Fuzzy–BKM Spectral Geometry: Axial Blocks, Positivity, and Reproducible Finite-Dimensional Validation

We study a finite-dimensional generalized spectral problem combining Bogoliubov–Kubo–Mori (BKM) operator geometry with the fuzzy sphere. For the irreducible N-dimensional representation of su(2), we introduce the faithful axial state ρ0,N(γ)=e−γJ3/Tr(e−γJ3) and decompose Mat(N,C) into fixed-weight sectors q of dimension N−|q|. The covariant BKM kernel induces a positive Gram matrix G(q)(γ), while adjoint su(2) commutators define an adjoint-Casimir stiffness Kadj(q)(γ). We prove G(q)>0 for every finite N, q, and real γ. Under the assumptions β≥0 and μ≥0, the adjoint stiffness obeys Kadj(q)≥0; the full stiffness K(q)=Kadj(q)+μ2G(q) is strictly positive for μ>0. Thus the generalized spectrum is real and nonnegative, and strictly positive when μ>0. We also establish the exact isotropic spectrum, the protected constant mode Ω002=μ2, and exact q↔−q isospectrality. A local analytic quadratic law for the lowest nonconstant axial branches yields a negative q=1 versus q=0 splitting near isotropy when N>2 and β>0. Reproducible computations distinguish the historical restricted N=4,5 regressions from the full-space spectra and validate the construction for N=8 and N=12 under algebraic, spectral, conditioning, determinant, and residual diagnostics. The results are deliberately finite-dimensional: no N→∞ convergence or spacetime-emergence claim is made.