= Solution
For every integer $m\in\{0,\ldots,N\}$,
$$
m+2\sum_{K=1}^N(m-K)^+
=m+2\sum_{K=1}^{m-1}(m-K)
=m+m(m-1)=m^2.
$$
Thus the <static replication on a finite terminal support> consists of \b[one share and two calls of every listed strike], held to maturity:
$$
\boxed{S_T^2=S_T+2\sum_{K=1}^N(S_T-K)^+.}
$$
The strike-$N$ call contributes zero at maturity but is harmless in this identity. The portfolio costs $S_0+2\sum_{K=1}^NC_0(K)$. No distributional assumption on the <stock> is needed beyond its stated terminal support.
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