Geometric Rules for Lunar Month Commencement: A Deterministic Astronomical Framework

This study presents a deterministic astronomical framework based on geometric rules for determining the first day of the lunar month (Lunar Month Commencement) over Makkah, Saudi Arabia (2010–2012). The framework compares two decision models: IDRAK-VIS-EL, based on geometric crescent visibility criteria (elongation ≥ 10°, altitude ≥ 5°, age ≥ 10 hours), and UQ-Proxy, a simplified computational approximation of the Umm al-Qura calendar. Using a Reference Swap Test, the lag metric is shown to be algebraically invariant under model label exchange, yielding a mean difference of 0.00 hours across 36 cases under deterministic construction. The observed invariance arises from the formal structure of the lag definition under reference inversion and should not be interpreted as an independent astronomical symmetry. This work is a methodological and computational validation exercise within a strictly defined temporal and geographic scope.