Mechanical construction of functions realizing arbitrary computable complexities

Determine whether there exists a mechanical method that, for every given computable round-complexity function t(n), explicitly constructs a function f such that the largest integer k satisfying f^{(k)}(0)\leq n equals t(n), equivalently providing a systematic solution of Equation (\ref{eq:generic-simple}) for arbitrary computable t(n).

Background

The Increasing Degree problem \Pi_f has complexity governed by the largest number of iterations of f that can be applied to 0 while remaining at most n. The paper constructs several functions f that realize specific complexity classes, including iterated logarithmic, log-sharp, polynomial, polylogarithmic, and related complexities.

The conclusion asks whether these constructions can be generalized mechanically to every computable target complexity, rather than requiring a separate ad hoc design of f for each asymptotic function. This would characterize how broadly the Increasing Degree framework can realize complexity functions in arbitrary graphs.