Armor Augment Crystals on Shoes?

[Ursus Spelaeus]

Premise 1: Armor Augment Crystals can be attached to armors, shields, and any magic items that grant an armor bonus to AC.
Premise 2: Shoes (and other foot gear) grant a +2 armor bonus to AC against attacks from caltrops.
Conclusion: I can attach an armor augment crystal to any magical shoe.

[TheCorsairMalac]

Premise 1: Heavy armor includes armored boot-covers similar to how it includes gauntlets.
Premise 2: Light armors do not provide boots.

Conclusion: Armored boots do count as armor, but not as a separate set. A character can make use of only a single armor crystal.

[Necroticplague]

You're missing the first conclusion/minor premise of "therefore, shoes (and other footwear) are at least one of armor, shields, or a magic item that grants a bonus to AC". The part about armor crystals on them is a secondary conclusion.

However, regardless, syllogisms only work if all their premises are correct. Yours isn't. You missed an important line later down on the page:

Each augment crystal’s Body Slot entry gives the appropriate item to which it can be attached.
Thus, in order for this to have to work, you would either have to find a boot-slot magic item that specifically is armor (note that something can provide an armor bonus without being armor, such as Bracers of Protection, which are a Wondrous Magic Item) for ones that say 'armor', or find an Armor Crystal that says it works with the Feet magic item slot. I know of no such crystals or footwear that meet those requirements.

[sleepyphoenixx]

As explained above it doesn't work that way.
You can however attach armor crystals to dastana (A&EG or OA) and chahar-aina (OA), for a maximum of 3. Which should really be all you need.