  We establish new non-uniqueness results for the Euler equations with external
force on $\mathbb{T}^{d}$ $(d\geq3)$. By introducing a novel alternating convex
integration scheme, we construct non-unique, almost-everywhere smooth,
H\"older-continuous solutions with regularity $\frac{1}{2}-$, which is notably
above the Onsager threshold of $\frac{1}{3}$.
  The solutions we construct differ significantly in nature from those which
arise from the recent unstable vortex construction of Vishik; in particular,
our solutions are genuinely $d$-dimensional ($d\geq3$), and give non-uniqueness
results for any smooth data. To the best of our knowledge, this is the first
instance of a convex integration construction above the Onsager exponent.
