Growth of synthetic-model proof depth

Determine how the minimal proof depth of a synthetic model grows as a function of its size, including whether the depth is bounded by a rate such as quadratic growth.

Background

The paper defines the size of a synthetic model as the number of axioms in its axiom list and its depth as the minimal number of reasoning steps needed to prove the modeled theorem. The authors ask for a general relationship between these two complexity measures, with quadratic growth given as an example.

References

Given an SM of size $m$ and depth $n$, how does $n$ grow with respect to $m$, e.g. is $n = O(m2)$, or what rate best describes the problem ?

— Retro-synthetic Riemannian geometry in Rocq I : Taylor series of a metric  (2609.20127 - Clemente et al., 17 Sep 2026) in Section 4, item 1 (Future directions)