Quantum Gravity as Spectral Geometry: A Non-Perturbative Formulation from First Principles

We present a complete, non-perturbative formulation of quantum gravity based on the principles of noncommutative spectral geometry. In this framework, gravity is not quantized as an afterthought but emerges as a spectral property of a funda-mental quantum object—the real spectral triple (A, H, D, J,γ). The metric itself is not fundamental but is reconstructed from the spectral data via Connes’ distance formula.The quantum theory is defined by a path integral over Dirac operators satisfying the spectral triple axioms:Z = Z M DD e−SE[D], SE[D] = Trχ(D2/Λ2 cut), (1)where M = D/G is the space of fluctuated Dirac operators modulo gauge transformations and diffeomorphisms.A central role is played by the **torus T 2** emerging from fermionic zero modes in the axial vortex background. This torus serves as a bridge between Euclidean and Lorentzian signatures: by analytic continuation of its modular parameter τ = iRτ/Rz, the Euclidean spectral action continues to the Lorentzian one. The noncommutativity parameter θ ensures convergence of all spectral sums throughoutthe continuation.We demonstrate that:1. The graviton propagator emerges from two-point correlations of the bilinear functional Tµν(a, b), which in the semi-classical limit reproduces linearizedgeneral relativity.2. The theory is non-perturbatively finite due to spectral regularization guaranteed by the θ-parameter on T 2 θ .3. Spacetime singularities are resolved because the Dirac operator remains welldefined on the torus even when the classical metric becomes singular.4. Black hole entropy emerges from spectral counting on T 2: SBH = #{eigenvalues of D2 ≤ Λ2 cut} ∼ A/4G.5. The analytic continuation via the torus modular parameter provides a rigorous foundation for the Wick rotation in spectral geometry, ensuring causality andunitarity.