The Ramanujan bound is the tempered bound: the Hecke operator on the Bruhat-Tits tree, and a no-go theorem for condition (C4)

The Hecke operator T_p on the Maharam extension of the Hurwitz quaternion system is the adjacency operator of the (p+1)-regular Bruhat-Tits tree up to a central unitary, and the self-adjoint operator eps_p^{-1/2} T_p has spectrum exactly [-2 sqrt(p), 2 sqrt(p)] on the complement of the invariants: the Ramanujan bound is attained. It is also worthless for condition (C4): attaining it certifies temperedness, which by Kuhn's theorem is the signature of amenability. Indeed every element of the monoid is a unit at almost every place, so the orbit relation is an increasing union of relations of proper actions, hence hyperfinite: (C4) is false, the factor is injective, and conditions (C1)-(C3) are incompatible with (C4) for every order in every finite-dimensional Q-algebra, since (C3) forces a finite unit group.