uk – What documents to show for "Proof that family members are staying in your country of origin while traveling on a standard visit visa"

I am applying for a transit visa to the United Kingdom. One of my parents lives with me and is my direct dependent. They do not have a passport at the moment and all their documents indicate an old address in a different city that we lived in a few years ago. What we currently use as proof of address is a gas connection registration document.

What documents can I submit for my caase Evidence of family members remaining in your home country whilst you travel for Standard visiting visa in the visa application?

Does the paper "Proof of Collatz's Conjecture" by Agelos Kratimenos contain valid proof?

The document "Proof of Collatz's conjecture" (link arXiv) by Agelos Kratimenos was submitted to arXiv a few days ago (4 Nov. 2019).

Is it really a valid proof?

I mean, I know that arXiv is a prestigious newspaper. Do they have peer reviews before an article is accepted? Or are submissions possible without that?

It would be amazing if it was really a proof of Collatz's conjecture.

The exercise "Proof of Key" was it "The Ultimate Strength Test"?

The day yesterday was announced as a "Proof of Keys" day when Bitcoin trading account holders were urged to withdraw their balance in a stand-alone portfolio for the day to check if trading actually held all Bitcoin bought theoretically. them or deposited with them.

Are there verifiable figures for the proportion of total assets withdrawn from the most popular exchanges?