Can the verb 'to be' be used purely intransitively, like 'to exist'?

Can the verb 'to be' be used intransitively, as in "I am." OR "It was."?

I don't mean as an answer to a question, when there is all kinds of implied content. I mean as a complete sentence. And I also don't mean in a philosophical context, like "I think, therefore, I am."

We can't say something like "I am happily." We would say "I exist happily." So can we ever use 'to be' in an intransitive sense like 'to exist'?

Note that I mean 'without auxiliaries'. "It cannot be!" uses auxiliaries.

If we can't use 'to be' intransitively, then it shouldn't be able to take adverbs without an object. But oxforddictionaries.com/definition/english/here considers 'here' to be only ever an adverb. So "I am here" would be subject/intransitive verb/adverb.

Can someone unwind this conundrum?

Not all adverbs in English behave the same.

The verb to be doesn't just take here; it can also take many other adverbs of place.

He's downstairs.
I am home.
It's far away.
He's abroad.
The store is nearby.
They're outside.
Are the missing documents anywhere?

And finally, I don't agree with your reasoning:

If we can't use 'to be' intransitively, then it shouldn't be able to take adverbs without an object.

You sound like you're reasoning about mathematical objects, not about languages. Languages aren't logical.

I just noticed that this doesn't answer the actual question: "Can the verb 'to be' be used purely intransitively, like 'to exist'?" I'm inclined to answer "no". Even with auxiliaries, as in "it cannot be", there is usually something implicit that is being referred to. We wouldn't say "unicorns cannot be"; we'd use "exist".

But this depends on exactly what you mean by "purely intransitively". When something is implicitly referred to, an intransitive it is is fine. For example,

Many people think computers are smart. They aren't.