We establish unconditionally that for every integer $k \geq 1$ there is a language $L \in \mbox{P}$ such that it is consistent with Cook's theory PV that $L \notin Size(n^k)$. Our argument is non-constructive and does not provide an explicit description of this language.

@article{paperbot206,
  title = {Unprovability of circuit upper bounds in Cook's theory PV},
  author = {Jan Krajicek and Igor C. Oliveira},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 13, Issue 1},
  year = {2017},
  doi = {10.23638/lmcs-13(1:4)2017}
}