...So the term cycle of energy going from the hind end through the back to the poll and to the hand is in fact the half halt affecting the swing to the hind leg so that the forward thrust energy is converted to upward support energy...
Ummm...that is a visualization thing, not reality. There is no energy going in a cycle from the hindquarters thru the back, thru the poll and back into your hand.
Power from the horse's rear can be turned into forward motion or weight support, or not used at all. The power never goes in a cycle. Power from the front can likewise turn into forward motion (inefficiently), weight support, or not be used.
Maximum efficiency in moving the horse comes when the hind end provides thrust, and the front end supports weight (lift), both to the maximum extant possible.
At slower speeds, the horse can mix and match as it desires, but it may not do so efficiently. When the front end stretches out and tries to pull forward while the rear end is resting, the horse becomes strung out. With training, the horse learns the pressure on the bit means to stretch out less with his front end, while using his rear for more thrust. Carried a bit further, he'll shift some of his weight to the rear, which would limit his acceleration - but you don't do half-halts to accelerate, so that is OK.
For some reason, I don't think that explanation will satisfy many...