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Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
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The Riemann Hypothesis is Not Well-Typed: Completion-by-Clause, Strictness, and Gödel Constraints for "Total Analytic Universes"

Authors: Emmerson (Yaohushuason), Parker;

The Riemann Hypothesis is Not Well-Typed: Completion-by-Clause, Strictness, and Gödel Constraints for "Total Analytic Universes"

Abstract

This version/draft of the paper is more involved and conceptual. It was significantly tightened and cl3aned up here: https://zenodo.org/records/18895874 We give a semantics-first diagnosis of a formation/typing mismatch in the naive statement ofthe Riemann Hypothesis (RH) when $\zeta$ is treated as a partial operator given only by itsDirichlet-series evaluation rule. The diagnosis is not about the truth of RH but aboutwhether the usual atomic clause ``$\zeta(s)=0$'' is \emph{type-correct / assertable} under astrict denotation discipline. Mathematics is treated as an evaluation system generating judgments$\mathcal{E};\rho\vdash t\Downarrow q$ (term $t$ evaluates to value $q$ under environment$\rho$). Atomic predicates are interpreted \emph{strictly}: if an argument term fails todenote, the atomic formula is \emph{undefined} (neither true nor false). We use strongKleene ($\mathbf{K}_3$) connectives so undefinedness propagates in a controlled, explicit way. Under the Dirichlet-series evaluator, $\zeta(s)$ has no value judgment on the critical strip,so $\zeta(s)=0$ becomes undefined there; consequently the naive universally-quantified RHsentence evaluates to undefined and is not assertable. A typed/guarded RH formulation usingan explicit definedness predicate $\mathrm{Def}(\zeta(s))$ \emph{is} assertable in the basesystem but becomes vacuous/trivial. To recover the classical (non-vacuous) problem statementone must add non-neutral completion-by-clause rules (analytic continuation) that extenddenotation for the old term $\zeta(s)$. We separate analytic continuation from the distinct ``failure-as-zero'' pathology(reifying undefinedness as the value $0$). Finally, we record two orthogonal Gödel constraintsfor any consistent, effectively axiomatized, arithmetically strong formalization of a``completed analytic universe'': it cannot decide every sentence (Gödel/Rosser) and itcannot prove its own consistency (Gödel II).

Keywords

three-valued logic, analytic continuation, Gödel incompleteness, strong Kleene logic, formal arithmetic, partial functions, evaluation judgments, definedness predicate, strict semantics, Riemann Hypothesis

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
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