We complete a multi-stage program establishing the Riemann Hypothesis as a forced admissible consequence of the normalized Weil explicit formula [1, 2, 3, 4]. Building on the normalization and kernel-forcing results of Stage VII [5] and the independent construction of a conserved, nonnegative spectral density in Stage IX [6], Stage X proves that any admissible spectral realization agreeing with the explicit formula on the determining cone must coincide with the Stage VII distribution. No appeal is made to Weil positivity as an axiom [3], no spectral assumptions on the zeros are imposed [7], and no auxiliary test functions beyond the determining cone are used. Positivity of the admissibility kernel arises solely from conservation and Kirchhoff-type balance [6], independent of the Riemann Hypothesis. We show that any zero off the critical line would introduce a signed contribution incompatible with kernel admissibility, rendering such configurations inadmissible. This yields a closed proof of the Riemann Hypothesis as the unique admissible closure of the normalized explicit formula.