If \(p=3\), the group \(C^+\) is the full unit group, and the index is \(1\). For \(p\ge 5\), theorem 3.7 identifies the Bernoulli nonvanishing hypothesis with nonvanishing of the Kummer logarithm determinant. By theorem 3.12, the subgroup \(C^+\) is then \(p\)-saturated in the full unit group. The index nondivisibility follows from theorem 2.7.

By theorem 3.5, the hypothesis \(p\nmid h^-(K)\) implies that no Bernoulli numerator in Kummer’s range is divisible by \(p\). Hence theorem 1.1 gives

If \(p\mid h^+(K)\), then the prime-conductor index theorem theorem 4.5 gives divisibility of the normalized cyclotomic-unit index. By theorem 1.6, this is equivalent to divisibility of the ordinary \(C^+\) index, a contradiction.

This is the contrapositive of theorem 1.2. Indeed, if \(p\nmid h^-(K)\), then that theorem gives \(p\nmid h^+(K)\).