We consider the fate of $1/N$ expansions in unstable many-body quantum systems, as realized by a quench across criticality, and show the emergence of ${rm e}^{2lambda t}/N$ as a renormalized parameter ruling the quantum-classical transition and accounting nonperturbatively for the local divergence rate $lambda$ of mean-field solutions. In terms of ${rm e}^{2lambda t}/N$, quasiclassical expansions of paradigmatic examples of criticality, like the self-trapping transition in an integrable Bose-Hubbard dimer and the generic instability of attractive bosonic systems toward soliton formation, are pushed to arbitrarily high orders. The agreement with numerical simulations supports the general nature of our results in the appropriately combined long-time $lambda tto infty$ quasiclassical $Nto infty$ regime, out of reach of expansions in the bare parameter $1/N$. For scrambling in many-body hyperbolic systems, our results provide formal grounds to a conjectured multiexponential form of out-of-time-ordered correlators.
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