That would be the most awesome explanation for adamant cloth ever.
So, how about this:
Natual adamantine should be flexible fibers, that harden under force into the axiomatically inflexible, invincible metal, that we see in the raws.
Heat can't really melt these fibers, but it can make them stick together into intricately patterned wafers, which can then be further bonded to make armor plates or sword blades.
The light weight could be explained by ultra fine threads, with lots of empty space.
The incredible edge is trickier, but it could be a natural consequence of the weaving and the shape of the strands. If the tip of each thread is actually ultra sharp (so demons that try to tear through it lacerate themselves, like on barbed wire), or even a mono-atom point, you could weave the blade so that the actual edge was a row of exposed tips.
I'm imagining something that looks like metal, but when examined with a jewelers glass, is actually a dense pack of threads, with ultra sharp ends that are woven to lie flat or stand upright, according to the needs of the craftsman. Unskilled smiths would just gently press and fold the wafers in proscribed ways with their hammers, to get a blade shaped object that stiffens up properly, but a masterwork would have every thread in perfect arrangement.
Sounds like we need a way to make adamantine whips. Cut a goblin right in half if you lash him with the edge.
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Should I feel guilty that now I want raw tags for non-newtonian fluids and item support for treated fabrics?