Select the correct option:

Only a complete paradigm shift can ___ the deeply entrenched bureaucratic inertia.

The correct option is dislodge.

Step-by-Step Explanation:

1. Analyze the Context of the Sentence:
The sentence reads: "Only a complete paradigm shift can ___ the deeply entrenched bureaucratic inertia."
- "Deeply entrenched" means firmly established and difficult to change or remove.
- "Bureaucratic inertia" refers to resistance to change within a administrative or official system.
- Therefore, the blank requires a verb that means to knock out, remove, or force out something that is firmly established or fixed in position.

2. Evaluate the Options:

- discourage: Means to cause someone to lose confidence or enthusiasm. It does not fit logically with an abstract concept like "entrenched inertia."
- dislodge: Means to knock or force out of position, especially a firmly fixed or established one. This fits the context of breaking or removing "deeply entrenched" inertia perfectly.
- disillusion: Means to cause someone to realize that a belief is mistaken. It applies to people, not inertia.
- persuasion: This is a noun (meaning the act of persuading), whereas the sentence structure requires a verb after the modal auxiliary verb "can".

Conclusion:
The word dislodge accurately completes the sentence both grammatically and semantically.