For doubled rāʾ (rāʾ with shaddah), the book treats the pair as following the vowel of the pronounced doubled unit; the first component follows the second.
In الرَّحْمَٰنِ, the doubled rāʾ has fatḥah and is heavy. In الْبِرِّ, the relevant kasrah gives lightness. Shaddah requires proper doubling, not a new inserted vowel or exaggerated rolling.
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The rule for doubled rāʾ
Tafkhīm and Tarqīq — Alif, Divine-Name Lām and Rāʾ
For doubled rāʾ (rāʾ with shaddah), the book treats the pair as following the vowel of the pronounced doubled unit; the first component follows the second.
In الرَّحْمَٰنِ, the doubled rāʾ has fatḥah and is heavy. In الْبِرِّ, the relevant kasrah gives lightness. Shaddah requires proper doubling, not a new inserted vowel or exaggerated rolling.