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

1. Introduction

The logical and certificate-theoretic status of the Riemann Hypothesis depends on which object is being discussed. This paper separates four objects and then connects them.

The framework is explicitly a theory of RH-specific analytic and number-theoretic reasoning. Its point is not to discard such reasoning, but to locate it precisely. The actual theta kernel, actual $\Xi$-values, winding certificates, Li coefficients, Lagarias inequalities, zero-count certificates, and half-plane-cover certificates are coordinate-bearing RH-specific data. Once made exact, they become sign fibres of an enriched selected RH package. A successful use of one of these criteria is therefore fibre occupation: finite negative occupation is exactly $\neg \mathsf{RH}$, and all-stage positive occupation is exactly $\mathsf{RH}$. Universal selected-logical nullity concerns the pure invariant residue in which such concrete RH-specific fibres and coordinates are not observed.

First, there is the raw first- formula containing a symbol $\zeta (·)$:

$${\mathsf{RH}}_0:\forall s\left((0<Re\left(s\right)<1\wedge \zeta \left(s\right)=0)\Rightarrow Re\left(s\right)=\frac{1}{2}\right).$$

As a raw formula, this has no fixed mathematical truth value until the symbol $\zeta$ is interpreted. Under the Dirichlet-series clause alone, $\zeta \left(s\right)$ is defined only in the half-plane $Re\left(s\right)>1$, not in the critical strip. In strict atomic strong Kleene semantics, ${\mathsf{RH}}_0$ is gappy. Moreover arbitrary strip completions can realize arbitrary strip zero sets. Therefore there is no bivalent RH sign prior to the analytic selection of the meromorphic continuation.

Second, there is the analytically selected classical RH proposition, where $\zeta$ denotes the meromorphic continuation to $\mathbb{C}\setminus \left\{1\right\}$. This proposition is denoted

$$\mathsf{RH}.$$

Third, there is the exact selected $\Xi$-certificate calculus. This version of the paper strengthens that layer by adding an explicit theta-kernel atlas criterion. With

$$\Xi \left(z\right)=\xi (1/2+\mathrm{i}z),\mathcal{U}=\{x+\mathrm{i}y:x>0,0<y<1/2\},$$

classical RH is equivalent to

$$Z(\Xi ;\mathcal{U})=⌀.$$

For $N\ge 5$, define

$$K_N=\left[{\displaystyle \frac{1}{N}},N\right]\times \left[{\displaystyle \frac{1}{N}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{N}}\right]\subset \mathcal{U}.$$

The compact windows $K_N$ exhaust $\mathcal{U}$ by interiors. We define a finite rational certificate predicate

$$\mathsf{ThetaAtlas}(N,c)$$

which certifies zero-freeness on $K_N$. Its construction uses the theta-kernel formula

$$\Xi \left(z\right)=4{\int }_0^{\infty }{\Phi }_{+}\left(u\right)cos\left(zu\right)\mathrm{d}u$$

and explicit rational majorants for $|{\Xi }^{\prime }|$.

The resulting positive channel is

$$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

This is stronger than a merely qualitative compactness assertion: if $K_N$ is zero-free and

$${\delta }_N=\underset{z\in K_N}{min}|\Xi \left(z\right)|>0,$$

then an accepted certificate exists with explicitly bounded cell count and precision in terms of ${\delta }_N$ and a certified majorant for

$$\underset{\left|z\right|\le N+2}{sup}\left|{\Xi }^{\prime }\left(z\right)\right|.$$

The displayed cell-count bounds use $\sqrt{2}$ as a mathematical shorthand; in finite rational certificate records one supplies a rational upper bound $q_2>0$ with $q_2^2>2$. The quantitative theorem is existential unless a rational lower bound for ${\delta }_N$ is supplied; a later lower-bound-to-atlas conversion theorem isolates the effective construction from such supplied positive lower-bound data.

The negative side remains finite:

$$\neg \mathsf{RH}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right),$$

where ${\mathsf{Bad}}_{\Xi }\left(w\right)$ is a rational winding-number certificate proving a positive zero count for $\Xi$ in an admissible rational rectangle contained in $\mathcal{U}$. The quantitative winding mesh bound depends both on the separation of $\Xi (\partial Q)$ from 0 and on a certified tube radius keeping the boundary-subarc disks inside the compact guard. The paper also compares this $\Xi$-negative channel with classical criteria: Li-negative and Lagarias-negative certificates are exact finite negative channels for $\neg \mathsf{RH}$, and all these negative witness fibres are pairwise computably replay-equivalent.

