Starting from theorem 2.5, the proof indexes the odd characters by powers of the Teichmuller character. The boundary character \(\omega ^{p-2}\) contributes a factor congruent to \(1\) modulo \(p\), so it can be absorbed into the error term \(pz\).

Replace every generalized Bernoulli factor in theorem 3.1 by the congruent classical factor from theorem 1.7. The integrality statement lemma 1.6 keeps each replacement inside \(\mathbb {Z}_p\) and collects all errors into one multiple of \(p\).

If \(j\) is odd, write \(j=2k-1\), so \(j+1=2k\). Conversely, an even index \(2k\) in the Kummer range corresponds to \(j=2k-1\). The remaining inequalities are elementary arithmetic in natural numbers.

By theorem 3.2, \(h^-\) is congruent modulo \(p\) to a finite product of \(p\)-adic integers

The factor \(-1/2\) and the denominator \(j+1\) are \(p\)-adic units in the range. Thus the product is divisible by \(p\) exactly when at least one Bernoulli numerator is divisible by \(p\). Finally lemma 3.3 rewrites \(j+1\) as \(2k\) and gives the usual range \(1\le k\), \(2k\le p-3\).

This is the contrapositive direction of theorem 3.4. If some numerator were divisible by \(p\), then the theorem would imply \(p\mid h^-(K)\).