Structure of the exceptions (descriptive), and a falsified pattern: every i=3 exception has j = n/2 exactly and n of the form 3^k+1 with prime exponent k ∈ {2,3,5,7}, while every composite-exponent 3^k+1 ≤ 10^5 (k = 4,6,8,9,10) is clean — yet the pattern is NOT predictive: the next prime exponent, n = 3^11+1 = 177148, certifies clean across all 3,922,632,451 of its pairs (zero weak, zero strong). The i=2 exceptions are exactly n = 2^4, 2^9, 2^11 among all powers of 2 up to 2^16.