meaning - "Be yet to do" vs "have yet to do"

This theory has yet to be proven.
This theory is yet to be proven.

This is yet to be done.
This has yet to be done.

I have yet to spend summer in the mountains.

I am yet to spend summer in the mountains.

Are they all correct? How are "have yet to do" and "be yet to do" different?

Thank you!

Answer

The "has/have" forms of these can be used if the writer/speaker is commenting about the state of something, such as the theory in the first sentence, whatever it is to be done in the second sentence, and spending summer in the mountains in the third.

The "is/am" forms of these can be used if the writer/speaker want to imply that there's something like a currently undergoing process that is somehow involving the writer/speaker regarding these things.

But you'll probably hear/read these phrases used to describe either situation, there's not a lot of difference in meaning between them in many cases.

This theory has yet to be proven.

We're talking about an unproven theory. We aren't trying to prove the theory, or involved in proving the theory.

This theory is yet to be proven.

A scientist who is researching or trying to prove this theory may say this over the first sentence.

This is yet to be done.

This might be said if someone asked the speaker how far along he/she is on a project.

This has yet to be done.

This might be said if someone asked about the state of a project, but not one where the listener is responsible.

I have yet to spend summer in the mountains.

I haven't ever spent summer in the mountains. That's all I'm saying.

I am yet to spend summer in the mountains.

I haven't spent summer in the mountains. By saying this, I may be planning to or may want to, or intend to, or I'm doing something like vacationing in different places every year and this is on my list.