The universal multiplicity-norm mismatch, and the reduction of (C4) to the norm-one subgroup

In every localized Bost-Connes system over a global field the number of sublattices or ideals of a given norm exceeds the norm: d_p = (q^d - 1)/(q - 1) > q, a universal multiplicity-norm mismatch (Theorem 1.2) that empties boundary quotients and displaces equilibrium ranges throughout the series. Condition (C4) of Paper X is reduced to a single representation-theoretic question about the norm-one subgroup, with the warning, via Kuhn's theorem and the properness of translation actions, that a spectral gap would not suffice.