Energy of codes with forbidden distances in 48 dimensions

We prove the universal optimality of four remarkable spherical 11-designs in 48 dimensions either among all antipodal codes, or all spherical 3-designs, whose inner-products avoid the set $T_1=(-1/3,-1/6) \cup (1/6,1/3)$. We also prove the universal optimality of these configurations among all codes whose distance-avoiding set is $T_2=(-1/2,-1/3) \cup (1/3,1/2)$.