Fourth, these exact number-theoretic criteria are absorbed into an enriched selected RH criterion package. The number theory does not escape nullity; it supplies the exact fibres to which nullity applies. The finite negative fibres include:

$$\exists w{B}_{\Xi }\left(w\right)\iff \exists v{B}_{\mathrm{Li}}\left(v\right)\iff \exists v{B}_{\mathrm{Lag}}\left(v\right)\iff \neg \mathsf{RH},$$

where $B_{\Xi }={\mathsf{Bad}}_{\Xi }$, $B_{\mathrm{Li}}$ is the packaged Li-negative predicate, and $B_{\mathrm{Lag}}$ is the packaged Lagarias-negative predicate. The all-stage positive fibres include:

$$\forall n\exists c{C}_{\Theta }(n,c)\iff \forall n\exists c{C}_0(n,c)\iff \forall n\exists c{C}_{\mathrm{hp}}(n,c)\iff \mathsf{RH},$$

where

$$C_{\Theta }(n,c)=\mathsf{ThetaAtlas}(n+5,c),C_0(n,c)=\mathsf{ZeroCert}(n,c),C_{\mathrm{hp}}(n,c)=\mathsf{HalfPlaneCoverCert}(n,c).$$

Thus Li, Lagarias, theta-atlas, winding, zero-count, and half-plane-cover criteria are not external routes around selected-logical nullity. Once made exact, they become sign fibres of the same enriched selected RH package. A finite negative certificate is literal occupation of the negative fibre and is therefore exactly $\neg \mathsf{RH}$. A complete positive certificate stream is literal occupation of the positive fibre and is therefore exactly $\mathsf{RH}$. Before such occupation is supplied, the mere availability, exactness, and replay equivalence of the criteria remain selected-logically null.

Fifth, there is the universal selected-logical level. At this level, one asks which fixed-instance labels are forced purely by exact selected-package form, or by enriched exact criterion-package form. The labels considered here are

$${\mathcal{L}}_T=\{\mathbf{True},\mathbf{False},{\mathbf{Prov}}_T,{\mathbf{Ref}}_T,{\mathbf{Unprov}}_T,{\mathbf{Unref}}_T,{\mathbf{Ind}}_T\}.$$

The endpoint of the paper is not merely a conditional statement about an arbitrary resolver that happens to be invariant. There are two linked layers.

At the semantic resolver layer, for arbitrary assignments

$$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right),$$

one proves the universal selected-resolution nullity theorem in the following unconditional metatheoretic form: for every consistent $T\supseteq \mathsf{Q}$,

$${\mathsf{Sound}}_T\left(\mathcal{R}\right)\wedge {\mathsf{Inv}}_{\mathrm{sel}}\left(\mathcal{R}\right)⟹\forall \mathcal{P}\in \mathfrak{Pkg}\mathcal{R}\left(\mathcal{P}\right)=⌀.$$

Here ${\mathsf{Inv}}_{\mathrm{sel}}$ is full selected-package replacement invariance:

$$\mathcal{R}\left(\mathcal{P}\right)=\mathcal{R}\left(\mathcal{Q}\right)(\mathcal{P},\mathcal{Q}\in \mathfrak{Pkg}).$$

This condition is intentionally strong. It says that the resolver depends only on exact selected-package form and not on accidental package identity. The consistency hypothesis on T is not a loss of metatheoretic unconditionality: the theorem is an unconditional statement about all consistent extensions of $\mathsf{Q}$. It is necessary because for inconsistent T, proof-status labels such as ${\mathbf{Prov}}_T$ and ${\mathbf{Ref}}_T$ are correct for every package under the label semantics used in this paper.

At the pure selected-parametric layer, this invariance is derived. The pure resolver language is stipulated explicitly as an opaque-package language. It contains selected bridge forms valid for every exact selected package, but it contains no eliminator for concrete fibre facts

$$B_P\left(w\right),C_P(n,c),D_P(w,u),$$

no package-code equality or reflection operation, no semantic truth oracle, no proof-status oracle, and no coordinate-bearing analytic data of a particular package. The selected parametricity theorem proves, by induction on the generation of resolver terms in this language,

$$\mathsf{PureSel}\left(\mathcal{R}\right)⟹{\mathsf{Inv}}_{\mathrm{sel}}\left(\mathcal{R}\right).$$

Consequently, for every consistent $T\supseteq \mathsf{Q}$,

$${\mathsf{Sound}}_T\left(\mathcal{R}\right)\wedge \mathsf{PureSel}\left(\mathcal{R}\right)⟹\forall \mathcal{P}\in \mathfrak{Pkg}\mathcal{R}\left(\mathcal{P}\right)=⌀.$$

Thus the invariance premise is a semantic hypothesis for arbitrary resolvers, but it is a theorem for the intended pure selected-logical regime.

