A variant of Radin forcing

Question

Suppose $\kappa$ is a large cardinal (strong cardinal seems to be enough). Is there a forcing notion $\mathbb{R}$ with the following properties:

$(1)$ Forcing with $\mathbb{R}$ adds a club $C$ into $\kappa$ of $V$-regular cardinals.

$(2)$ Forcing with $\mathbb{R}$ preserves cardinals and the regularity of $\kappa.$

$(3)$ Any condition in $\mathbb{R}$ is a finite sequence $p = (p_0, \dots, p_n)$, where to each $p_i, i<n$, a cardinal $\kappa(p_i) \in C$ is associated (we may assume $\kappa(p_n)=\kappa$).

$(4)$ Given a condition $p=(p_0, \dots, p_n)$ and $i<n,$ there exists $\gamma < \kappa$ such that $p$ forces $\kappa(p_i)=\dot{C}(\gamma)$, where $C(\gamma)$ denotes the $\gamma$-th element of $C$.

Note that if we remove clause $(4)$, then Radin forcing has the required properties.

Answer 1

A mild variant of the standard Radin forcing has property (4).

Let $\mathbb{R}$ be the forcing notion with conditions $p = (p_0, \dots, p_n)$ such that $p_i = (u_i, A_i)$, $u_i$ is a measure sequence and if $\mathrm{len}( u_i) > 0$ then $A_i$ is large relative to all measures in $u_i$. We require also that if $\mathrm{len}(u_i) < \kappa(u_i)$ then $A_i \cap V_{\mathrm{len}(u_i) + 1} = \emptyset$ and for all $v\in A_i$, $\mathrm{len}(v) < \mathrm{len}(u_i)$. This requirement is the only difference between this forcing and the standard Radin forcing.

We require $\kappa(u_i)$ to be increasing and $\kappa(u_n) = \kappa$, $\mathrm{len}(u_n) = \kappa^{+}$.

The order on this poset is the same as in the standard Radin forcing.

Let $p = (p_0, \dots, p_n)\in \mathbb{R}$ be a condition. Let $\gamma_i$ be the position of $\kappa(p_i)$ in the generic club. Then we can compute $\gamma_i$ recursively as follows:

If $\mathrm{len}(u_0) \geq \kappa(u_0)$ then $\gamma_0 = \kappa(u_0)$. Otherwise, $\gamma_0 = \omega^{\mathrm{len}(u_0)}$ (ordinals exponentiation).

For $i > 0$, if $\mathrm{len}(u_i) \geq \kappa(u_i)$ then $\gamma_i = \kappa(u_i)$. Otherwise, $\gamma_i = \gamma_{i-1} + \omega^{\mathrm{len}(u_i)}$.