  Renormalization-group theory stands, since over 40 years, as one of the
pillars of modern physics. As such, there should be no remaining doubt
regarding its validity. However, finite-size scaling, which derives from it,
has long been poorly understood above the upper critical dimension $d_c$ in
models with free boundary conditions. Besides its fundamental significance for
scaling theories, the issue is important at a practical level because
finite-size, statistical-physics systems, with free boundaries above $d_c$, are
experimentally accessible with long-range interactions. Here we address the
roles played by Fourier modes for such systems and show that the current
phenomenological picture is not supported for all thermodynamic observables
either with free or periodic boundaries. Instead, the correct picture emerges
from a sector of the renormalization group hitherto considered unphysical.
