with twisted boundary conditions with a
very particular choice of twist: both the number of colours and the 't Hooft
$\mathbf{Z}_N$ fluxes piercing the box are taken within the Fibonacci sequence,
i.e. $N=F_n$ (the $nth$ number in the series) and $|\vec m| =
|\vec{k}|=F_{n-2}$. Various arguments based on previous works and in particular
on ref. \cite{Chamizo:2016msz}, indicate that this choice of twist avoids the
breakdown of volume independence in the large $N$ limit. These solutions become
relevant on a Hamiltonian formulation of the gauge theory, where they represent
vacuum-to-vacuum tunneling events lifting the degeneracy between electric flux
sectors present in perturbation theory. We discuss the large $N$ scaling
properties of the solutions and evaluate various gauge invariant quantities
like the action density or Wilson and Polyakov loop operators.