The same theorem holds for enriched exact selected packages. Applying it to the enriched selected RH criterion package gives

$${\mathcal{R}}^{*}\left({\mathcal{P}}_{\mathsf{RH}}^{*}\right)=⌀$$

for every sound pure enriched selected-parametric resolver, again for every consistent $T\supseteq \mathsf{Q}$. Taking $\mathcal{R}$ to be the canonical resolver induced by the full invariant selected-logical closure ${\Gamma }_{\mathrm{sel}}^{max}$ yields

$${LogRes}_T^{\mathrm{unif}}({\Gamma }_{\mathrm{sel}}^{max};{\mathsf{RH}}^{♯})=⌀$$

for every consistent $T\supseteq \mathsf{Q}$. The canonical ${\Gamma }_{\mathrm{sel}}^{max}$-resolver is invariant by construction: its forcing relation quantifies uniformly over exact selected packages and does not inspect the coordinate-bearing identity of ${\mathsf{RH}}^{♯}$.

Nullity matching.

The paper isolates a matched nullity pattern:

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,$$

$${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀,$$

and, for every consistent $T\supseteq \mathsf{Q}$,

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀,$$

$$\bigcap _{{\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}}{Act}_T\left({\mathcal{P}}^{*}\right)=⌀.$$

The first says that the raw Dirichlet-clause formula has no bivalent sign. The second says that arbitrary strip completions have no invariant bivalent sign. The third says that no fixed-instance label is common to all exact selected packages. The fourth says that adjoining exact criterion fibres does not change this: no fixed-instance label is common to all enriched exact selected packages. Pure selected-parametric resolvers factor through these common-label intersections, hence are null if sound.

Criterion absorption, not escape.

The number-theoretic criteria used in the paper are not exceptions to the final nullity theorem. They are absorbed into the selected RH package as exact sign-fibre channels. For example, a $\Xi$-winding bad witness, a negative Li coefficient certificate, and a Lagarias violation are different finite presentations of the same negative fibre:

$$\exists w{\mathsf{Bad}}_{\Xi }\left(w\right)\iff \exists v{B}_{\mathrm{Li}}\left(v\right)\iff \exists v{B}_{\mathrm{Lag}}\left(v\right)\iff \neg \mathsf{RH}.$$

Similarly, theta-atlas, zero-count, and half-plane-cover streams are different all-stage positive presentations of the same positive fibre:

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c)\iff \forall n\exists c\mathsf{ZeroCert}(n,c)\iff \forall n\exists c\mathsf{HalfPlaneCoverCert}(n,c)\iff \mathsf{RH}.$$

Thus the number theory grounds the exact fibres to which selected-logical nullity applies. It does not bypass that nullity. A non-null sign outcome occurs only when one of the exact sign fibres is occupied, and such occupation is already equivalent to the target sign.

Observation effects.

A non-null sound resolver must leave the pure invariant regime. In this framework, that is precisely how RH-specific analytic and number-theoretic reasoning enters. Such a resolver may do so by observing a finite negative fibre, by observing a complete all-stage positive fibre, by using coordinate-bearing RH data such as the actual theta kernel, actual $\Xi$-values, winding certificates, Li coefficients, Lagarias inequalities, Euler-product data, or explicit-formula data, or by adding an explicit oracle/sign axiom. Such a resolver may be mathematically meaningful, and indeed it is RH-specific in the intended sense; but it is not pure selected-parametric and is not invariant under arbitrary package replacement. If it succeeds, it occupies a sign fibre rather than extracting a fixed-instance label from selected-package form alone.

What is not claimed.

