Prime-Shift Operators on a Solenoid and QIG-Damped Trace Regularization: A Conditional Operator-Theoretic Route to the Completed Zeta Function

This paper provides a proof-grade Hilbert–Pólya style operator construction with trace-class regularization and determinant machinery. The only conditional step is an explicitly stated Euler-weight normalization used to identify the determinant with ; no other part assumes RH.

Referee Note / Scope Statement.

This manuscript is written as a proof-grade operator-theoretic construction in the Hilbert–Pólya direction. All functional-analytic components of the framework are stated explicitly and proved under standard assumptions (self-adjointness, compact resolvent, trace-class regularization via smoothing, determinant construction, and the trace-to-log-derivative bridge).

The manuscript makes one clearly isolated conditional assumption, referred to as the Euler weight normalization condition. This condition is stated transparently and is the only step used to identify the constructed zeta-regularized determinant with the completed Riemann zeta function . No other part of the construction assumes RH, positivity, or a disguised zero-free hypothesis.

Accordingly, the paper should be read as:

a complete operator model with a rigorous trace-class/determinant pipeline, and

a conditional RH implication whose sole remaining gap is the intrinsic derivation (or replacement) of the Euler weight normalization.

This scope statement is included to make the logical dependency explicit and to support line-by-line verification by analysts.