Referring to the second circuit:
1) The current drain from the positive output must be more than that from the negative output for the circuit to work correctly, if not the negative rail will reduce and the positive rail increase.
Thanks, I'm not aware of this, but I think its safe to assume that normally the pos output will be loaded a lot more than the neg output. As I mentioned, the neg output is just for low current usage, like biasing neg rail of the op-amp for example.
2) The voltage across the capacitor does not reverse, ideally it would remain fixed and equal to the peak AC voltage (less diode drops).
Though sounds so simple, this is it that I want to hear, makes perfect sense, thank you.
3) This circuit does not cause DC in the transformer winding, a capacitor cannot pass DC!
I'm assuming this to counter the post made by Hero999, lets hear from him.
Is it possible this problem seems counter intuitive just because you assume GND is the same as the lower tap of the transformer? It's not. This makes me feel like building the circuit just to scope it. Maybe I'll deliver some scope measurements if I can find the parts.
Share it please once you built and tested it.