Using Kolmogorov complexity to establish proof complexity lower bounds?
Arvind, Köbler, Mundhenk, and Torán introduced the notion of time-bounded nondeterministic instance complexity. Based on a quick reading, It seems they use Kolmogorov complexity measure that depends on the size of shortest nondeterministic TM. They were able to prove the existence of hard to prove Tautologies under a notion of hardness based on nondeterministic instance complexity.

Vikraman Arvind, Johannes Köbler, Martin Mundhenk, Jacobo Torán,
Nondeterministic Instance Complexity and Hard-to-Prove Tautologies,