A little bit about science, proofs and buddhism bellow.
"Check out my reasoning and tell me where it's flawed."
1) To know that nothing stays the same you have to have a reliable, permanent memory.
2) If memory is impermanent, then it cannot be used to measure change.
3) If you cannot measure change then you can't know if anything is impermanent.
4) Impermanence is impossible to prove.
Domain and definitions
"to know", "the same", "reliable", "permanent" and "memory", all of these terms can be, individually, subject of many books. The same goes for "impermanent", "measure" and "change".
As they are, they can be very ambiguous, hard to understand their meanings and hard to understand how they relate. For example, does "permanent memory" mean inability to forget? Does "reliable memory" mean what is remembered is always precise and "the same"?
Playing this little game, if these questions are answered with "yes", then the requisites you posited seem to refer to some sort of ability not found in humans and animals, since we forget and our memory faculty is not reliable and does not work like reliable "data retrieval" mechanisms (which always "results" in the "same data"). Therefore, one can argue that if we don't have this sort of faculty, then what can we "know"?
And the really tricky part in the first and third propositions is the "to know" part: it imports the massive topic of epistemology into the problem. Pandora box opens with questions such as "what is knowledge?", and goes through "well, how do you know that our memories are unreliable and we forget?" (to which a witty person could reply: "I'm sorry, what are we talking about, again?").
At this point, you are in philosophical grounds and might feel like venturing into the nature of knowledge and what is possible to know, exploring the dungeons of theories of knowledge.
One might also age and die in this dungeon, ironically witnessing and suffering a lot of changes oneself, and still feeling uncertain of whether things are impermanent.
Science, Inferences and Proofs
Or, perhaps you might not want to philosophically penetrate every corner here and die without an answer. Then, some corners must be given up.
"to know" seems too philosophical, and perhaps you want to be scientific in some way. Interestingly, in science, "knowledge" is, roughly speaking, just something accepted, shared and rooted in what is public, accessible, reproducible -- or simply, "empirical". All that is needed is two observers agreeing on their descriptions of what they observe.
In this path, you are effectively sacrificing all the philosophical problems of knowledge in order to move on, on a hopefully more manageable ground. By making this sacrifice, you will also be forced to change your propositions that refer to "knowledge". Likely, the best you can do is talk about what is observed.
"Memory" is very problematic as well (do you really want to talk about whether there's knowledge or not for a being with imprecise memory such as humans? This puts you right back into philosophy asking what knowledge is...).
But I see other problems with your reasoning too, if you want to explore this in the domain of science.
The first problem are the inferences. Taking your propositions as they are for illustration, to undertake this project it will be clear you need to rigorously justify things like: how properties of memory affect possibility of knowledge (p.1), how they affect ability of measurement (p.2), how this ability relates to knowledge (p.3), and how the impossibility of measuring a property entails impossibility of proving same property (p.4).
For example, in p.1, if "to know" is simply what is agreed by two or more people, then how does a unrelated change in memory affects knowledge? According to the answer, your reasoning and formulations must be revised.
The second problem is "proof". And what is a proof? In mathematics and logic they are roughly structures drawn from rules of inference and axioms. In journalism, it's how journalists and media refer to scientific work. And in science (to the surprised of journalists) it's not advised to use it: "evidence" and "arguments" are commonly used in contexts of findings and theories, since as a matter of principle, "proofs" must be subject to refutation.
So, to insist on irrefutable proofs, you must sacrifice science. And to insist on science, you must sacrifice proofs. The same goes, roughly, for what is "possible" or "impossible" (p.4).
Reality and Buddhism
If you turned around from the philosophy dungeon trying to answer this question and preferred to stick with reality, at this point, I believe you concluded that the nature of knowledge won't help you decide if impermanence is experienced or not.
Also -- and if I was convincing enough so far -- I hope you have concluded that elaborating a proof that impermanence can't be proved is a futile exercise.
What is left, then, is a job much easier. To establish whether something is impermanent or not:
- start with an observation.
- If any alteration is perceived, then Q.E.D.