What sigma-equivariance gives for free, and why the range projections cannot give the rest

The modular flow of a localized Bost-Connes system determines, by sigma-equivariance alone, the point spectrum Sp_p attached to each prime and the resulting spectral invariants; this comes for free. The paper then shows the limits of the method: the range projections of the generating isometries cannot recover the remaining structure, for the same nesting reason that blocks the transfer-operator route of Paper XI.