[Christ]

By opening the throttle more, you're inducing less pumping loss in the engine. The pumping loss isn't the sole source of the reduction in consumption, but it's one piece of the puzzle.

By adding warm air, you are in fact changing the volumetric efficiency of the engine, which, in this case, means you're creating less power (with less fuel) for the same amount of engine speed, with more throttle angle. A higher throttle angle decreases parasitic load on the engine, again, due to a reduction in pumping losses.

Hotter air flows with increased velocity, due to the heat induced expansion of gasses. Increased velocity allows for less necessary vacuum to achieve the same result. Again, less pumping losses.

If the intake air is already hot, less calories (energy, not fat) are consumed to heat the intake charge to the necessary temperature for combustion, and to keep the engine at the proper operating temperature. The decreased differential between engine temps and intake temps means, effectively, that less energy calories must be consumed in heating/expanding the air in the combustion charge, so that more of the energy exchange can be focused on pushing on the piston, rather than heat-exchange taking place with the cylinder walls. (Yes, this does happen. When you eat cold food, your body's first exercise is to heat the food to your body temperature, which takes more calories to maintain your core temperature. If you eat food that is at your core temperature or warmer, less calories are necessary to digest/fully consume the food, so you get more energy content from it, comparatively.)

HAI's work on the "less is more" principal. You're only getting a percentage of the charge, but for that percentage, a higher percentage of the exchanged energy (see: law of conservation of energy) can be used to perform the desired task, with less waste generated.

In essence, a hot cycle is a more efficient cycle, although producing less power per cycle.

I hope that helped, I think it made my carpal tunnel flare up.