Just going by the first section, I can anticipate a problem here. The last paragraph of the first section suggests that the problem of finding a Hamilton cycle

Just going by the first section, I can anticipate a problem here. The last paragraph of the first section suggests that the problem of finding a Hamilton cycle is not in P because some cycles are maximal solutions without being a maximum solution. However, the same could be said of the problem of finding a maximum matching -- a maximal matching need not be a maximum matching -- and yet that problem is in P. 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.