Unfortunately, tests can never be a complete specification unless the system is simple enough to have a finite set of possible inputs.

For all real-world software, a test suite tests a number of points in the space of possible inputs and we hope that those points generalize to pinning down the overall behavior of the implementation.

But there's no guarantee of that generalization. An implementation that fails a test is guaranteed to not implement the spec, but an implementation that passes all of the tests is not guaranteed to implement it.