Explicit form of the primitive recursive bound \(\mathcal{E}\) in \(\mathbb{R}_{\mathrm{LN}}\)

Derive an explicit expression for the primitive recursive function \(\mathcal{E}\) that converts the format of definable sets in the Log–Noetherian structure \(\mathbb{R}_{\mathrm{LN}}\) (and its Pfaffian extension) into a universal bound on the number of connected components, including the complex-graph modification used in this work.

Background

Effective o-minimality requires a primitive recursive function $\mathcal{E}$ that bounds the number of connected components of sets in terms of their format. The authors adapt the framework to include complex LN-function graphs and absorb the corresponding bound into $\mathcal{E}$.

They explicitly note that they currently lack an explicit expression for $\mathcal{E}$, leaving the quantitative conversion from format to component bounds unspecified. An explicit $\mathcal{E}$ would strengthen the quantitative aspects of their complexity framework.