  We introduce a mechanism stabilizing a one-dimensional quantum many-body
phase, characterized by a certain wave vector $k_0$, from a $k_0$-modulated
coupling to a center site, via the protection of an emergent $\mathbb Z_2$
symmetry. We illustrate this mechanism by constructing the solution of the full
quantum many-body problem of hardcore bosons on a wheel geometry, which are
known to form a Bose-Einstein condensate. The robustness of the condensate is
shown numerically by adding nearest-neighbor interactions to the wheel
Hamiltonian. We identify the energy scale that controls the protection of the
emergent $\mathbb Z_2$ symmetry. We discuss further applications such as
geometrically inducing finite-momentum condensates. Since our solution strategy
is based on a generic mapping from a wheel geometry to a projected ladder, our
analysis can be applied to various related problems with extensively scaling
coordination numbers.
