What is this 'gerund' or 'noun'

In the sentence:

1. I saw your busting of the bug busters.

We can make the following structural observations and analysis:

• The sentence's subject is I; the predicate is everything else after it.
• The sentence's finite/tensed verb is the transitive verb, saw; its direct object (DO) is everything else after it.
• The direct object comprises two Noun Phrases (NPs) connected to each other by the preposition of.
• The DO's first part is the NP your busting.
• The object of the preposition of is everything else after it, the NP the bug buster.
• The first NP in the DO comprises the possessive determiner your followed by the noun, busting.
• The second NP in the DO comprises the definite article the followed by the noun bug used attributively to restrict the final noun, busters.

The thing that guarantees that busting is a noun here not a verb is that you have to use a preposition to connect it to another noun.

In contrast, if you could connect it directly without a preposition, then the first would necessarily be a transitive verb with its own direct object, as in:

2. I saw you busting the bug busters.

Now there is no preposition, so it is a verb in its non-finite -ing inflection. That non-finite verb's own direct object is the bug busters.

There is also the possibility of using a bare infinitive phrase with sense verbs like this:

3. I saw you bust the bug busters.

Here bust is a verb in a different non-finite form, that of the bare infinitive. This non-finite verb's own direct object is also the bug busters.