lydianchromaticconcept.com

Not to be too "nit-picky", but another matter is confusing me. On page 15 of the LCC it states that both the Lyd Aug and Lyd dim scales represent the "Nine Tone Vertical Order" of the LC scale. Yet, the +5 degree is the eighth tone listed in the hierarchy. The flat 3 is the ninth. I can see how the Lyd dim is therefore of the Nine Tone order. However, as the +5 allows for all of the notes of the Lyd Aug scale, and occurs in the 8th position, how is the Lyd Aug scale a representative of the Nine Tone order instead of the Eight Tone order?

The +5 is in the 8-tone order. If the set of notes includes the b3 too then the +5 is in the 9 tone order. Another way to look at it is that 123#4567 are all in the 8 and 9 tone order.
This could be a good time to look at the section of the book about 'official scales'.