This paper completes the five-part canonical QMS–spectral program and establishes a universality theorem for quantum gravity within the standard physical problem class. We prove that any relativistic quantum theory satisfying five minimal and widely accepted structural requirements must reduce, at low energy, to Einstein gravity with cosmological constant. The five requirements are: Positive-definite Hilbert space and unitary time evolution Microcausality (local observable net structure) Existence of stable, jointly consultable macroscopic records A local finite-derivative low-energy effective field theory limit Local fermionic degrees of freedom No geometric manifold, metric, Dirac operator, spectral triple, or Einstein equations are assumed at the outset. The proof proceeds in three stages: Universal reduction: the minimal physical requirements force a contractive completely positive semigroup structure on the macroscopic record algebra. Emergent spectral geometry: contractive QMS structure, reflection positivity, primitive completeness, and capacity constraints force: commutativity of the fixed-point algebra, emergence of a smooth four-dimensional manifold, spin structure, a Dirac-type first-order generator, and assembly of a spectral triple via the Connes reconstruction theorem. Dynamical forcing: heat-kernel universality and Lovelock uniqueness in four dimensions force the Einstein–Hilbert action at leading order, and uniquely fix the second-order field equations to be the Einstein equations with cosmological constant. Higher-curvature corrections appear but are determined by the same spectral data and introduce no independent tunable couplings at leading order. In addition, five technical verification obligations required for unconditionality are discharged within this paper: Primitivity of the record-sector quantum Markov semigroup Nondegeneracy of the constrained mass Hessian Uniform Schatten trace-class control Positivity of the internal spectral gap Contraction of the metric self-consistency map Every link in the forcing chain relies only on: Proved results from the preceding four papers in the series Established theorems of operator algebras, spectral geometry, and mathematical physics Internal consistency of the QMS framework No phenomenological assumptions, fitted parameters, or ad hoc tunings are introduced. Within the defined physical class, Einstein dynamics are not postulated but structurally forced. This paper provides the terminal universality statement of the canonical QMS–spectral program.