Spectral Quantum Gravity: Emergent Spacetime, Matter, and Gravity from a Timeless Two-Dimensional Spectral Substrate

This paper presents Spectral Quantum Gravity (SQG), a phenomenological framework in which classical spacetime, matter, and gravitation emerge from a timeless two-dimensional resonant spectral substrate Q. The central result is a spectral uniqueness theorem: the gravitational kernel J (a) = a·exp (a) ·E₁ (a) is the unique ghost-free single-scale non-local modification of General Relativity consistent with four physical axioms (ghost-freedom, single coherence scale, UV recovery of GR, and maximum spectral entropy via the Jaynes principle). Key results include: A rigorous spectral uniqueness theorem for the gravitational propagator modification Parameter-free derivation of the microscopic coherence scale Lₘicro = 2√ (ln2) · lP ≈ 1. 665 lP from the Bekenstein–Hawking entropy Reproduction of the Hawking temperature from a vortex-dissociation mechanism Covariant field equations and Friedmann equations derived from the non-local action A full Einstein–Boltzmann perturbation hierarchy with three distinct CMB signatures Perturbative stability bound for vortex solutions (a < 1. 25) With one fitted cosmological parameter L ≈ 46 Mpc, SQG potentially alleviates the S₈ tension, predicting S₈ ≈ 0. 777 A falsifiable prediction for the lensing consistency parameter Σ (k) = J (k²L²) ≠ 1, with peak deviation near k ~ 1/L, testable by Euclid, LSST and DESI The third spatial dimension emerges via a Young-Laplace membrane mechanism driven by vortex pressure. Physical time emerges subsequently as a collective coherence ordering parameter. The Sine-Gordon model on the substrate yields fermions as topological solitons and provides a heuristic geometric motivation for baryon structure. The framework is ghost-free by construction through a positive Källén–Lehmann spectral representation. Dark energy is interpreted as residual global membrane tension. Many aspects remain prospective, and a full likelihood analysis against existing cosmological data is required before observational viability can be claimed. Keywords: quantum gravity, non-local gravity, modified gravity, emergent spacetime, spectral geometry, S8 tension, weak lensing, Euclid, cosmological perturbations, topological vortices, Sine-Gordon, Bekenstein-Hawking entropy, Källén-Lehmann representation, maximum entropy