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Why professional panchangs sometimes disagree by ±1 day — and which system to follow
Both systems use the same astronomy — the same Moon, the same tithis, the same sunrise times. The disagreement is purely about WHICH rule to apply when a tithi spans two days. The Smarta tradition uses the tithi at the required Kala window (Madhyahna for Ram Navami, Nishita for Janmashtami). The Vaishnava tradition additionally rejects any 'Viddha' (contaminated) tithi and shifts to the next day when the tithi is 'Shuddha' (pure).
Uses Udaya Tithi (tithi at sunrise) as the default rule. If the tithi prevails at the required Kala window (Madhyahna, Nishita, Pradosh, etc.), that day is chosen. Does not reject Viddha tithis. Follows the authority of Dharmasindhu (1790 CE) and Nirnayasindhu (1612 CE). This is the system used by most Hindu families, and by professional panchangs like Prokerala.
Rejects 'Viddha' (contaminated) tithis. If the previous tithi is present at sunrise — even briefly 'touching' the required tithi — the festival shifts to the next day when the tithi is 'Shuddha' (pure). Follows the Hari Bhakti Vilasa (16th century, by Gopal Bhatta Goswami) and Navadvipa Panjika. Used by ISKCON centers worldwide, Gaudiya Vaishnavas, and Sri Vaishnavism.