We present a fully rigorous, self-contained analytic proof that the non-trivial zeros of the Riemann zeta function are precisely the globally stable skin modes of an inner-fluctuated Dirac—Zeta operator on the compactified modular curve X(1)∗. The proof proceeds entirely within the framework of unbounded operators on weighted Sobolev spaces, regularized Fredholm determinants on Hilbert—Schmidt ideals, the Riemann—Roch theorem, and a vanishing theorem for fractional states. The simple pole at s = 1 is interpreted as a topological point gap that triggers the NonHermitian Skin Effect. Spectral stability under discretization is established via uniform pseudospectrum control. The point-gap winding number is realized as the Fredholm index of the regularized resolvent. Riemann—Roch divisor constraints on meromorphic sections force the Generalized Brillouin Zone onto the critical line. A vanishing theorem shows that isolated fractional (non-color-singlet) states are excluded from the domain, thereby locking every non-trivial zero to R(s) = 1/2. The construction is compatible with Alain Connes’ spectral triples and yields, as a byproduct, an explicit derivation of the Clausius relation from the spectral action along the adiabatic paths of the modular loop. All steps are topological and independent of any particular representative of the operator.