This question is about the notion of a companion for a Spector class, as defined in Moschovakis's book Elementary Induction on Abstract Structures. I am interested in Spector classes on $\mathbb{R}$, which are just a type of boldface pointclass. The smallest one is IND, the class of (boldface) inductive sets, which I will consider as a typical example.

The companion of a Spector class $\bf \Gamma$ on $\mathbb{R}$ is a structure $(M,\in,R)$ with certain properties (listed in the book) such that $\bf \Gamma$ is the class of pointsets that are $\Sigma_1$-definable in $M$ with real parameters from the relation $R$. The companion of $\bf \Gamma$ is not unique, but its underlying set $M$ is unique and also the class of relations on $M$ that are $\Sigma_1$-definable from $R$ in $M$ with real parameters is unique.

The pointclass IND can also be described as the class of pointsets that are $\Sigma_1$-definable over $M$ from parameters in $\mathbb{R} \cup \lbrace\mathbb{R} \rbrace$ where $M = L_\kappa(\mathbb{R})$ is the least admissible level of $L(\mathbb{R})$. We must allow $\lbrace\mathbb{R} \rbrace$ itself as a parameter here or we would just get the ${\bf \Sigma}^1_2$ sets. For any companion $(M,\in,R)$ of IND we must have $M = L_\kappa(\mathbb{R})$. I am interested in why allowing $\lbrace\mathbb{R} \rbrace$ as a parameter in $\Sigma_1^M$ definitions of pointsets has the same effect as allowing $R$ as a predicate.

(1) Is there a companion $(M,\in,R)$ of IND where the relation $R$ has a simple form (simpler than in Moschovakis's general construction of a companion?)

(2) Is $\lbrace{\mathbb{R}\rbrace}$ definable over $M$ from $R$ by a $\Delta_1$ formula, and if so, can this be proved for general $\bf \Gamma$ using abstract properties of the companion?

This question is about the notion of a companion for a Spector class, as defined in Moschovakis's book Elementary Induction on Abstract Structures. I am interested in Spector classes on $\mathbb{R}$, which are just a type of boldface pointclass. The smallest one is IND, the class of (boldface) inductive sets, which I will consider as a typical example.

The companion of a Spector class $\bf \Gamma$ on $\mathbb{R}$ is a structure $(M,\in,R)$ with certain properties (listed in the book) such that $\bf \Gamma$ is the class of pointsets that are $\Sigma_1$-definable in $M$ with real parameters from the relation $R$. The companion of $\bf \Gamma$ is not unique, but its underlying set $M$ is unique and also the class of relations on $M$ that are $\Sigma_1$-definable from $R$ in $M$ with real parameters is unique.

The pointclass IND can also be described as the class of pointsets that are $\Sigma_1$-definable over $M$ from parameters in $\mathbb{R} \cup \lbrace\mathbb{R} \rbrace$ where $M = L_\kappa(\mathbb{R})$ is the least admissible level of $L(\mathbb{R})$. We must allow $\lbrace\mathbb{R} \rbrace$ itself as a parameter here or we would just get the ${\bf \Sigma}^1_2$ sets. For any companion $(M,\in,R)$ of IND we must have $M = L_\kappa(\mathbb{R})$.

Question: Is there a companion $(M,\in,R)$ of IND where the relation $R$ has a simple definition over $M = L_\kappa(\mathbb{R})$ (simpler than in Moschovakis's general construction of a companion?) Maybe something that is already studied in the fine structure of $L(\mathbb{R})$?