Is there proof that the stone of the ring is to be turned downwards in the prayer or outside of it?
It has come in the hadith which is on the authority of ‘Ibn ‘Umar, may Allah be pleased with him and his father, in Sahih Muslim, that the Prophet ﷺ would when wearing it, turn the stone towards the palm of his hands.
There are those from the people of knowledge that understood this to be that the Prophet ﷺ did this due to it having the inscription of the Majestic word, because he would use it as a stamp.
And Allah knows best.
And I do not know of anything -as far as I am aware- of any proof specifying this with the prayer, unless it causes a disturbance in attention, then yes.
Shaykh Abu Hatim Yusuf Al-‘Inaabi Al-Jazaa’iree – may Allah preserve him.
Abu ‘Abdirrahman ‘Abdullaah bin Ahmed Ash-Shingaani