This preprint develops an operator-theoretic framework for analyzing faithful positivity in Hilbert–Pólya-type approaches to the Riemann Hypothesis. The paper introduces the concepts of Null-Pair Gauge Compensation (NGC) and Faithful Arithmetic Norm–GNS Factorization (FANG), and proves that gauge compensation can regularize auxiliary channels without altering the genuine prime-side kernel. A shifted-root defect hierarchy is established, reducing positivity obstructions to a projection-valued classification and a residual fractional spectral branch. Under a faithful FANG construction and an independent determinant realization hypothesis for the completed xi-function, a conditional Hilbert–Pólya closure is obtained. The work is presented as a conditional structural reduction rather than an unconditional proof of the Riemann Hypothesis.