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The relation $R$ can be empty; i.e., if $\kappa$ is the least ordinal such that $L_\kappa(\mathbb{R})$ is admissible, then the structure $(L_\kappa; \in, \emptyset)$ is a companion of the pointclass $\mathrm{IND}$ of inductive sets. We can prove a somewhat more general statement.

Assume that $\kappa$ is an ordinal such that

• $J_\kappa(\mathbb{R})$ is admissible (in which case $J_\kappa(\mathbb{R}) = L_\kappa(\mathbb{R})$, but there may be points later in the argument where it is necessary to use the Jensen hierarchy) and

• $\kappa$ begins a $\Sigma_1$-gap in $L(\mathbb{R})$, meaning that $J_\alpha(\mathbb{R})$ is not a $\Sigma_1(\mathbb{R} \cup \{\mathbb{R}\})$-elementary substructure of $J_\kappa(\mathbb{R})$ for any ordinal $\alpha <\kappa$.

Then the structure $(L_\kappa; \in, \emptyset)$ is a companion for the pointclass $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. (So letting $J_\kappa(\mathbb{R})$ be the least admissible level of $L(\mathbb{R})$ we obtain the desired result for the pointclass $\mathrm{IND}$.)

The criterion for "companionship" that was not clear to me when posting the question was projectability: the existence of a $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$.

The existence of a $\Sigma_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ is well-known. Moreover, from this we can easily get a $\Delta_1^{J_\kappa(\mathbb{R})}$ partial surjection as follows.

(I have edited my answer below to replace the more convoluted argument that I wrote before.)

Let $F: \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ be a $\Sigma_1^{J_\kappa(\mathbb{R})}$ partial surjection and let $\theta$ be a $\Sigma_1$ formula defining it over $J_\kappa(\mathbb{R})$. Then define $G: \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ by letting $(x,a) \in G$ if and only if there is an $\alpha < \kappa$ such that $$J_\alpha(\mathbb{R}) \models \theta(x,a) \quad\And\quad \forall \xi < \alpha\, J_\xi(\mathbb{R}) \models \forall y \in \mathbb{R}\, \neg\varphi(y,a).$$ Then $F \subset G$ and it is easy to check that $G$ is $\Delta_1^{J_\kappa(\mathbb{R})}$ and that the range of $G$ is equal to the range of $F$, which is all of $J_\kappa(\mathbb{R})$.

The relation $R$ can be empty; i.e., if $\kappa$ is the least ordinal such that $L_\kappa(\mathbb{R})$ is admissible, then the structure $(L_\kappa; \in, \emptyset)$ is a companion of the pointclass $\mathrm{IND}$ of inductive sets. We can prove a somewhat more general statement.

Assume that $\kappa$ is an ordinal such that

• $J_\kappa(\mathbb{R})$ is admissible (in which case $J_\kappa(\mathbb{R}) = L_\kappa(\mathbb{R})$, but there may be points later in the argument where it is necessary to use the Jensen hierarchy) and

• $\kappa$ begins a $\Sigma_1$-gap in $L(\mathbb{R})$, meaning that $J_\alpha(\mathbb{R})$ is not a $\Sigma_1(\mathbb{R} \cup \{\mathbb{R}\})$-elementary substructure of $J_\kappa(\mathbb{R})$ for any ordinal $\alpha <\kappa$.

Then the structure $(L_\kappa; \in, \emptyset)$ is a companion for the pointclass $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. (So letting $J_\kappa(\mathbb{R})$ be the least admissible level of $L(\mathbb{R})$ we obtain the desired result for the pointclass $\mathrm{IND}$.)

The criterion for "companionship" that was not clear to me when posting the question was projectability: the existence of a $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$.

The existence of a $\Sigma_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ is well-known. Moreover, from this we can easily get a $\Delta_1^{J_\kappa(\mathbb{R})}$ partial surjection as follows.

(I have edited my answer below to replace the more convoluted argument that I wrote before.)

