You just can't keep any system bug free, the only thing you can do is improve your detection systems, codebase, and documentation and over time bugs will pop up less and less
* Make illegal states unrepresentable
* pure, static functions wherever possible
* use test generators (a la quick check)
* if you can't unit test important code, try refactoring until you can, rather than immediately reaching for integration tests and mocks.
I don't think you'll be able to model your system well enough to be able to run it through a formal language (I'm thinking about TLA+). But Leslie Lamport did say something about "the algorithm is not the code". That is, if you have any tricky algorithms that you invented for this project, you can formally prove those algorithms in TLA+ and then translate them to code.
Count yourself lucky. Most teams find it impossible! :)