We know how to approach this. You use a brutally simple microkernel such as SEL4 and do full proofs of correctness on it. There's a performance penalty for microkernels, maybe 20%, because there's more copying. There's a huge cost to making modifications, so modifications are rare.
The trouble with SEL4 is that it's not much more than a hypervisor. People tend to run Linux on top of it, which loses most of the security benefits.
Well, yeah, that's a problem.
But the bigger problem is that this works for jets, as long as you don't need updates. It doesn't work for general purpose computers, for office productivity software, for databases (is there an RDBMS with a correctness proof?), etc. It's not that one couldn't build such things, it's that the cost would be absolutely prohibitive.
Diagrams like this scare me.[1]
[1] https://new.abb.com/docs/librariesprovider78/eventos/jjtts-2...
Well, but those need new features from time to time, and certification would make that nigh impossible. I'd settle for memory-safe languages as the happy middle of the road.
And it proves nothing. And it's as expensive for every update, so forget updates.