Rigorous Functional Measure, Non-Singular Spectral Regularization, and Horizon Modular Duality in SQG

We construct a mathematically rigorous functional measure dµ SQG for Spectral Quantum Gravity (SQG) over distribution spaces S ′ utilizing the Krein-Bochner-Minlos theorem. By implementing the fundamental scale boundaries—the Planck scale L Pl and the de Sitter horizon scale L Λ —we prove that the variance kernel operator K −1/2 governing substrate mode correlations is of trace class. The Trans-Planckian regime acts as an intrinsic, indefinite ghost-regulator, resolving geometric spacetime singularities (r → 0) and UV divergences. Upon scale-symmetry breaking, these ghost modes dynamically decouple, leaving a strictly positive-definite Källén-Lehmann measure. Finally, we establish that the induced physical state on the local horizon von Neumann algebra M(L Λ ) satisfies the Kubo-Martin-Schwinger (KMS) condition at Hawking temperature T dS = (2πL Λ ) −1 with respect to the TomitaTakesaki modular automorphism group.