Leopoldt and the p-local Bost-Connes groupoid: the rank of the isotropy, not its triviality, and why log_p(Np) vanishes

The isotropy of the p-local Bost-Connes groupoid at an intermediate stratum Y_T is identified exactly: it is the kernel of the map from the lattice of T-supported principal divisors to the quotient of the complementary unit factors by the projected closure of the global units. Consequently, whenever the unit rank is at least one and the complementary block is one-dimensional the isotropy has full rank |T| - already for real quadratic fields, unconditionally - so the proposed equivalence between Leopoldt's conjecture and trivial isotropy is false; Leopoldt enters only as the rank budget r - delta_p. The proposed L-invariant vanishes identically because log_p kills p, and the correct invariant is the dimension r_2 + 1 + delta_p of the unit-quotient, recovering Paper XXII.