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Proca stars and their frozen states in an infinite tower of higher-derivative gravity

In this work, we investigate the five-dimensional Proca star under gravity with the infinite tower of higher curvature corrections. We find that when the coupling constant exceeds a critical value, solutions with a frequency approaching zero appear. In the finite-order corrections case $n=2$ (Gauss-Bonnet gravity), the matter field and energy density diverge near the origin as $ω\to 0$. In contrast, for $n\geq 3$, the divergence is efficiently suppressed, both the field and the energy density remain finite everywhere, and both the matter field and energy density remain finite everywhere. In the limit $ω\to 0$, a class of horizonless frozen star solutions emerges, which are referred to ``frozen stars". Importantly, frozen stars contain neither curvature singularities nor event horizons. These frozen stars develop a critical horizon at a finite radius $r_c$, where $-g_{tt}$ and $1/g_{rr}$ approach zero. The frozen star is indistinguishable from that of an extremal black hole outside $r_c$, and its compactness can reach the extremal black hole value.