No they don't, and no Infusible tag, but Alla and Lucy are showing they do have Infusible tag, but 0 Purity. I thought it was strange, as I don't remember any datum for Infusible tag, other than a Purity rating of non-0.
Thus, I assumed there is some algorhythm Lucy is using to calculate an Infusible tag that is incorrect.
To rephrase it again, as it still doesn't sound clear:
In-game there is no Infusible tag nor Purity Rating.
On Lucy, it shows an Infusible tag, but no Purity rating. I'm not sure how Lucy calculates an Infusible tag, but I assumed the in-game method was to check for a non-zero Purity rating.
Yther Ore.