A lemma on a total function defined over the Baker-Gill-Solovay set of polynomial Turing machines

Abstract

If we establish that the counterexample function for P=NP, if total, overtakes all total recursive functions when extended over all Turing machines, then what happens to the same counterexample function when defined over the so-called Baker-Gill-Solovay (BGS) set of poly machines? We state and prove here a lemma that tries to answer this query.