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Notice that if the third term is positive (and it usually is)

Do they teach the difference between endothermic and exothermic reactions at your number 1 school? I just have a MSME from a lowly State University but I could grasp that concept.

Again your first statement is ludicrous

"The laws of physics show that it will never be possible to get cheap hydrogen and therefore it is a waste of time to even look for a solution.

Tell me the law of physics, and I'll tell you where you you're wrong. There is not cost-benefit analysis in any one of them. "

The hydrogen in water has been oxidized and, catalyst or not, it will require at least the energy that was liberated when the bond was formed to break it. There is no free lunch.

Look, I don't know where you got the idea that I said it takes less than the bond energy to break a molecular bond. (Unless you are using that as the red herring for your argument.) All I said was that we can put energy into a substance, initiating a reaction which will release more energy than we put in. This happens every day in a car cylinder. You, on the other hand, said that the laws of physics prove it is impossible to "get" cheap hydrogen. Then you quoted energy conservation, but didn't indicate where that proved your point.

Refer to thermonuclear fusion: Two particles have a given amount of internal energy by themselves. When these particles are together, and provided with so much energy that the nuclei are fused, a new particle is formed out of the two, which has a much lower internal energy than the two on there own. This difference in energy is released, and is MUCH greater than the energy required to heat up the particles in order to fuse them together. Hence the concept of fusion as an energy source. Does this violate the first law of thermo? NO.

It is possible that you and I are arguing over different things. But it is definitely not a waste of time to look for a good way of extracting hydrogen from water. If a natural source of energy (like the sun) can consistently provide enough energy so that what we have to provide in the form of electricity or heat is less than that provided by the hydrogen product, it would indeed be worth it.