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Abstract We perform numerical comparison of the fragmentation mechanism of charmonium production ($$g\,g\rightarrow c\,{\bar{c}}$$ gg→cc¯ followed by $$c\rightarrow \psi \,c$$ c→ψc ) with the full leading order calculation ($$g\,g\rightarrow \psi \,c\,{\bar{c}}$$ gg→ψcc¯ at $${{\mathcal {O}}}(\alpha _s^4)$$ O(αs4) ). We conclude that the non-fragmentation contributions remain important up to $$J/\psi $$ J/ψ transverse momenta about as large as 40 GeV, thus making questionable the applicability of the fragmentation approximation at smaller transverse momenta.