Quantum Gravity from Noncommutative Geometry: Black Holes, Gravastars, and the Resolution of Singularities

We present a complete theory of quantum gravity emerging from the noncom-
mutative torus T 2
θ that previously gave us the masses of elementary particles, the
nature of time, and the origin of CP symmetry. The spectral action principle on
T 2
θ yields modified Einstein equations with a dynamical θ-field that regularizes all
classical singularities.
Key results:
1. Effective action: From the spectral action derived
2. Regular solutions: Spherically symmetric static solutions reveal a critical value Θcrit 
of the θ-field at the center, separating two phases:
· Phase I (Θ0 < Θcrit 0 ): Quasi-classical black holes with horizons, but with the central singularity replaced by a regular Planckian core of radius `Pl.
· Phase II (Θ0 > Θcrit 0 ): Gravastars -- compact objects without horizons, entirely regular, with matter concentrated in a thin shell.
3. Phase transition: The transition between phases is second-order, with critical exponents derived analytically.
4. Thermodynamics: Black hole thermodynamics is modified

5. Observational signatures: Primordial black holes in the mass range 10^10­~10^15 kg exhibit deviations from classical predictions, potentially observable as gamma-ray bursts or gravitational wave echoes.
This work completes the geometric unification of physics: the same noncommutative torus that determines particle masses and the arrow of time also governs the structure of compact objects and resolves all singularities.