Sharp o-minimality of the Log–Noetherian Pfaffian structure

Prove that the Pfaffian extension of the Log–Noetherian structure, \(\mathbb{R}_{\mathrm{LN,PF}}\), is sharply o-minimal, thereby providing a two-parameter sharp complexity theory for period integrals and related functions.

Background

Throughout the paper the authors work within the effective o-minimal structure $\mathbb{R}_{\mathrm{LN,PF}}$ to assign finite complexities (formats) to period maps and effective couplings.

They note that a stronger, sharp o-minimal structure would refine complexity into two integers $(F,D)$, enabling more precise classification of asymptotic behaviors of period integrals. They reference a conjecture in the literature that $\mathbb{R}_{\mathrm{LN,PF}}$ is sharply o-minimal, highlighting this as an open direction.