Is there a way to submit privately to claim intellectual property Hypothetically, if I had an idea to improve an algorithm/theorem but

I wasn't sure it was correct, or
I wasn't sure it was significant or meaningful or hasn't already been published in a different form/somewhere else already or wasn't sure where to submit something like this .

is there somewhere I could submit a manuscript first privately to secure any intellectual property rights while I think about whether it is worth submitting or trying to do some more work on this and try to understand the field a bit better?
Please note that this does not substitute proper legal advice. In the US, I believe that patent protection falls under the 'first inventor to file' rule. So it is not relevant who the first inventor is, but who the first inventor is who files for patent protection. So if you want to claim IP protection in the US then the only relevant place to file is at the USPTO. This is indeed private at first, since patents are made public a year after filing.
Also, please note the publishing your data before filing for a patent may constitute 'prior art', which could mean that any future attempt at filing a patent in the US would be denied.
The safest (and possibly only) option for you to protect the IP for your idea is to file for (and get awarded) a patent.
Renewed disclaimer: This constitutes my personal opinion and should not be taken as legal advice.