Perhaps Phantom registers first of your core exercises (5-5)
I believe the answer is:
rolls-royce
'perhaps phantom' is the definition.
I know nothing about this answer so I cannot judge whether it can be defined by this definition.
'registers first of your core exercises' is the wordplay.
'registers' becomes 'rolls' (lists of names).
'first of your' becomes 'y' (1st letter of 'your').
'exercises' indicates anagramming the letters (letters exercise or move around).
'y'+'core'='ycore'
'ycore' anagrammed gives 'royce'.
'rolls'+'royce'='ROLLS-ROYCE'
Can you help me to learn more?
(Another definition for rolls-royce that I've seen is " Vehicle".)