Solving this problem would unlock a unified framework connecting proof theory, model theory, and foundations of arithmetic in a novel way. It would clarify whether there is a natural correspondence between proof complexity in minimalist systems and the depth of a concept in the symmetry-definability hierarchy, potentially giving a new semantic interpretation of cut rank or normalization depth. This could lead to new independence results by showing that certain arithmetical statements are not definable below a given symmetry level and therefore not provable below a corresponding proof-theoretic threshold, strengthening connections between descriptive complexity, proof complexity, and reverse mathematics.