Categorification of Spectral Action Functionals: Non-Commutative Geometry and Topological Phase Transitions in Spin-Foam Manifolds

We present an exhaustive derivation of the spectral action functional within the rigorous framework of non-commutative Riemannian manifolds (A,H,D). By employing a non-perturbative heat kernel expansion for Dirac-type operators, we demonstrate that the Einstein-Hilbert-Palatini action is an emergent property of the spectral zeta function at its principal meromorphic poles. We extend this formalism to include the dynamical Barbero-Immirzi parameter γ as a pseudoscalar field coupled to the Nieh-Yan topological invariant. The paper further investigates the categorification of spin-foam vertex amplitudes using SU(2)q quantum group invariants. We rigorously prove that the transition from Lorentzian to Euclidean geometry is a KMS-state thermalization process within von Neumann algebras of type III1. Finally, we discuss the role of Mukai-Fourier transforms in Calabi-Yau fibers as a mechanism for generating particle masses in the spectral Standard Model.