For Logicians unfamiliar with Number Theory issues, I give below one quick example of why Theorem 4 (with two attached conditions, as stated in my papers) has major implications in Number Theory and thus should be decided in a serious and final manner.
From theorem 4 (with the two conditions attached), follows theorem 2; and from theorem 2 it follows immediately that, as stated on the opening page, is an integer and will give an infinite series that converges to an irrational number. Consider the excitement that was generated when a mathematician named Roger Apéry in 1978-79 proved the above irrationality for k=3. See http://www.ega-math.narod.ru/Apery1.htm (this site sometimes loads slowly). The irrationality for series with k odd, k>3 has been an open problem.
Further consider the links at the bottom of my opening page. Thus it should be clear that any Logician contemplating the issue of validity of the arguments in my proof would be helping decide a worthy matter.