On the existence of certain total recursive functions in nontrivial axiom systems, I

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We investigate the existence of a class of ZFC-provably total recursive unary functions, given certain constraints, and apply some of those results to show that, for $Σ_1$-sound set theory, ZFC$\not\vdash P<NP$.
LaTeX, 16 pages, no figures. This paper was submitted to a major journal in the field and rejected. The referee somehow misundesrtood Corollary 3.8 and wrongly concluded that the proof had either a gap or an error. Can you find whether that error exists?

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