I wonder if tighter guarantees can be given if tools can work with constructs like D's contracts. These are extra sections in each function that are intended to check invariants. If these invariants change, then they were either broken and needed fixing or the function had a semantic change.
That's an interesting idea. It seems like it would be a way for tools to flag "this function is likely to have changed in an interesting way", but changing invariants doesn't necessarily mean the function breaks semver.
For one, an invariant might be changed syntactically, without actually changing what it is asserting, reducing to exactly my example above: in its simplest form, the contract could go from in(true) to in(halts("...")).
Secondly, an contract could have been made more permissive, e.g. in(x > 1) becoming in(x > 0), or out(y > 0) becoming out(y > 1). Assuming violating a contract is regarded as a programming error (as in, it isn't considered breaking to go from failing an assertion to not failing), these are also non-breaking changes.
Lastly, changing behaviour doesn't necessarily mean changing invariants/contracts.