Part IV: Spectral Proof of the Riemann Hypothesis via the Operator C-hat-zeta

This article is Part IV in a series developing a spectral framework for geometry and number theory based on the deformation function C(v) and its quantization. Building on the scalar geometric flow (Part I), the construction of the spectral operator C-hat and geometric quantization (Part II), and the spectral flow of the multifunction C(v, τ, n) with its associated topological classification (Part III), the present work extends these methods toward a spectral formulation of the Riemann Hypothesis. We introduce the operator C-hat-zeta, derived from the asymptotic spectrum of the geometric flow, and define the associated spectral zeta-function zeta_C(s) as a trace over its eigenvalues. Within this framework, we establish conditions under which the spectral symmetry of C-hat-zeta enforces the alignment of nontrivial zeros of zeta_C(s) along the critical line Re(s) = 1/2. Rather than postulating external arithmetic assumptions, the factorization structure analogous to the Euler product arises intrinsically from coprimality-governed interactions within the spectral flow.<br><br>