Spectral Realization of the Riemann Zeta Function from the Indefinite Metric Quantization of Euclidean Quantum Gravity

We present a non-perturbative framework for quantum gravity that resolves the conformal factor instability through the topology of indefinite metric Krein spaces. By identifying the conformal mode with a PT-symmetric sector HΦ, we demonstrate that the vacuum admits a unitary time evolution within the physical BRST cohomology. We derive the low-energy coupling constants from the symplectic geometry of the fundamental causal cell, proving that the geometric trace of the vacuum polarization tensor yields a bare renormalization group coefficient b₀ = 8/3. This establishes a topological hierarchy that constrains the fine structure constant and identifies the electron mass scale as the necessary geometric infrared cutoff of the vacuum flux. Finally, we demonstrate that the spectral determinant of the vacuum transfer operator is isomorphic to the completed Riemann Zeta function ξ(s). Within this framework, the Riemann Hypothesis is equivalent to local vacuum stability under the Einstein Equivalence Principle.