Original sentence :
My question is, why not :
When do we need to convert the verb to
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I think this Ta means "completion". The job has been stabilized.
Present form in Japanese language can mean the future things, so 一生安定する may be able to use for it like 人工知能に関連する職業は、一生安定する職業になるだろう. In this case, 一生安定した also can be used.
For example, 結婚する人 means "a person who will get married", 結婚した人 means "a person who got married".
Putting that part of the phrase in the past tense shows that this is a quality of the profession, not the purpose of the profession (intransitive vs. transitive). In other words, the occupation is a stable one, rather than a stabilizing one.
In the past tense, the action is shown to be completed and is therefore an intrinsic part of the following noun.
閉店する店 - A closing shop (A shop which will close)
閉店しない店 - A non-closing shop (A shop which will not close) i.e. open 24 hours.
閉店した店 - A closed shop
閉店していない店 - A shop which is currently not closed.
顧客の満足する品質を提供します。We offer quality which satisfies our clients.
満足したお客様は （お客様が満足したことは）何の職業でもメインの望みである。Having satisfied customers is the main aspiration of any occupation.
Hopefully someone else might have a better explanation if this is still unclear.