This paper does not assert $\mathsf{RH}$, $\neg \mathsf{RH}$, or ${Ind}_T\left(\widehat{\mathsf{RH}}\right)$ for any specified T. It proves:

(i)

raw and completion-level nullity statements;

(ii)

an explicit quantitative theta-atlas certificate criterion for RH;

(iii)

finite negative $\Xi$-winding certificates;

(iv)

exact comparison with selected finite negative channels from classical RH criteria;

(v)

absorption of the exact number-theoretic criteria into one enriched selected RH sign-fibre package;

(vi)

literal ${\Pi }_1$ arithmetization and proof-status calibration;

(vii)

global nonclosure/no-transfer theorems under their explicitly stated hypotheses;

(viii)

universal selected-resolution nullity at the full invariant level for every consistent $T\supseteq \mathsf{Q}$;

(ix)

selected-parametric derivation of full invariance for pure selected-logical resolvers;

(x)

enriched selected-resolution nullity for exact criterion packages.

It also does not claim that a finite prefix of the all-stage positive channel determines RH. A finite prefix proves zero-freeness only on the corresponding finite collection of windows; all-stage closure requires either all atlas certificates or additional information producing them. In the terminology of this paper, such all-stage closure is sign-fibre occupation.

Nor does the universal nullity theorem claim that all RH-specific or coordinate-bearing reasoning is null. On the contrary, the paper is explicitly a theory of such reasoning once it is made exact. It applies automatically to pure selected-parametric resolvers and conditionally to arbitrary resolvers satisfying full replacement invariance, always under the theorem’s consistency hypothesis on the background theory T when proof-status labels are included. A resolver, proof, or certificate construction that uses the actual theta kernel, actual $\Xi$-values, winding certificates, Li coefficients, Lagarias inequalities, or any other coordinate-bearing RH data is not pure selected-parametric; it is coordinate-bearing RH-specific reasoning in the sense of this paper. Such an argument may occupy a sign fibre. It lies outside the pure fully invariant resolver regime, but it lies inside the enriched RH-specific fibre calculus.

Nature of the contribution.

The paper does not claim a new zero-free region, a new verified height, or a proof/disproof of RH. Its contribution is a theory of RH-specific analytic and number-theoretic reasoning organized as exact selected sign-fibre reasoning. The theta-atlas, winding, Li, Lagarias, zero-count, and half-plane-cover channels are grounded in standard number-theoretic facts, but here they are placed into one exact enriched selected package and compared at the level of finite negative fibres, all-stage positive fibres, arithmetized traces, replay maps, fibre complexity, nullity matching, and selected-parametric invariant nullity. The central nullity theorem is formal in its invariant core, but its RH application is to an enriched RH-specific package whose fibres are supplied by coordinate-bearing number-theoretic criteria.

Why the theta-atlas theorem matters for the logical endpoint.

The theta-atlas theorem supplies one exact number-theoretic positive fibre for the selected RH package:

$$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

Its role is not to stand outside the nullity theorem, but to become part of the enriched package to which nullity is applied. The same is true on the negative side for winding certificates, Li-negative certificates, and Lagarias-negative certificates. These criteria are exact number-theoretic channels; their occupation would decide the sign, but their mere availability as equivalent criteria does not produce an invariant selected-logical label. The final nullity theorem says that the unoccupied enriched criterion architecture has no truth, falsity, proof, refutation, nonproof, nonrefutation, or independence label forced at the pure selected-logical level.

Claim hierarchy.

The following table records the status of the main claim types.

