Admissible Closure of the Weil Explicit Formula and Closure of the Riemann Hypothesis

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.