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Riemann-Hypothesis Witness Fibres, Quantitative Θ-Atlas Ξ-Certificates, Criterion Absorption, Nullity Matching, and Universal Selected-Logical Nullity

Year: 2026
Type: preprint
Abstract: correspondence at [email protected] This paper develops a selected-witness metatheory for the Riemann Hypothesiswhich separates four levels that are often conflated: the raw formula containing\(\zeta(\cdot)\), the analytically selected classical RH proposition, the exact\(\Xi\)-certificate calculus, and the global metatheory of proof, disproof,independence, and resolution procedures. At the clause-prior level, the raw strip formula\[\RHzero:\quad\forall s\Bigl((0<\Rea(s)<1\wedge \zeta(s)=0)\Rightarrow\Rea(s)=\tfrac12\Bigr)\]is interpreted in the Dirichlet evaluator \(\EDir\), whose only zeta clause isthe Dirichlet-series clause on \(\Rea(s)>1\). Under strict atomic strong Kleenesemantics,\[\Sem{\RHzero}{\EDir}=\Gap,\]while the definedness-guarded formula is true by definedness-vacuity. Acrossarbitrary strip-total completions every subset of the critical strip is realizableas a zero-set of the raw \(\zeta\)-symbol. Hence the raw formula has noclause-prior bivalent sign and no completion-invariant bivalent sign. Analytic continuation is therefore treated as a selector and formation operation:it forms the selected classical proposition\[\RH:\Longleftrightarrow \EAC\models\RHzero,\]where \(\EAC\) is the unique analytically coherent evaluator agreeing with theDirichlet series on \(\Rea(s)>1\). The selected RH package constructed here has exact positive and negative witnessfibres. The negative fibre is finite-certificate based:\[\neg\RH\iff \exists w\,\BadXi(w),\]where \(\BadXi(w)\) is a decidable finite exact-count certificate predicate foran off-critical zero of the Riemann \(\Xi\)-function. The positive fibre isall-stage:\[\RH\iff \forall n\,\exists c\,\HalfPlaneCoverCert(n,c),\]where \(\HalfPlaneCoverCert(n,c)\) is a decidable half-plane zero-free certificatefor the \(n\)-th stage window in the RH-relevant \(\Xi\)-plane. A finitecomputation-history checker gives a literal \(\Pi_1\) arithmetic representative\[\RHarith:\Longleftrightarrow\forall w\,\forall u\,\neg\BadXiCheckArith(w,u),\]with\[\mathbb N\models\RHarith\iff\RH.\] The proof-status calibration of this representative is sharp on the negativeside. If \(\neg\RH\), then every \(T\supseteq\Qarith\) proves\[\neg\RHarith.\]Consequently\[T\nvdash\neg\RHarith\Rightarrow\RH,\qquad\Ind_T(\RHarith)\Rightarrow\RH.\]If \(T\) is \(\Sigma_1\)-sound, then\[T\vdash\neg\RHarith\iff\neg\RH.\] The validated-numerics backend is included theorem-level and constructively. Thecompleted \(\xi\)-function is defined as the entire removable extension of\[\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\]and the backend gives uniform rational enclosures for \(\Xi^{(j)}\),\(0\le j\le3\), on compact subsets of the RH-relevant region. The correctedenclosure theorem is a shrinking-domain theorem: on compact guards, an enclosureradius for \(\Xi^{(j)}(D)\), \(j\le2\), is bounded by a compact derivative boundfor \(\Xi^{(j+1)}\) times \(\rad(D)\), plus a precision term. The appendices giveexplicit rational interval algorithms, Euler--Maclaurin and gamma-polygamma tailbounds, theta-kernel truncation bounds, and validated quadrature estimates. The winding certificate is formulated as a parameterized tube-homotopycertificate. A rational polygonal loop \(P\) is supplied together with a boundarypartition and zero-avoiding rational image disks \(E_i\) such that both\(\Xi(\gamma([t_i,t_{i+1}]))\) and \(P([t_i,t_{i+1}])\) lie in \(E_i\). Thestraight-line homotopy inside the convex disks \(E_i\subset\mathbb C^\times\)gives rigorous winding soundness. The paper proves global metatheorems: Selection Jump for occupied stagewise-localsuccess classes, computable triage failure, reflection collapse, Tarski anddiagonal barriers, finite-prefix nonclosure, finite-interface nonclosure, andlegacy support-filter laundering. The no-transfer theorem is sharpened to a maximal a-priori selected-logicaluniform-nullity theorem. The Dirichlet-clause formula has no bivalent resolvent,\[\BivRes_{\EDir}(\RHzero)=\varnothing;\]no bivalent sign is invariant across strip completions,\[\CompInvRes_{\mathscr C_{\Strip}}(\RHzero)=\varnothing;\]and, for every consistent recursively axiomatizable\(T\supseteq\Qarith\), after analytic selection and exact witness packaging, themaximal selected-logical invariant \emph{uniform fixed-instance} resolvent remainsempty,\[\LogRes_T^{\Unif}(\Gamselmax;\RHsharp)=\varnothing.\]Here \(\Gamselmax\) denotes the full selected-logical invariant closure of exactselected packages, not merely a list of theorem schemes proved up to a stage. Thesuperscript \(\Unif\) emphasizes that this is a uniform consequence relation overall exact selected packages: it is not the set of all actual metamathematicalfacts about the particular sentence \(\RHarith\). Thus, after all a-prioriselected-logical invariant consequences are included, the intersection with theuniformly forced fixed-instance label set\[\{\mathbf{True},\mathbf{False},\mathbf{Prov}_T,\mathbf{Ref}_T,\mathbf{Unprov}_T,\mathbf{Unref}_T,\mathbf{Ind}_T\}\]is empty. This is an unconditional no-forcing theorem for the maximal invariantselected-logical closure, not a statement of failed proof search.
DOI: 10.5281/zenodo.19933344
Language: English
Open Access status: green
OpenAlex ID: W7159052891
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