Claim type	Status in this paper
Classical RH or $\neg \mathsf{RH}$	Not asserted.
Independence of RH from a specific T	Not asserted.
Raw Dirichlet-clause bivalence	Refuted: $${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀.$$
Completion-invariant bivalence over arbitrary strip completions	Refuted: $${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀.$$
Theta-kernel majorants for ${\Xi }^{\left(j\right)}$	Proved: $$\underset{\left|z\right|\le R}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le {\mathfrak{M}}_j\left(R\right).$$ For rational radii, or for supplied rational upper radii, the upper bounds admit finite rational certificates.
Quantitative theta-atlas positive channel	Proved: $$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$
Finite negative $\Xi$-certificate channel	Proved: $$\neg \mathsf{RH}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right).$$ Quantitative winding meshes require a certified tube-radius parameter.
Classical-criterion negative channel transfer	Proved for Li and Lagarias negative channels: all are finite-witness equivalent to $\neg \mathsf{RH}$.
Number-theoretic criterion absorption	Proved: $\Xi$-winding, Li-negative, and Lagarias-negative channels are replay-equivalent finite negative fibres for $\neg \mathsf{RH}$, and $\Theta$-atlas, zero-count, and half-plane-cover streams are all-stage positive fibres for $\mathsf{RH}$. These criteria are absorbed into the enriched selected RH package and are subject to enriched selected-logical nullity.
Literal ${\Pi }_1$ arithmetic representative	Proved: $$\mathbb{N}\vDash \widehat{\mathsf{RH}}\iff \mathsf{RH}.$$ The bounded matrix is a low-level accepting-tableau checker with concrete bounded access coding and no hidden halting assertion.
Negative proof-status calibration	Proved: $$\neg \mathsf{RH}\Rightarrow T⊢\neg \widehat{\mathsf{RH}}$$ for every $T\supseteq \mathsf{Q}$.
Universal selected-resolution label output for arbitrary resolvers	Proved under the full replacement-invariance hypothesis and the consistency hypothesis on T: $$T\supseteq \mathsf{Q}\mathrm{consistent}⟹\left({\mathsf{Sound}}_T\left(\mathcal{R}\right)\wedge {\mathsf{Inv}}_{\mathrm{sel}}\left(\mathcal{R}\right)\Rightarrow \mathcal{R}\equiv ⌀\right).$$ Full replacement invariance is not automatic for arbitrary resolvers.
Pure selected-parametric label output	Proved null from soundness and consistency of T, because selected parametricity derives full replacement invariance for resolvers generated in the explicitly stipulated opaque-package pure resolver language: $$\mathsf{PureSel}\left(\mathcal{R}\right)\Rightarrow {\mathsf{Inv}}_{\mathrm{sel}}\left(\mathcal{R}\right),$$ hence, for consistent $T\supseteq \mathsf{Q}$, $${\mathsf{Sound}}_T\left(\mathcal{R}\right)\wedge \mathsf{PureSel}\left(\mathcal{R}\right)\Rightarrow \mathcal{R}\equiv ⌀.$$
Enriched criterion-package nullity	Proved in the same two forms: conditional on full enriched-package invariance for arbitrary enriched resolvers, and derived from pure enriched parametricity for pure enriched selected-logical resolvers, always for consistent $T\supseteq \mathsf{Q}$ when proof-status labels are included.

Proof architecture.

The proof has the following dependency structure.

Layer	Output
Partial zeta semantics	The Dirichlet-clause formula is gappy; arbitrary strip completions have no invariant bivalent RH sign.
Analytic selection	Analytic continuation selects the classical proposition $\mathsf{RH}$.
$\Xi$-plane reduction	$$\mathsf{RH}\iff Z(\Xi ;\mathcal{U})=⌀.$$
Theta-kernel analytic layer	The representation $$\Xi \left(z\right)=4{\int }_0^{\infty }{\Phi }_{+}\left(u\right)cos\left(zu\right)\mathrm{d}u$$ gives explicit majorants for ${\Xi }^{\left(j\right)}$.
Theta-atlas positive certificates	Finite rational atlas certificates prove zero-freeness on $$K_N=[1/N,N]\times [1/N,1/2-1/N].$$
Quantitative positive channel	If ${\delta }_N={min}_{K_N}|\Xi |$, then a certificate exists with explicit cell-count and precision bounds. If rational lower bounds for ${\delta }_N$ are supplied, the certificates are effectively constructible from those bounds and certified derivative majorants.
Winding certificates	Finite tube-homotopy certificates give exact zero counts in rational rectangles. Quantitative mesh bounds require both value-separation and a certified tube radius.
Witness fibres	$$\neg \mathsf{RH}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right),\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$
Classical criteria comparison	Li-negative and Lagarias-negative certificates are exact finite negative channels for $\neg \mathsf{RH}$, replay-equivalent to $\Xi$-bad witnesses.
Criterion absorption	The exact number-theoretic criteria are bundled into an enriched selected RH package. Their non-null force occurs only by sign-fibre occupation: finite negative occupation is $\neg \mathsf{RH}$, and all-stage positive occupation is $\mathsf{RH}$.
Arithmetization	A bounded accepting-trace relation gives a literal ${\Pi }_1$ representative $\widehat{\mathsf{RH}}$. The traces are complete low-level Turing/register-machine histories with concrete bounded access coding.
Proof-status calibration	If $\neg \mathsf{RH}$, every $T\supseteq \mathsf{Q}$ proves $\neg \widehat{\mathsf{RH}}$; under ${\Sigma }_1$-soundness, $$T⊢\neg \widehat{\mathsf{RH}}\iff \neg \mathsf{RH}.$$
Global barriers	Selection Jump, computable triage failure, reflection collapse under its stated proof-theoretic hypotheses, Tarski and diagonal barriers, finite-prefix and finite-interface nonclosure, support-filter laundering.
Nullity matching	Raw bivalence nullity, completion-invariant nullity, ordinary exact-package common-label nullity for consistent T, and enriched exact-package common-label nullity for consistent T are matched instances of the same invariant-nullity pattern.
Selected parametricity	Pure selected-parametric resolvers are fully replacement invariant because the package variable is opaque in the stipulated pure resolver language. Therefore sound pure selected-parametric resolvers are identically null for every consistent $T\supseteq \mathsf{Q}$.
Universal nullity	>Every sound fully selected-package invariant resolver is identically empty for every consistent $T\supseteq \mathsf{Q}$; every sound pure selected-parametric resolver is therefore identically empty. The same holds for enriched exact selected packages. Hence $${LogRes}_T^{\mathrm{unif}}({\Gamma }_{\mathrm{sel}}^{max};{\mathsf{RH}}^{♯})=⌀$$ for every consistent $T\supseteq \mathsf{Q}$.

