Short version: does anyone know of any good sources on class-forcing over [E-closed](http://www.math.harvard.edu/~sacks/erecint.pdf), non-admissible sets?

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Longer version: A problem I'm working on has reached an interesting conclusion - I've managed to show that, under Appropriate Hypotheses, there is a countable ordinal $\gamma$ which is the height of a transitive set with some surprisingly nice determinacy properties. The ordinal in question, though, is annoyingly high, and so I'm trying to bring it down a bit. The problem is that building the transitive set in question involves a somewhat messy iterated forcing over $L_\gamma$. If all the pieces of the iteration are *sets* in $L_\gamma$, then everything is nice and easy - but ensuring that of course is what makes $\gamma$ so big. I *think* I've figured out how to bump $\gamma$ down a fair ways via class forcing over admissible sets; however, the real accomplishment would be to bump $\gamma$ down even further and show that the forcing behaves nicely even when $L_\gamma$ is non-admissible but E-closed!

. . . Which, unfortunately, is a bit of a non-starter, since I can find *absolutely nothing* about class forcing over E-closed, non-admissible sets.

To give some context for why I don't want to attack this entirely from scratch, let me briefly describe the situation. Set forcing, of course, preserves ZFC and ZF - which raises the question of what fragments T $\subset $ ZFC are also preserved by set forcing. With some care, but not too much difficulty, we can show that set forcing preserves the very tiny fragment KP - that is, a set forcing extension of an *admissible set* is again admissible. (I believe the first person to observe this [was Ershov](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0ahUKEwjFlKH5rePPAhVh2IMKHcvGAWUQFggsMAM&url=http%3A%2F%2Flink.springer.com%2Fcontent%2Fpdf%2F10.1007%2FBF01978554.pdf&usg=AFQjCNHbJU1OoYJYRaUC_Wf9qBq3MpPIzw&sig2=0QU6LwemR__qj2TVx4rOmQ).) Class forcing [is more complicated](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0ahUKEwi5x5_tr-PPAhWp8YMKHahMD2sQFggzMAM&url=http%3A%2F%2Fprojecteuclid.org%2Feuclid.bsl%2F1182353827&usg=AFQjCNFjVFryCDaDXsyinvBpLYvDeejqXA&sig2=UBYQma-fBHVc2kzJBlXlaw&bvm=bv.135974163,d.amc), but not too terrible.

At the E-closed but not admissible level, however, everything goes bonkers. Let $E(\omega_1)$ denote the least E-closed set containing $\omega_1$. Then there is a *set* forcing in $E(\omega_1)$ which does not preserve E-closedness! 

*(That forcing is the usual collapse of $\omega_1$ to $\omega$. The reason this does not preserve E-closedness is roughly: if $b$ is the real coding the generic collapse $G$ of $\omega_1$, then $E(\omega_1)[G]=E(b)$ if the former is E-closed. But $E(b)=L_{\omega_1^b}(b)$ is admissible since $b\subset\omega$; so $E(\omega_1)$, which has the same height as $E(b)$, must also be admissible. However, $E(\omega_1)$ is not admissible, since it admits divergence witnesses.)*

Now, there are sources on class forcing over *admissible* sets; [Jensen has some notes](https://www.mathematik.hu-berlin.de/~raesch/org/jensen/pdf/AS_6.pdf) about this, and [Friedman's article](https://books.google.com/books?id=5-2gnWhwzmsC&pg=PA129&lpg=PA129&dq=Sy+Friedman+introduction+admissibility+spectrum&source=bl&ots=h1Ie3_fpgP&sig=PFhcTN8MvZLqcWyoVpDCBI904_s&hl=en&sa=X&ved=0ahUKEwjHmaLJquPPAhUF2IMKHWyWDFAQ6AEIIzAB#v=onepage&q=Sy%20Friedman%20introduction%20admissibility%20spectrum&f=false) in an old "Philosophy, Logic, and Methodology of Science" volume gives a good description of almost disjoint coding in this context. Similarly, there are sources on forcing over E-closed sets - besides [an article by Sacks](http://www.sciencedirect.com/science/article/pii/0168007295000526) and [an article by Sacks and Slaman](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwj52IDZquPPAhVq1oMKHfLVBgQQFggcMAA&url=http%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2F0001870887900284&usg=AFQjCNFm14Offulcqi7yidb8uDnH2B1yvA&sig2=aadiJrbARODPirvBHgEwbA), there is [the thesis of Sherry Marcus](https://dspace.mit.edu/handle/1721.1/28018). However, as far as I can tell, *nobody* treats the problem of class forcing in an E-closed, non-admissible context.

So the question stands: does anyone know a source on class forcing over E-closed but non-admissible sets?