So I have been playing around with some similar ideas but from a symbolic dynamics perspective. In that setting one can define a free energy over strings. There also exists a fractal dimension related to compressibility. The free energy and this fractal dimension are seemingly related through SI, and I suspect this fractal dimension is related to the LC. Happy to share if you think this might be useful for you.
So I have been playing around with some similar ideas but from a symbolic dynamics perspective. In that setting one can define a free energy over strings. There also exists a fractal dimension related to compressibility. The free energy and this fractal dimension are seemingly related through SI, and I suspect this fractal dimension is related to the LC. Happy to share if you think this might be useful for you.