It's necessary to have a formal language to describe this design. Such a formal language needs to have one and only one interpretation. We should be able to take a document written with that formal language and verify that it meets our assumptions. We should also be able to verify that the final implementation is isomorphic to the design.
Luckily such tools are ubiquitous. You can use whatever programming language you like to specify your design. In the case where the programming language is defined poorly, then you must also decide what build system and run time environments you will support ahead of time.
You can document your assumptions using the same programming language. This is made easier if you use a "test framework", but if needed you can also roll your own. You can validate your design by running the tests against your design. Normally it is best to break up your design into "units" that make it more clear how to validate your assumptions. You can then add some extra validation that the composition of "units" transitively validates your assumptions.
You can verify that your final produce adheres to that design because there will be a 1:1 mapping from the design to the final product (i.e. the final product will be implemented using the same code as your design). There are wonderful tools that will tell you the "coverage" of code used in the final product that has been validated against the assumptions.
Finally, you can even specify your requirements using the same formal language and verify that the design meets the requirements. Normally you do this by validating that the "integration" of "units" meets your assumptions. This should not be done without individually validating the assumptions on the "units", though, because the "integration" can lead to exponentially growing alternatives. Normally it is infeasible to validate each alternative.
Yes, this reply is tongue in cheek, but I am not ignorant of formal specification methods. They have their place. That place is currently not in a professional software development shop. We have better methods at the moment. Possibly formal specifications methods will improve to the point where we can reasonably use them, but we aren't there yet.
