I haven't yet checked the formal conditions to see whether they'll apply to the matching problems, but I bet they will. (The fact that the paper contains no instances of the word "matching", meaning it never explains why this condition doesn't apply to the matching problem even though it looks like it should, is not encouraging.) If so this proof cannot be correct.
Here's a link to a list of P vs NP proofs that didn't work out. "Making it to arxiv" and "sounding reasonable" are usually pretty good filters but for some reason this problem has attracted a lot of trouble.
The fact that there are no inequalities anywhere throughout the paper is the real tip-off that this is not a serious attempt.