Spectral Quantum Gravity: A Falsifiable Single-Scale Modification of General Relativity

This paper introduces Spectral Quantum Gravity (SQG), a novel non-local extension of General Relativity (GR) derived from a fundamental spectral fixed-point structure. Unlike traditional modified gravity theories that introduce additional fields or dark components, SQG is characterized by a single emergent infrared scale L. The framework operates on the principle that gravitation is the linear response of a timeless, two-dimensional spectral medium. The resulting effective field equations reproduce General Relativity in the ultraviolet (kL 1) while providing controlled, scale-dependent modifications in the infrared. Key Features: Ghost-Free & Stable: The theory satisfies spectral positivity and the Källén–Lehmann representation, ensuring vacuum stability and the absence of Ostrogradsky ghosts. Falsifiable Predictions: SQG predicts a characteristic, scale-dependent deviation from the GR lensing consistency relation (1) peaking at the spectral scale L. Minimalist Approach: It provides a unified explanation for phenomena typically attributed to Dark Matter and Dark Energy using only one additional parameter (L). Consistency: Full compatibility with diffeomorphism invariance and covariant conservation of the energy-momentum tensor. This work provides a rigorous mathematical foundation for SQG and outlines a six-level experimental verification framework for future cosmological surveys.