The spectral action selects the Kostant point for the spinor part of the Dirac operator on the fuzzy S²×S²

We consider a Barrett–Glaser-type 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 scale s. We show that the spectral action selects the Kostant cubic operator as the spinor part of the Dirac operator. This happens at the joint minimum of the action over the matrix scale r and the Casimir, treated as a continuous variable. There r/s = 3/2 for all g₄ > 0 and g₂ < −(13/4) g₄ s². For every block sector we prove a sharp lower bound on the action. The coupling g₂ is taken linear in n, with a parameter fac normalised so that the irreducible fuzzy S²×S², n = (2j+1)², reaches the Kostant point at fac = 1. At this value it attains the lower bound for every such n. Hence it is a ground state among block sectors. For m = 2j+1 ≤ 3000 this ground state is unique. Hence, for these n, the irreducible sector stays the ground state on an interval of fac around 1. For n ≥ 5625 and fac ∈ [1/4, 2] we prove that the irreducible sector can be a ground state only if |fac − 1| = O(n^{-1/2}). In that case the minimising matrix scale satisfies |r*/s − 3/2| = O(n^{-1/2}). At equal scales the Kostant operator on the monopole module has the spectrum of the continuum Dirac operator, truncated and with doubled multiplicity. We also show that there is no SU(2)-covariant two-dimensional calculus on the fuzzy sphere M_N = End(V_j), N = 2j+1. The calculus of the model is the unique minimal connected SU(2)×SU(2)-covariant calculus. All numerical claims are reproducible with the code package kostant_scripts, version v5 (concept DOI 10.5281/zenodo.23137077).