Let $F: \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ be a $\Sigma_1^{J_\kappa(\mathbb{R})}$ partial surjection and let $\theta$ be a $\Sigma_1$ formula defining it over $J_\kappa(\mathbb{R})$. Then define $G: \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ by letting $(x,a) \in G$ if and only if there is an $\alpha < \kappa$ such that $$J_\alpha(\mathbb{R}) \models \theta(x,a) \quad\And\quad \forall \xi < \alpha\, J_\xi(\mathbb{R}) \models \forall y \in \mathbb{R}\, \neg\theta(y,a).$$ Then $F \subset G$ and it is easy to check that $G$ is $\Delta_1^{J_\kappa(\mathbb{R})}$ and that the range of $G$ is equal to the range of $F$, which is all of $J_\kappa(\mathbb{R})$.

The relation $R$ can be empty; i.e., if $\kappa$ is the least ordinal such that $L_\kappa(\mathbb{R})$ is admissible, then the structure $(L_\kappa; \in, \emptyset)$ is a companion of the pointclass $\mathrm{IND}$ of inductive sets. We can prove a somewhat more general statement.

Assume that $\kappa$ is an ordinal such that

• $J_\kappa(\mathbb{R})$ is admissible (in which case $J_\kappa(\mathbb{R}) = L_\kappa(\mathbb{R})$, but there may be points later in the argument where it is necessary to use the Jensen hierarchy) and

• $\kappa$ begins a $\Sigma_1$-gap in $L(\mathbb{R})$, meaning that $J_\alpha(\mathbb{R})$ is not a $\Sigma_1(\mathbb{R} \cup \{\mathbb{R}\})$-elementary substructure of $J_\kappa(\mathbb{R})$ for any ordinal $\alpha <\kappa$.

Then the structure $(L_\kappa; \in, \emptyset)$ is a companion for the pointclass $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. (So letting $J_\kappa(\mathbb{R})$ be the least admissible level of $L(\mathbb{R})$ we obtain the desired result for the pointclass $\mathrm{IND}$.)

The criterion for "companionship" that was not clear to me when posting the question was projectability: the existence of a $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$. The existence of a $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ is well-known. It turns out that if we use a bit of care when defining the partial surjection then it will be $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ (although its domain can only be $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$.)

Letting $(\varphi_n : n<\omega)$ be an effective enumeration of $\Sigma_1$ formulas, we define a partial surjection $F:\omega \times \mathbb{R}\times \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ by setting $F(n,x,y) = S$ if there is an ordinal $\alpha < \kappa$ such that

1. $\alpha+1$ begins a gap as witnessed by $\varphi_n$ and $x$, meaning that $\varphi_n[x,\mathbb{R}]$ holds in $J_{\alpha+1}(\mathbb{R})$ but not in any earlier level of the Jensen hierarchy.

2. $S \in J_{\alpha+1}(\mathbb{R})$.

3. For all $\beta < \alpha$, if $\beta+1$ begins a gap then $S \notin J_{\beta+1}(\mathbb{R})$

4. $f_{\alpha+1}(y) = S$ where $f_{\alpha+1}$ denotes the canonical partial surjection $\mathbb{R} \dashrightarrow J_{\alpha+1}(\mathbb{R})$ that exists because $J_{\alpha+1}(\mathbb{R})$ must project to $\mathbb{R}$ in order for $\alpha+1$ to begin a gap (the level of definability of $f_{\alpha+1}$ over $J_{\alpha+1}(\mathbb{R})$ is not important.)

It is straightforward to show that we have indeed defined a partial surjection $\omega \times \mathbb{R}\times \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ whose graph is $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. But in fact the graph is $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$: by condition 3, to check that $F(n,x,y) \ne S$ we only have to find some ordinal $\alpha < \kappa$ such that $\alpha+1$ begins a gap and $S \in J_{\alpha+1}(\mathbb{R})$ (which we can always do) and then verify some local statement about $J_{\alpha+1}(\mathbb{R})$. Identifying $\omega \times \mathbb{R}\times \mathbb{R}$ with $\mathbb{R}$, we have the desired $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ witnessing projectability.

The relation $R$ can be empty; i.e., if $\kappa$ is the least ordinal such that $L_\kappa(\mathbb{R})$ is admissible, then the structure $(L_\kappa; \in, \emptyset)$ is a companion of the pointclass $\mathrm{IND}$ of inductive sets. We can prove a somewhat more general statement.

