$Σ_1$ gaps as derived models and correctness of mice
Farmer Schlutzenberg, John Steel
TL;DR
The paper investigates $Σ_1$ gaps in $L(\mathbb{R})$ under $ZF+AD+V=L(R)$, framing $J_{β}(\mathbb{R})$ as a derived model of a premouse appearing in a generic extension and introducing the $\mathscr{M}$-hierarchy to realize these derived models within the codes. It develops inner-model techniques (tame projecting mice, P-construction, and mouse witnesses) to analyze the start and progression through admissible gaps, and constructs a systematic hierarchy to capture ordinal definability and definability hierarchies in this setting. A central goal is to advance a weak version of the Rudominer–Steel conjectures, showing that certain derived-model realizations exist and are definable under ω1-existence and related closure properties, and extending these insights through both admissible gaps and generic extensions. The work combines fine-structure, forcing, and iteration techniques to link derived models of premice with $L(\,\mathbb{R}\,)$-hierarchies, potentially resolving aspects of how $L(\mathbb{R})$-like universes arise as derived models of generic mice. The results provide a framework for identifying wellfounded derived models in the codes and for connecting these to the corresponding OT structures under determinacy hypotheses, with implications for the Rudominer–Steel theory of $L(\mathbb{R})$ in inner model theory.
Abstract
Assume ZF + AD + V=L(R). Let $[α,β]$ be a $Σ_1$ gap with $J_α(R)$ admissible. We analyze $J_β(R)$ 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 $\mathcal{P}(R)\cap J_β(R)=\mathcal{P}(R)\cap D$, and if $J_β(R)\models$ "$Θ$ exists", then $J_β(R)$ 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 $ω$-small mice $M$. We also establish some preliminary work toward this conjecture in the present paper.