An Invariant-Strip Barrier for Gevrey-1 Regularity

If a contracting family of maps has jointly Gevrey-1 derivatives, does its fixed-pointfamily inherit the same factorial derivative-growth bound? The classical answer to thisshape of question is Cauchy’s majorant method: build an explicit dominating series andcompare term by term. We show this classical route provably cannot work here — thenatural comparison recursion forces any constant-ratio geometric majorant’s rate to growwithout bound as more derivative orders are matched — and give a different argument thatsucceeds unconditionally. Rather than dominate the sequence term by term, we boundthe partial sums of its divided-coefficient series at a single, explicitly chosen point, using aself-consistency argument driven by the intermediate value theorem rather than induction onthe order. The result is a complete, elementary proof of Gevrey-1 regularity, assembled fromnothing beyond a finite Cauchy-product estimate, the exponential series, and continuity.