After your suggestion of an extra time dimension, I've been doing some rereading of Tegmark, Grandpierre, Penrose, the Space-Time-Matter consortium and Cumrun Vafa. It is increasingly clear to me that the key to the riddle is geometric.
A-G, it probably won't surprise you to learn that Grandpierre has studied and written on the Pythagorean basis of Plato's speculation on the structure of consciousness, psyche. Of course, the work is all in Magyar, so it won't do us any good! (What a strange and lovely language....)