The spectral action selects the Kostant cubic Dirac operator on the fuzzy S² × S²

We consider a matrix ensemble with weight e^{−S}, where S = g₂ Tr D² + g₄ Tr D⁴ is the spectral action of the Dirac operator of a fuzzy space. The variables are six n×n matrices. The structure constants of su(2)⊕su(2) are fixed. With the scaling g₂ ∝ n, the irreducible fuzzy S²×S² is the ground state in a finite window of the coupling. This is checked by enumerating block sectors up to n = 81. At the minimum of the action, the matrix scale adjusts so that the spinor part of the Dirac operator becomes the Kostant cubic operator. The ratio r/s = 3/2 is fixed by numbers from su(2) and the Clifford algebra and is not fitted. The Kostant point is reached exactly at one value of the coupling. This value lies inside the window for all values of n checked. On the monopole module the same operator gives the continuum Dirac operator. However, the ensemble selects equal embedding radii of the blocks. As a result, the charge of the module is screened completely. The reason for the disagreement with the continuum is the three-dimensional differential calculus on the fuzzy sphere. We show that there is no SU(2)-covariant two-dimensional calculus on M_n. We also show that the calculus of the model is the unique minimal connected SU(2)×SU(2)-covariant calculus. Scripts and logs reproducing all numerical results: https://doi.org/10.5281/zenodo.23137077