5 Final assembly

This chapter combines the minus criterion of Chapter 3 with the cyclotomic-unit results of Chapter 4. The bridge is the implication \(p \mid h^+ \Rightarrow p \mid h^-\): if \(p \nmid h^-\), the minus criterion clears every Bernoulli numerator in Kummer’s range, the Kummer logarithm determinant is then nonzero, \(C^+\) is \(p\)-saturated, and the prime-conductor index theorem forces \(p \nmid h^+\).