Waxing/Waning
The first thought in Rin's mind as she woke up was not 'I thought I was dead', but 'I wonder if his moon changes like the real one'.
The second one was 'I thought I was dead'.
She never asked him, but after having travelled with him for some time, and never seeing it change, she assumed it didn't.
Years later, when Rin had grown old and had a family of her own, but he still looked the same, the thought came back to her. Perhaps his moon would wax and wane with him?
She supposed she would never know.