Spectral Chain Quantum Gravity E: Quantum Cosmogenesis and the Spectral Origin of Initial Conditions

This paper establishes a rigorous spectral theory for the initial conditions ofquantum cosmology, reconstructing the Hartle-Hawking ”no-boundary” proposalas a topological constraint of the spectral chain. It is proved that the specificboundary condition of the wavefunction of the universe is equivalent to the causalcompleteness of the spectral chain in the imaginary time direction. The core innovation lies in discovering that the exponential expansion of de Sitter space, the nearlyscale-invariant spectrum of primordial perturbations, and the origin of baryon asymmetry can all be traced back to the algebraic structure of the first level of the spectral chain — in particular, the index of the initial Dirac operator and the chiralanomaly. We derive the ”cosmic spectral generation theorem,” showing that themacroscopic parameters of the observed universe (spatial curvature, dark energydensity, amplitude of the primordial power spectrum) are completely determinedby the finite-dimensional representation theory of the spectral chain, and predicta ”spectral fingerprint” in CMB B-mode polarization — a series of characteristicangular scales quantized by the spectral chain level N. This framework translatesthe anthropic principle into a falsifiable mathematical constraint: if the spectralchain level N deviates from the observed value Nobs ∼ 10122 by more than an order of magnitude, the causal structure cannot emerge in the classical limit. Allconstructions satisfy computability criteria and provide precise theoretical targetsfor primordial gravitational wave detection and CMB spectral distortion measurements.