Ah, yes, that's because the non-linear complexity of binary matrix multiplication (which CFG parsing can be reduced to) is strongly suspected to be a lower boun

Ah, yes, that's because the non-linear complexity of binary matrix multiplication (which CFG parsing can be reduced to) is strongly suspected to be a lower bound, but I don't think that there is a formal proof of it (I am not an expert in complexity theory, so I am just basing this on what some authors seem to suggest). So, the non-constructive argument is a "proof" in the same sense that some results can be proved by first assuming that P /= NP.