Growth beyond the golden-ratio exponent from finitely many analytic functions

Establish an additive-growth bound exceeding |A|^{(1+\sqrt{5})/2} for iterated sumsets of the images of finite sets under linearly independent analytic functions, or otherwise determine whether the golden-ratio exponent is an intrinsic barrier to the method.

Background

Theorem \ref{growthforsomef} produces, for one function among a finite family of analytic functions with linearly independent derivatives, an exponent \phi(n) defined recursively and converging from below to the golden ratio (1+\sqrt{5})/2. The author states that the available argument cannot prove growth beyond this limiting exponent, while it can approach the exponent arbitrarily closely by increasing the number of functions and the length of the iterated sum. This explicitly unresolved limitation motivates the problem of surpassing the golden-ratio threshold.