The proof states that the C code of the kernel is a refinement of an high-level "abstract" specification of the kernel. This means that the C code only exhibits behaviours that are also exhibited by the high-level specification. The high-level specification (by construction) doesn't have certain bad behaviours that you often see in C programs: NULL pointer dereferences, buffer overruns, signed integer overflow, etc. Thus this class of bugs is proven to be absent. A good primer is available at: http://www.ertos.nicta.com.au/research/l4.verified/proof.pml
The original proof didn't make any claims about the high-level behaviour of the kernel, such as security. So for instance, the original kernel might have had a bug where you set up a system so that two processes shouldn't be able to communicate with each other, but after some unexpected sequence of API calls they form a communications channel between them.
NICTA in the following years after the initial proof carried out two security proofs: an integrity proof and an information flow proof. These proofs state that the capabilities possessed by a process determine (i) what other processes a particular process can modify; and (ii) what other processes the process can communicate with. Like the original proof, these security proofs (especially the second) have assumptions. In particular, they can only talk about parts of the system that are modelled. Timing channels, hidden CPU state, etc. not modelled by the proof may still allow information flow between processes, for instance.
Sorry, I don't think you can claim you have a mathematical proof of the correctness of your kernel when you assume this much.
You need to assume something. Every operating system kernel makes these assumptions: seL4 is just explicit about them. Some of the assumptions are bigger than others (management of caches is a pain to get right and hard to test, for instance). Some assumptions remain just due to a lack of time, such as proving the initialisation code of the kernel.
Additionally, as time has gone on the assumptions have been reduced. For instance, the work of Thomas Sewell and Magnus Myreen mean that the compiler no longer needs to be trusted: https://www.cl.cam.ac.uk/~mom22/pldi13.pdf
And where is this proof? I can't seem to find it anywhere on their web site.
The proof will be part of the open-source release. It consists of around 200,000 lines of Isabelle/HOL proof script, which is machine-checked. Anyone will be able to inspect/modify/check the proofs after the release.