The Riemann Hypothesis as a Completion-Sensitive Sentence: Dirichlet-Clause Semantics, Neutral vs. Full-Weight Completion, Arbitrary Strip Zero-Set Realization, Analytic Selection, and Gödel Constraints on Completion-Totality Theories

This paper separates two clause packages that standard analytic discourse often conflates: the Dirichlet-series clause package for the symbol \(ζ(·)\), valid only on \((s)>1\), and the completed analytic package, in which \(ζ\) denotes on \(\{1\}\) by analytic continuation. The paper studies the first package in isolation and then analyzes the space of its possible completions. We work with an explicit base evaluator \(\) in which \(ζ(s)\) receives a value only when \((s)>1\). On the critical strip \(0<(s)<1\), evaluation of the term \(ζ(s)\) fails judgmentally: no value judgement \(ζ(s) q\) exists. A separate truth layer is then added, namely strong Kleene logic \(\) with strict atomic predicates. Under these semantics the atom \(ζ(s)=0\) is gappy on the strip, and the unguarded Riemann Hypothesis sentence\[∀ s((0<(s)<1 ζ(s)=0)⇒ (s)=12)\]evaluates to \(\) in the base evaluator. The central result is stronger than this local gap diagnosis. For every function \(κ:→\), there is a deterministic strip-total completion \( E_κ\) of \(\) obtained by adding strip clauses for \(ζ\). Across this completion family, every subset \(Z⊆\) can be realized as the strip zero-set of \(ζ\). Accordingly, the RH sentence has no robust truth value across arbitrary strip completions: there are completions making it true, completions making it false, and the base evaluator in which it is gappy. The completion spectrum of RH is therefore exactly \(\{,,\}\) once the base evaluator is considered alongside the full strip-completion family. To organize this phenomenon we import a relative taxonomy of completions. A completion can be neutral relative to a target usage class if varying the inserted values does not change the truth or value of the target expressions; this is the familiar role of zero-padding in many additive contexts. By contrast, strip completion for RH is full-weight: the inserted values directly determine the truth of the zero-predicate \(ζ(s)=0\). In fact the strip-completion family is maximally full-weight for zero-predication, since every subset of the strip is realizable as a zero-set. The classical analytic continuation then re-enters as a selection principle. Among arbitrary strip-total completions, imposing identity-preserving analytic coherence collapses the family to a singleton: any evaluator that agrees with the Dirichlet series on \((s)>1\) and interprets \(ζ\) holomorphically on \(\{1\}\) must coincide with the classical continuation. Finally, we formalize the sense in which completion-by-clause is already a completeness scheme. We define completion-totality theories: arithmetic theories that represent a completed evaluator and prove totality of \(ζ\) on a represented strip domain. Gödel--Rosser incompleteness then applies directly to the theory of the completion itself. Thus the completion is genuine and mathematically substantive, but any consistent effectively axiomatizable formal theory asserting that completion-totality remains deductively incomplete.