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

TL;DR

This paper develops a descriptive set theoretic framework for higher mouse models, specifically $0^{(n+1) ext{ extasciicircum}}$, and proves a level-4 Kechris-Martin theorem, advancing the understanding of inner model theory.

Contribution

It introduces a higher level EM blueprint for $0^{(n+1) ext{ extasciicircum}}$ and proves the level-4 Kechris-Martin Theorem, extending previous results to the case n=3.

Findings

Established the descriptive set theoretic representation of $0^{(n+1) ext{ extasciicircum}}$

Partially completed the case n=2 for the higher level EM blueprint

Proved the level-4 Kechris-Martin Theorem

Abstract

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