Starting from theorem 2.5, index the odd characters modulo \(p\) as \(\omega ^j\) with \(j\) odd, \(1 \le j \le p-2\). The boundary exponent \(j = p-2\) plays a special role: the combination of the prefactor \(2p\) with the boundary factor \(-\tfrac 12 B_{1,\omega ^{p-2}}\) is a \(p\)-adic integer congruent to \(1\) modulo \(p\), so it can be absorbed into the error term \(pz\), leaving the product over odd \(j {\lt} p-2\).

Replace each factor \(B_{1,\omega ^j}\) in theorem 3.1 by the congruent classical quotient \(B_{j+1}/(j+1)\) from theorem 1.7; for odd \(j {\lt} p-2\) the index \(j+1\) is even with \(0 {\lt} j+1 {\lt} p-1\), so the hypotheses of the congruence hold. Each replacement changes the product by an element of \(p\mathbb {Z}_p\) times a product of \(p\)-adic integers — integrality of the remaining factors is lemma 1.6 — so all errors collect into a single multiple of \(p\).

If \(j\) is odd, write \(j=2k-1\) with \(k \ge 1\), so \(j+1=2k\), and \(j {\lt} p-2\) becomes \(2k \le p-3\) after parity bookkeeping. Conversely, an even index \(2k\) in Kummer’s range corresponds to the odd index \(j=2k-1 {\lt} p-2\).

By theorem 3.2, \(h^-\) is congruent modulo \(p\) to the finite product of the \(p\)-adic integers \(-\tfrac 12 B_{j+1}/(j+1)\) over odd \(j {\lt} p-2\). The factor \(-\tfrac 12\) and the denominator \(j+1\) are \(p\)-adic units in this range, so by lemma 1.6 each factor is a non-unit exactly when \(p\) divides \(\operatorname {num}(B_{j+1})\). A product of \(p\)-adic integers lies in \(p\mathbb {Z}_p\) exactly when one of the factors does, so \(p \mid h^-\) holds precisely when some numerator in the range is divisible by \(p\). Finally lemma 3.3 converts the odd index \(j\) into the classical range \(1\le k\), \(2k\le p-3\) with \(2k = j+1\).

Contrapositive of one direction of theorem 3.4: a numerator divisible by \(p\) in the range would force \(p\mid h^-(K)\).