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We establish the descriptive set theoretic representation of the mouse$M_n^{\#}$, which is called $0^{(n+1)\#}$. This part partially finishes thecase $n=2$ by establishing the higher level analog of the EM blueprintdefinition of $0^{\#}$. From this, we prove the level-4 Kechris-Martin Theoremand deal with the case $n=3$.

The higher sharp III: An EM blueprint of $0^{3\#}$ and the level-4 Kechris-Martin