If the chain of dependent origination (Pratītyasamutpāda) ends, then isn't the last part independent?

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There is a cyclic dependency between ignorance and defilements (also called taints or fermentation).

Ignorance originates from defilement. Ignorance ceases when defilement ceases.
Āsavasamudayā avijjāsamudayo, āsavanirodhā avijjānirodho

Defilement originates from ignorance. Defilement ceases when ignorance ceases.
Avijjāsamudayā āsavasamudayo, avijjānirodhā āsavanirodho

MN 9

Why is there such a cyclic dependency? This is amply discussed in "Does MN9 imply a cyclic causation? How does it work?"

I quote from the best answer there:

It is like being born in a poor uneducated family: because you are poor, you can't get education, and because you can't get education you are poor.

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If the chain of dependent origination (Pratītyasamutpāda) ends, then isn't the last part independent?

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