This revised preprint develops a defect-theoretic operator framework for faithful positivity in Hilbert–Pólya-type approaches to the Riemann Hypothesis. Building on the original Null-Pair Gauge Compensation (NGC) and Faithful Arithmetic Norm–GNS Factorization (FANG) framework, the revision introduces a shifted-root defect analysis and a nilpotent confinement mechanism for the factorisation defect ΔK′ := KP − AA∗. The paper proves that pure-skew shifted-root defects admit a projection-valued classification, yielding null-channel branches rather than negative-spectrum obstructions. A self-adjoint nilpotent collapse theorem is established, showing that faithful nilpotent confinement of the shifted-root defect forces ΔK′ = 0, leading to KP = AA∗ and hence to a faithful square-norm factorization. In this revised formulation, FANG is no longer treated as a primitive positivity assumption, but instead arises as a consequence of faithful shifted-root construction together with faithful nilpotent defect confinement. The resulting framework reduces the positivity problem to three explicit arithmetic targets: • construction of a faithful shifted-root datum,• construction of a faithful nilpotent defect ideal,• establishment of a CRE determinant zero-set identity for the completed xi-function. Under these inputs, a conditional Hilbert–Pólya closure is obtained. The work is presented as a conditional structural reduction and positivity-refinement programme rather than an unconditional proof of the Riemann Hypothesis.
Junhu Park (Fri,) studied this question.