So for most objects of our experience, their definition, or essence, does not entail their existence. In other words, these objects are not the source of their own ongoing existence.

As others have pointed out, that first sentence is not remotely the same as the second sentence.

I will just add that that's some pretty bad map-territory confusion.

We don't mean a regress stretching back in time, but rather a hierarchical regress of dependent members here and now If object A does not entail its own, ongoing, existence, then it must depend on other factors for its own ongoing existence, as we saw.

The problem I have with this sort of invocation is that the examples of 'ongoing dependency' always seems to end up being a continuous stream of ordinary temporal regresses.

The lake won't immediately cease to exist if the Sun stops undergoing fusion. The relationship between the two is governed by normal temporal causation. You could describe it as a non-temporal enduring relationship, but that is an abstraction.

Now consider a chain of clamps that only stay closed if held by another clamp The only way this chain of clamps will stay closed if there is at least one "permanent" clamp holding shut one of the clamps, which then in turn holds together the rest of the clamps. One clamp must be "independent": not held shut by any further clamps

Nope. There are plenty of other ways of keeping all the clamps shut.

One could have a circular ring of clamps. Each clamp keeps the next clamp shut by virtue of it being kept shut by the clamp behind it.

Alternatively, one could have an infinite chain of clamps. There is nothing incoherent about each clamp in the chain being held shut. Of course, the system is somewhat underdetermined, so there is also nothing incoherent with each being open.

Similarly, if object A is receiving or dependent on further factors for its ongoing existence, and those factors are themselves dependent upon further factors, then this must terminate in something not dependent upon any further factors.

Or, it might not terminate at all.

The original explanation of Al Farabi's cosmological argument (with pictures) is found here

Oh, hey, /u/hammiesink's blog. I knew this sounded familiar.