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  1. Three-loop electronic vacuum-polarization corrections due to irreducible diagrams are evaluated for two-body muonic ions with nuclear charge numbers 1 Z 6 . The corrections are of order α 3 ( Z α ) 2 m r , where α is the fine-structure constant and m r is the reduced mass. Numerically, the energy corrections are found to be of the same order of magnitude as the largest of the order α 2 ( Z α ) 6 m r corrections, and are thus phenomenologically interesting. Our method of calculation eliminates numerical uncertainty encountered in other approaches. Published by the American Physical Society2025 
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    Free, publicly-accessible full text available March 1, 2026