Assume that $\kappa$ is an ordinal such that

• $J_\kappa(\mathbb{R})$ is admissible (in which case $J_\kappa(\mathbb{R}) = L_\kappa(\mathbb{R})$, but there may be points later in the argument where it is necessary to use the Jensen hierarchy) and

• $\kappa$ begins a $\Sigma_1$-gap in $L(\mathbb{R})$, meaning that $J_\alpha(\mathbb{R})$ is not a $\Sigma_1(\mathbb{R} \cup \{\mathbb{R}\})$-elementary substructure of $J_\kappa(\mathbb{R})$ for any ordinal $\alpha <\kappa$.

Then the structure $(L_\kappa; \in, \emptyset)$ is a companion for the pointclass $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. (So letting $J_\kappa(\mathbb{R})$ be the least admissible level of $L(\mathbb{R})$ we obtain the desired result for the pointclass $\mathrm{IND}$.)

The criterion for "companionship" that was not clear to me when posting the question was projectability: the existence of a $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$.

The existence of a $\Sigma_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ is well-known. Moreover, from this we can easily get a $\Delta_1^{J_\kappa(\mathbb{R})}$ partial surjection as follows.

(I have edited my answer below to replace the more convoluted argument that I wrote before.)

Let $F: \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ be a $\Sigma_1^{J_\kappa(\mathbb{R})}$ partial surjection and let $\theta$ be a $\Sigma_1$ formula defining it over $J_\kappa(\mathbb{R})$. Then define $G: \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ by letting $(x,a) \in G$ if and only if there is an $\alpha < \kappa$ such that $$J_\alpha(\mathbb{R}) \models \theta(x,a) \quad\And\quad \forall \xi < \alpha\, J_\xi(\mathbb{R}) \models \forall y \in \mathbb{R}\, \neg\varphi(y,a).$$ Then $F \subset G$ and it is easy to check that $G$ is $\Delta_1^{J_\kappa(\mathbb{R})}$ and that the range of $G$ is equal to the range of $F$, which is all of $J_\kappa(\mathbb{R})$.

The relation $R$ can be empty; i.e., if $\kappa$ is the least ordinal such that $L_\kappa(\mathbb{R})$ is admissible, then the structure $(L_\kappa; \in, \emptyset)$ is a companion of the pointclass $\mathrm{IND}$ of inductive sets. We can prove a somewhat more general statement.

Assume that $\kappa$ is an ordinal such that

• $J_\kappa(\mathbb{R})$ is admissible (in which case $J_\kappa(\mathbb{R}) = L_\kappa(\mathbb{R})$, but there may be points later in the argument where it is necessary to use the Jensen hierarchy) and

• $\kappa$ begins a $\Sigma_1$-gap in $L(\mathbb{R})$, meaning that $J_\alpha(\mathbb{R})$ is not a $\Sigma_1(\mathbb{R} \cup \{\mathbb{R}\})$-elementary substructure of $J_\kappa(\mathbb{R})$ for any ordinal $\alpha <\kappa$.

Then the structure $(L_\kappa; \in, \emptyset)$ is a companion for the pointclass $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. (So letting $J_\kappa(\mathbb{R})$ be the least admissible level of $L(\mathbb{R})$ we obtain the desired result for the pointclass $\mathrm{IND}$.)

The criterion for "companionship" that was not clear to me when posting the question was projectability: the existence of a $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$. The existence of a $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ is well-known. It turns out that if we use a bit of care when defining the partial surjection then it will be $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ (although its domain can only be $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$.)

Letting $(\varphi_n : n<\omega)$ be an effective enumeration of $\Sigma_1$ formulas, we define a partial surjection $F:\omega \times \mathbb{R}\times \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ by setting $F(n,x,y) = S$ if there is an ordinal $\alpha < \kappa$ such that

• $\alpha+1$ begins a gap as witnessed by $\varphi_n$ and $x$, meaning that $\varphi_n[x,\mathbb{R}]$ holds in $J_{\alpha+1}(\mathbb{R})$ but not in any earlier level of the Jensen hierarchy.

• $S \in J_{\alpha+1}(\mathbb{R})$.

