Fixed Points II: Projected Fixed Points and Equilibria in a 10D Modal Model

Corrected fourth edition. This paper applies the corrected Fixed Points I framework to a ten-dimensional control geometry M10 = Y4 x X6 with three compatible strongly commuting nonnegative vertical operators and their joint harmonic projector. Under explicit invariant-domain, compactness or condensing, spectral-gap, and coercivity or monotonicity hypotheses, it proves projected fixed-point existence, coherent-sector uniqueness, noncoherent damping estimates, Riesz-projector regularity, and promotion to a full equilibrium only through a strict Lyapunov identity. Scope boundary. The rank 1<2<3 flag acts on a separate lane factor in the q79 interpretation, the shared circle is not a seventh internal dimension, and neither the physical HYM endpoints, action, Lorentzian time, nor a physical Galerkin subspace is selected here.