What was the basis that Al-Hajjaj relied upon for dividing the Mushaf into sections (arba'), parts (ahzab), and tenths (a'shar), given that these divisions are unequal, and was this done according to a specific rule or randomly?

Perhaps the primary division of the Holy Quran into ajza' (parts), ahzab (sections), and arbāʿ (quarters) was based on quantity, as the ajza' usually comprise about twenty pages. This division is a matter of ijtihad (independent reasoning) and aims to facilitate memorization and recitation. It is not tawqīfī (divinely ordained) from the Prophet, peace be upon him; hence, the beginnings of the ajza' and ahzab vary in different mus-hafs (copies of the Quran) according to different regions and schools. This matter does not entail any legal ruling.