Part II: Spectral Quantization of Geometry via the Operator C-hat

This article is Part II in a sequence of works developing a geometric framework based on the deformation function C(v), previously introduced to describe relativistic curvature and topological evolution. Extending the scalar formulation of Part I, we construct here the corresponding deformation operator Ĉ, acting on the Hilbert space L²([-v_c, v_c]), where v_c is the critical velocity derived earlier. We rigorously prove the self-adjointness of this operator, establish its discrete spectrum {Cₙ}, and demonstrate the spectral completeness of the eigenbasis. This completeness ensures that the spectrum uniquely determines the underlying geometry within the class of compact rank-one symmetric spaces, providing a rigorous spectral foundation for geometric classification. The quantized spectral structure defines discrete geometric states, with entropy scaling and resonance phenomena near the lower spectral edge C → 0. This spectral formulation replaces the continuous deformation function by a sequence of discrete modes and prepares the analytical groundwork for Part III, where spectral flow and topological transitions will be developed through the evolution C(v, τ, n).<br><br>