Most operating systems do two things: they isolate processes from each other; and they allow some limited communication between processes. There have been multiple attempts over more than 30 years to prove that an operating system actually achieves these goals; this is the first really convincing such proof. The proof is about the seL4 microkernel that had already been formally verified as implementing its formal specification. The paper takes us step by step through the proof as well as being very clear about the assumptions and gaps in the verification and the effort required for the verification.
Besides the proof, the paper focusses on three tricky areas:
Information flow through the scheduler. To avoid information flow, they took the common step of replacing the scheduler with a simple static round-robin scheduler between partitions. They then had to prove that this was sufficient to eliminate information flow.
Directional communication between processes. They need to show that there is no back-channel that allows bidirectional information flow.
What is the formal security specification? Given a set of processes, communication channels between them, etc. how do we formally state what can and cannot happen? There are many subtleties here that the paper walk us through. (But reference  promises to have even more detail.)
The paper ends with a lengthy discussion of the strength of the proof. There is little concern about the proof since it is machine-checked and it is based on the actual C code rather than a hand-written model of the C code. (Around the same time, the group was also working on translation validation to show that the C code matched the compiled binary.) They acknowledge gaps around their model of the hardware. The most significant issue, of course, is around covert channels – a problem that we the field has no good solutions for at present.
They also discuss ways in which they changed the design of seL4, surprising information leaks they discovered and which parts of the proof were hardest to complete.
This paper should be compared with the later work on verifying information flow properties of the mCertiKOS separation kernel. The big difference between the two is that this paper proves results in the presence of communication between different processes while the mCertiKOS paper proves results about a system in which inter-process communication has been disabled.