A cars engine converts fuel into kinetic energy. Some of that kinetic energy goes into driving the wheels and some of it goes into driving the other components of the engine including the vehicles alternator.
The alternator spins and creates an electric field which charges that battery and powers all of the electrical components of the vehicle. This electric field varies in strength dependent on the load (amount of energy) required to power the all of the components. The higher the load the more the electric field will create resistance to the alternator spinning.
The energy needed to split 1 molecule of H2O is equal to the energy released when the hydrogen and oxygen molecules are ignited and recombined into H2O. This means that in a perfect world with a 100% efficient HHO generator 1 joule of energy applied to water will release 1 joule worth of HHO gas potential energy.
In order to apply this 1 joule of energy to the water your alternator has to generate it while being driven by your car’s engine. This increases the load and magnetic field which increases the alternators resistance to spinning.
Again in a perfect world to compensate for the increased resistance your engine will have to supply the alternator with 1 extra joule of energy thus reducing the amount available to drive the wheels.
After all of this your HHO generator has generated 1 joule of potential energy in the form of HHO gas. This gas travels into the intake then into the cylinder where it is ignited and releases 1 joule of energy driving the piston downward and powering the engine.
To put the entire process in simple terms you have taken 1 joule of energy in the form of fossil fuel and converted into 1 joule of kinetic (running engine) then 1 joule of electric (alternator) then 1 joule of potential (HHO gas) then finally back into 1 joule of kinetic (HHO ignition).
The key concept to understand is that nowhere in this process was excess energy produced as it is only converted. Fossil fuels on the other hand enter the vehicle already in the form of potential energy when combined with the oxygen in the air. This means that energy is produced and can drive the wheels of the car.
The only way HHO could work is to generate it off of a different power source and then load it on the vehicle in gas form. However this would require large tanks of pressurized explosive gas/oxygen mixture. You would be driving a bomb.
Finally one great analogy to put the last nail in the coffin for HHO. In a perfect world you could power a light using a solar panel that is producing energy from the light itself. X joules in X joules out. BUT you cannot tap into the system to draw out extra energy as no excess is being created.
Now HHO is a very cool mixture of gasses for other reasons such as cutting and welding steal but for increasing gas mileage it just doesnt add up.
Im not trying to start a fight but would love some discussion.
Note: I edited out the last part where I made some offensive comments. My apologies.