Integral equivariant K-theory of the inert local model: the ray-class entanglement and exact torsion

The first integral case of the Galois-descent K-theory programme for affine Bost-Connes boundaries is computed: K = Q(sqrt(-5)), Hilbert class field H = Q(i, sqrt 5), the prime above 3 inert, decomposition group Z/2. The ring of integers of H is a free module over the group ring, so the exterior tower carries explicit integral equivariant structure. A finite search produces the explicit generator Pi = -3 - i + 2 phi + i phi of norm +9 and trace -8, and Smith normal forms of 9 - Lambda^r(Pi) evaluate the intermediate layers exactly after prime-to-3 regularization: Z/5+Z/5+Z/61, Z/2+Z/2+Z/5+Z/7+Z/11 (containing the sign-twisted block), and Z/29. Assembled: K_0 contains Z^2 + (Z/8)^2 + the even torsion, K_1 contains Z^2 + the odd torsion, and the unit class is split by the spectral projections (1 +- U_sigma)/2 into a pair of classes of exact order 8, one per character of the class group. The entanglement law: the class group doubles or sign-twists exactly the non-induced strata, while free group-ring blocks count once by Shapiro - it changes representation labels, never the Smith orders. A structural by-product: the fixed locus of the Galois action is the bottom h-step system, null in measure but not in topology, so the fixed-point-versus-crossed-product question is a direct summand of the present computation. Two honest gaps are recorded: the cocycle-twisted character bookkeeping and the degree-four extension data.