Have I proven P equals NP if I find an amortized O(n) algorithm for Subset Sum
This is not what amortized analysis means: you can not infer the worst case runtime of an algorithm by plotting the actual execution time on larger and larger instances of some specific structure. Sadly, that doesn't prove you have a polynomial-time algorithm. Even when you want to do experimental work, there are other parameters to vary besides the size of the input that will have a major impact on the runtime of the proposed solution. If you do so, you will have a result that roughly says &quot;here's an algorithm that performs well in practice for instances that look like this, but I can't give any theoretical guarantees&quot;.
To summarize, it's not impressive to have an algorithm which solves huge instances belonging to a specific family of instances of an NP-hard problem quickly. What is impressive (or rather extremely remarkable), is to have an algorithm that you can prove runs in time that is polynomial in the size of the input regardless of the structure of the instance.