Projection-Valued Defects and Faithful Positivity in Hilbert–Pólya-Type Operator Frameworks

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.