• For all $\beta < \alpha$, if $\beta+1$ begins a gap then $S \notin J_{\beta+1}(\mathbb{R})$

• $f_{\alpha+1}(y) = S$ where $f_{\alpha+1}$ denotes the canonical partial surjection $\mathbb{R} \dashrightarrow J_{\alpha+1}(\mathbb{R})$ that exists because $J_{\alpha+1}(\mathbb{R})$ must project to $\mathbb{R}$ in order for $\alpha+1$ to begin a gap (the level of definability of $f_{\alpha+1}$ over $J_{\alpha+1}(\mathbb{R})$ is not important.)

It is straightforward to show that this $F$ is indeed a partial surjection and is $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. But in fact it is $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$, because to check that $F(n,x,y) \ne S$ we only have to find some $\Sigma_1$-gap that begins past the place in the Jensen hierarchy where $S$ appears and verify some local statement there. Identifying $\omega \times \mathbb{R}\times \mathbb{R}$ with $\mathbb{R}$, we have the desired $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ witnessing projectability.

The relation $R$ can be empty; i.e., if $\kappa$ is the least ordinal such that $L_\kappa(\mathbb{R})$ is admissible, then the structure $(L_\kappa; \in, \emptyset)$ is a companion of the pointclass $\mathrm{IND}$ of inductive sets. We can prove a somewhat more general statement.

Assume that $\kappa$ is an ordinal such that

• $J_\kappa(\mathbb{R})$ is admissible (in which case $J_\kappa(\mathbb{R}) = L_\kappa(\mathbb{R})$, but there may be points later in the argument where it is necessary to use the Jensen hierarchy) and

• $\kappa$ begins a $\Sigma_1$-gap in $L(\mathbb{R})$, meaning that $J_\alpha(\mathbb{R})$ is not a $\Sigma_1(\mathbb{R} \cup \{\mathbb{R}\})$-elementary substructure of $J_\kappa(\mathbb{R})$ for any ordinal $\alpha <\kappa$.

Then the structure $(L_\kappa; \in, \emptyset)$ is a companion for the pointclass $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. (So letting $J_\kappa(\mathbb{R})$ be the least admissible level of $L(\mathbb{R})$ we obtain the desired result for the pointclass $\mathrm{IND}$.)

The criterion for "companionship" that was not clear to me when posting the question was projectability: the existence of a $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$. The existence of a $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ is well-known. It turns out that if we use a bit of care when defining the partial surjection then it will be $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ (although its domain can only be $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$.)

Letting $(\varphi_n : n<\omega)$ be an effective enumeration of $\Sigma_1$ formulas, we define a partial surjection $F:\omega \times \mathbb{R}\times \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ by setting $F(n,x,y) = S$ if there is an ordinal $\alpha < \kappa$ such that

1. $\alpha+1$ begins a gap as witnessed by $\varphi_n$ and $x$, meaning that $\varphi_n[x,\mathbb{R}]$ holds in $J_{\alpha+1}(\mathbb{R})$ but not in any earlier level of the Jensen hierarchy.

2. $S \in J_{\alpha+1}(\mathbb{R})$.

3. For all $\beta < \alpha$, if $\beta+1$ begins a gap then $S \notin J_{\beta+1}(\mathbb{R})$

4. $f_{\alpha+1}(y) = S$ where $f_{\alpha+1}$ denotes the canonical partial surjection $\mathbb{R} \dashrightarrow J_{\alpha+1}(\mathbb{R})$ that exists because $J_{\alpha+1}(\mathbb{R})$ must project to $\mathbb{R}$ in order for $\alpha+1$ to begin a gap (the level of definability of $f_{\alpha+1}$ over $J_{\alpha+1}(\mathbb{R})$ is not important.)

It is straightforward to show that we have indeed defined a partial surjection $\omega \times \mathbb{R}\times \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ whose graph is $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. But in fact the graph is $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$: by condition 3, to check that $F(n,x,y) \ne S$ we only have to find some ordinal $\alpha < \kappa$ such that $\alpha+1$ begins a gap and $S \in J_{\alpha+1}(\mathbb{R})$ (which we can always do) and then verify some local statement about $J_{\alpha+1}(\mathbb{R})$. Identifying $\omega \times \mathbb{R}\times \mathbb{R}$ with $\mathbb{R}$, we have the desired $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ witnessing projectability.