We don't plan to take notes on mushaf, but we have noticed some parts of the Arabic writing has faded in our copy of the Qur'an.
Is it permissible to add the faded letter back in?
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Note that the disapproval, if any, is on adding something to a mushaf which is not part of the Quranic text, not on writing the text itself.
If a letter has faded then it means that the mushaf now has a potential mistake, and it is obligatory not just permissible to correct it when you find it.
ينص الحنفية والشافعية على أن إصلاح ما قد يقع في بعض المصاحف من الخطأ في كتابتها واجب، وإن ترك إصلاحه أثم، حتى لو كان المصحف ليس له بل كان عارية عنده، فعليه إصلاحه ولو لم يعلم رضا صاحبه بذلك، وقال ابن حجر: لا يجوز ذلك إلا برضا مالكه، وقال القليوبي: محل الجواز إذا كان بخط مناسب وإلا فلا
[ Personal translation so may contain paraphrasing etc. ]
The Hanafis and Shafi'is have documented that correcting a mistake that has occurred in the writing of a mushaf is wajib, and to leave it is sinful, even so if the mushaf is not one's own property but is borrowed and the owner's consent is not known.
And Ibn Hajar said: It is not permissible except with the consent of the owner. And al-Qalyubi said: The condition of permissibility is if it is with an appropriate handwriting and otherwise not (without consent of the owner)