  For each large enough $m\in\mathbb{N}$ we construct by PDE gluing methods a
closed embedded minimal hypersurface ${\breve{M}_m}$ doubling the equatorial
three-sphere $\mathbb{S}_{eq}^3$ in $\mathbb{S}^4(1)$, with ${\breve{M}_m}$
containing $m^2$ bridges modelled after the three-dimensional catenoid and
centered at the points of a square $m\times m$ lattice $L$ contained in the
Clifford torus $\mathbb{T}^2\subset \mathbb{S}_{eq}^3$. The construction
respects the symmetries of the lattice $L$ as a subset of $\mathbb{S}^4(1)$ and
is based on the Linearized Doubling (LD) methodology which was first introduced
in the construction of minimal surface doublings of $\mathbb{S}_{eq}^2$ in
$\mathbb{S}^3(1)$.
