I'm at a lost for words.
My assumptions pertaining to the mathematical impossibility of protein folding and synthesis being a random process seems to be correct, there must be a guiding hand.
Alpha2Go, a sub-project of Googles deep mind, seems to have found the pattern that searches the space for valid combinations of amino acids and their folding processes. In this sense, that process is essentially an algorithm and from that algorithm further complexity of the system manifests.
https://deepmind.com/blog/article/alphafold-a-solution-to-a-50-year-old-grand-challenge-in-biology