PERMANENT GRAVITY: A RENORMALIZABLE THEORY OF QUANTUM GRAVITY FROM BOSONIC STATISTICS

We present a unified geometric theory where gravity emerges from the permanent—the symmetric analog of the determinant—reflecting the bosonic natureof the graviton. The theory is formulated as a spectral action combining a fermionic determinant det(f(DF/Λ)) and a bosonic permanent perm(f(DB/Λ)). The fermionic determinant yields the Standard Model of particle physics, while the bosonic permanent gives an additional gravitational contribution. We prove a theorem establishing that the Λ4 term (cosmological constant) vanishes when the total KO-dimension of the spectral triple is 0 (mod 8), with explicit computation of the KO-dimension for the bosonic sector. We then perform a detailed two-loop calculation of the beta functions for quadratic gravity and demonstrate that the theory is renormalizable. Numerical analysis of the renormalization group flow reveals a non-trivial ultraviolet fixed point, indicating asymptotic safety. The graviton propagator coincides with general relativity at low energies, while the ultraviolet behavior is controlled by the fixed point. This work establishes the permanent as the natural geometric object for bosonic gravity and provides a rigorous foundation for a renormalizable quantum theory of gravity.