Russ, you're right, the homopolar generator can't go against the 1st law of thermodinamics (as any device can go against it).
I thought you was talking about the second one, but later I decided to let my error there to get a faster and stronger response.
So we are in agreement that the homopolar generator can go against the second one... but how?
Take a look to the image i attached at the initial post on this page.
Consider the working of the homopolar generator like a serial to pararel conversion on a transfer of charges.
On the circuit, the charges have a serial transfer fashion, and on the whole disc, the charges have a paralel transfer fashion.
Now, you can see that given a certain time step, the speed needed on the serial path is much greater than the speed needed on the paralel paths to carry the same amount of charges to the same place. The only difference is that on serial path, the charges goes arriving one by one, and on the parallel path, they arrive all at the same time.
You can use this fact , and keep a high velocity transfer of charges on a serial path (the circuit), using the parallel paths to go back to initial conditions (the disc axis and periphery).
The trick is that only one (or some, but not all) of the parallel paths taken by the charges oppose to the prime mover that keeps the whole thing working.
We can get volts by storing one kind of charges on some place, and other kind on other place, but the current through this places will make the store finally get empty, so the volts dissapear.
But the homopolar generator does not use the charges that create the volts to create also the current. The charges that creates the volts are different ones than the ones taken into the circuit current, so the volts can be maintained, and the current can run also. To keep the voltage, you only must maintain the speed.
The current on circuit is on a serial path (so high amount of current as charge/time) and the current on the disk is taken a lot of paralel paths, where only a few ones make resistance to the rotation that keeps the voltage.
That is where the difference on energy cames from. the charges that creates the EMF are different than the charges collected to make the circuit current. The motor speed gives the EMF created, but the current is marked by the circuit resistance, and the collection of this current charges from the disk does not affect the EMF generation. So we are "de-coupling" the correlation between current on circuit and EMF source. Once you can do that, the currents on the source of the EMF and on the circuit can be different (as explained by the serial and paralel paths examples), and the output energy be greater than the input.
I've got to think more the way to explain this fact, but now i think i understand the proccess very well, and sure it works. I only have to get an easy example to explain myself.