= Solution
\b[No threshold can work uniformly over all <holomorphic vector bundles> $E$.] Given a proposed threshold $M$, choose an integer $m\geq M$ and let $E=F^m$. The <canonical trivialization of a line bundle tensored with its dual> gives
$$
E\otimes F^{-m}\cong\mathcal O_X,
\qquad H^0(X,\mathcal O_X)\ni1\ne0.
$$
This contradicts vanishing at that value of $m$. In the proof above, the curvature of $E=F^m$ exactly offsets the curvature contribution from $F^{-m}$, explaining why the bound cannot be independent of $E$.
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