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 \ ( () \), valid only on \ ( (s) >1\), and the completed analytic package, in which \ (\) 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 \ ( (s) \) receives a value only when \ ( (s) >1\). On the critical strip \ (01\) and interprets \ (\) 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 \ (\) 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.