If someone manages to formally prove the existence of these correlations, that would settle it for me.
I've seen very complex attempts at extracting these correlations, through artificial intelligence algorithms such as high-dimensional support vector methods. These algorithms can find extremely complex correlations in the data that would be very hard for us humans to grasp, or completely unintuitive. If these methods fail at detecting correlations, I have a hard time with the credibility of "toy functions" used in classical TA.
Wrong on two counts, I'd say:
(a) there are a number of statistically well established predictive factors (not many, but some), so I'm not sure where you're getting the idea from that nobody ever managed to find any correlations that enable market decisions/risk control better than 'guessing'.
(b) your claim that "if formal algorithm X can't do it, the human brain most certainly can't do it" is pretty off, imo. As Lo et al. put it:
Therefore, technical analysis employs the tools of geometry and pattern recognition, while quantitative finance employs the tools of mathematical analysis and probability and statistics. In the wake of recent breakthroughs in financial engineering, computer technology, and numerical algorithms, it is no wonder that quantitative finance has overtaken technical analysis in popularity—the principles of portfolio optimization are far easier to program into a computer than the basic tenets of technical analysis.
Nevertheless, technical analysis has survived through the years, perhaps because its visual mode of analysis is more conducive to human cognition, and because pattern recognition is one of the few repetitive activities for which computers do not have an absolute advantage (yet)
(source)