SKI calculus golf: Half of a Church numeral
40 combinators

S(S(SI(K(S(S(KS)K)(K(S(S(KS)(S(KK)S))(K(S(KK)(S(S(KS)K)))))))))(K(S(SI(K(KI)))(K(KI)))))(KK)


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Generated with a little help from a slightly modified version of my answer to Combinatory Conundrum:

$$\begin{align*}
\textit{zero} &amp;= λf. λx. x = KI \\
\textit{succ} &amp;= λn. λf. λx. f\,(n\,f\,x) = S(S(KS)K) \\
\textit{zero-pair} &amp;= λf. f\,\textit{zero}\,\textit{zero} = S(SI(K\,\textit{zero}))(K\,\textit{zero}) \\
\textit{next-pair-helper} &amp;= λf. λm. λn. f\,n\,(\textit{succ}\,m) \\ &amp;= S(S(KS)(S(KK)S))(K(S(KK)\,\textit{succ})) \\
\textit{next-pair} &amp;= λp. λf. p\,(\textit{next-pair-helper}\,f) \\ &amp;= S(S(KS)K)(K\,\textit{next-pair-helper}) \\
\textit{half} &amp;= λn. n\,\textit{next-pair}\,\textit{zero-pair}\,K \\ &amp;= S(S(SI(K\,\textit{next-pair}))(K\,\textit{zero-pair}))(KK)
\end{align*}$$