In this blog post by Gowers on Borel determinacy, Andres Caicedo says the following in a comment (slightly rephrased).

Let $\mathsf{ZFC^-}$ be $\mathsf{ZFC}$ without power set and $\mathsf{ZC^-}$ be $\mathsf{ZFC^-}$ without replacement, then working in $\mathsf{ZC^-}+\Sigma_1\text{-replacement}$ one can show that, if $\alpha$ is a recursive ordinal such that all $\Sigma_{\alpha+3}$-games on $\mathbb{N}^{<\mathbb{N}}$ are determined, then there is an ordinal $\beta_\alpha$ such that $L_{\beta_\alpha}\models\mathsf{ZFC^-}+\mathcal{P}^\alpha(\omega)\text{ exists}$, where $\mathcal{P}^\alpha(\omega)$ means the $\alpha$-th iteration of power set.

My questions are:

1. Caicedo attributes this to unpublished work of Martin. Is there a published account now?

2. $\Sigma_1\text{-replacement}$ is needed for basic theory of ordinals, and the result clearly fails in $\mathsf{ZC^-}$ because $V_{\omega+\omega}$ satisfies full Borel determinacy. Is it nevertheless possible to prove in $\mathsf{ZC^-}+\text{Borel Determinacy}$ the consistency of $\mathsf{ZFC^-}+\mathcal{P}^\alpha(\omega)\text{ exists}$ or something similar? What if $\mathsf{ZC^-}$ is further weakened to subsystems of second order arithemtic? I am aware of some results regarding lower levels like $\Delta^0_n$-determinacy for $n\approx 4$, but couldn't find anything about strength of full Borel determinacy.

In this blog post by Gowers on Borel determinacy, Andres Caicedo says the following in a comment (slightly rephrased).

Let $\mathsf{ZFC^-}$ be $\mathsf{ZFC}$ without power set and $\mathsf{ZC^-}$ be $\mathsf{ZFC^-}$ without replacement, then working in $\mathsf{ZC^-}+\Sigma_1\text{-replacement}$ one can show that, if $\alpha$ is a recursive ordinal such that all $\Sigma_{\alpha+3}$-games on $\mathbb{N}^{<\mathbb{N}}$ are determined, then there is an ordinal $\beta_\alpha$ such that $L_{\beta_\alpha}\models\mathsf{ZFC^-}+\mathcal{P}^\alpha(\omega)\text{ exists}$, where $\mathcal{P}^\alpha(\omega)$ means the $\alpha$-th iteration of power set.

My questions are:

1. Caicedo attributes this to unpublished work of Martin. Is there a published account now?

2. $\Sigma_1\text{-replacement}$ is needed for basic theory of ordinals, and the result clearly fails in $\mathsf{ZC^-}$ because $V_{\omega+\omega}$ satisfies full Borel determinacy. Is it nevertheless possible to prove in $\mathsf{ZC^-}+\text{Borel Determinacy}$ the consistency of $\mathsf{ZFC^-}+\mathcal{P}^\alpha(\omega)\text{ exists}$ or something similar? What if $\mathsf{ZC^-}$ is further weakened to subsystems of second order arithemtic? I am aware of some results regarding lower levels like $\Delta^0_n$-determinacy for $n\approx 4$, but couldn't find anything about strength of full Borel determinacy. A natural question is whether we can define in $\mathsf{ZC^-}$ an $L$-like inner model in the first place; apparently we can.