Will,
Not being an attorney, I can only go on a few thoughts in this regard.
The HAL-GPT3 Chatbot output is not written to any tables so Hal does not keep a record of your exchange. (you might be able to do so but not Hal).
In theory, your questions, and Hal's output, you could claim the work since the conversation is just between you two (plus maybe the cloud people at the GPT-3 servers...who knows?
One has to consider the legal status with respect to valid or actual responses done by or made by said interaction with GPT-3. I'm sure there is no claim of certainty or validity in GPT-3's answers being truthful, useful, or factual by any stretch of the word.
Ask them about proprietary licenses or who owns any formulas, ideas, rights, or potential money from the use of the said program (GPT-3, etc.)
You are certainly free to explore your notes from the exchange of ideas and responses between you and Hal/GPT-3 but again, how factual might those answers be? I for one, would not be willing to risk any inventions based on GPT-3 formulas or methods.
Lastly, upon further exploration, one might find, some disclaimers in the fine print of the ChatAI/GPT-3 website or its agents, etc. It might state that you are on your own if or when using any output from GPT-3 or their platforms.
I understand your eagerness to dive into this exciting knowledge system further but my friend, please proceed with caution and ask those people lots of pertinent questions.
Make good choices and be well!
- Art -