The answer is slightly complicated. All of the Java code is written for Java 7 right now, so you could not
compile it with Java 6. It would not be too hard to change the parts that don't compile in 6 to do so, if you want to go that way.
core/ however will run in Java 6 since it works in Android, which is effectively the Java 6 API. For example, try-with-resources is not used since that would actually cause the code to refer to AutoCloseable, which is only in 7. javase/ however does use try-with-closeable; it's only going to run in 7, as-is.
I am still not sure if core/ works in the standard Java 6 VM. It won't unless you compile it for 6 since the major/minor bytecode version won't match. And I forget whether javac itself lets you cross-compile from 7 to 6. It works on Android since dex/aapt will consume Java 7 bytecode even though the API level is only 6.