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.



A variant of Radin forcing
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, $\gamma\_i = \gamma\_{i-1} + \omega^{\mathrm{len}(u\_i)}$.


