Cuticle
The thin dead tissue stuck to the plate at the base of the nail.
The real cuticle is the clear film on the plate itself, and that is what is pushed back and taken off during prep. The living fold of skin above it is the eponychium, and cutting into it is what causes hangnails and infections.
Most people use "cuticle" for both, which is why the two words are worth separating here.
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