What am I missing Chaos? Did a post get deleted?
If I am not mistaken Eschers and coticules do not break down.
The finisher really does not matter only the quality o the edge. All different stones have differently shaped grit particles so they can't be compared in terms of sizes , rahter in terms of quality of finish. Even this comparison is very subjective because there are no two identical natural stones from the same source. At least that is how I see it.
You asked if all four were Awaseto... where did you get that from? I feel like I'm missing something obvious.
That is how I understand it too. The abrasive in Eschers and Coticules are rhomboid garnets. Being crystals, they may cleave to form the slurry. If they continue to break down after that, in my experience it is not in a fashion similar to Japanese naturals.
Of course, I could be wrong.
PC - I think the abrasives in the Eschers are quartz grains. Neither quartz nor garnets have cleavage (maybe Mrs. quartz and Mrs. garnet...). They may fracture under a lot of pressure (e.g., grinding) but I doubt honing would cause much of that. I have no experience w/ the Eschers, but that's essentially why you have to dilute with coticules.