$\Sigma_1$ gaps as derived models and correctness of mice

TL;DR

This paper investigates $oldsymbol{}$-gaps in models of set theory with determinacy and constructibility, analyzing derived models of premice and their universes, with implications for conjectures on inner models and mice.

Contribution

It introduces a new analysis of $J_eta(R)$ as derived models of premice in generic extensions, advancing understanding of their structure and relation to conjectures on mice.

Findings

$J_eta(R)$ can be viewed as a derived model of a premouse $P$.

Under certain conditions, $J_eta(R)$ and a derived model $D$ share the same universe.

Preliminary results towards a conjecture of Rudominer and Steel on $(L(R))^M$ for $oldsymbol{}$-small mice.

Abstract

Assume ZF + AD + V=L(R). Let $[\alpha,\beta]$ be a $\Sigma_1$ gap with $J_\alpha(R)$ admissible. We analyze $J_\beta(R)$ as a natural form of "derived model" of a premouse $P$, where $P$ is found in a generic extension of $V$. In particular, we will have $\mathcal{P}(R)\cap J_\beta(R)=\mathcal{P}(R)\cap D$, and if $J_\beta(R)\models$ "$\Theta$ exists", then $J_\beta(R)$ and $D$ in fact have the same universe. This analysis will be employed in further work, yet to appear, toward a resolution of a conjecture of Rudominer and Steel on the nature of $(L(R))^M$, for $\omega$-small mice $M$. We also establish some preliminary work toward this conjecture in the present paper.