Related work

Classical formulations of RH.

We use the standard completed zeta function

$$\xi \left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)$$

and the function

$$\Xi \left(z\right)=\xi (1/2+\mathrm{i}z).$$

The functional equation, the symmetries of $\xi$, and the equivalence between RH and zero-freeness in the corresponding $\Xi$-region are classical; see Titchmarsh [30], Edwards [9], Apostol [2], and Conrey [7]. Many equivalent formulations of RH are known, including Robin’s criterion [23], Lagarias’s elementary criterion [14], Li’s criterion [16], and the Nyman–Beurling–Báez-Duarte circle of criteria [3,4,18]. The present paper does not claim novelty for the mere existence of arithmetical reformulations of RH. Its analytic/certification contribution is the explicit quantitative theta-atlas certificate criterion and its integration into the exact selected witness framework. Its metalogical contribution is the absorption of exact number-theoretic criteria into enriched selected sign fibres, nullity matching, and the derivation of selected-logical nullity for pure selected-parametric resolvers.

Theta kernels.

The theta-kernel representation of $\Xi$ is classical and follows from the Jacobi theta transformation and the Mellin representation of $\xi$; see Whittaker–Watson [36], Titchmarsh [30], and Edwards [9]. The contribution here is not the identity itself, but the finite rational certificate system and explicit quantitative cell-count and precision bounds derived from it.

Rigorous zero computations and the argument principle.

The argument principle and winding-number zero counts are standard tools in complex analysis; see Ahlfors [1]. Rigorous computations for zeros of $\zeta \left(s\right)$ have a long history, including Turing’s method [33], the computations of van de Lune, te Riele, and Winter [34], Odlyzko’s computations [19], and recent large-scale verifications such as Platt and Trudgian [20]. Our goal is not to improve the numerical height to which zeros are verified. Instead, we isolate finite rational certificate formats whose correctness can be checked independently and then arithmetized.

Validated numerics and computable analysis.

The backend uses standard ideas from interval arithmetic and validated numerical analysis: rational interval enclosures, ball arithmetic, rigorous tail bounds, and validated quadrature. General references include Moore–Kearfott–Cloud [17], Tucker [31], Rump [25], and Johansson’s Arb system [11]. The computable-analysis background is represented by Pour-El and Richards [21], Weihrauch [35], and Ko [13]. The appendices give the specific constructive backend needed for the finite certificate interface.

Computability, arithmetization, and proof theory.

The computability formalism is based on Turing’s machine model [32], expressed in the paper through an acceptable numbering of partial computable functions and a primitive recursive Kleene computation predicate ${\mathsf{T}}_{\mathrm{K}}$. Thus the index set

$$\mathsf{TOT}=\{e:\forall n\exists t{\mathsf{T}}_{\mathrm{K}}(e,n,t)\}$$

is the usual totality problem for Turing-computable partial functions, written in Kleene normal form. The arithmetization of finite computation histories, the arithmetical hierarchy, ${\Pi }_2^0$-completeness, and standard proof predicates are classical; see Turing [32], Kleene [12], Shoenfield [26], Rogers [24], Soare [28], Boolos–Burgess–Jeffrey [6], and Hájek–Pudlák [10]. This computability use of Turing is distinct from Turing’s method for verifying zeros of the zeta function [33], which is cited separately in the discussion of rigorous zero computations.

Truth, reflection, and diagonal barriers.

The global nonclosure results use standard barriers: Tarski’s undefinability of truth [29], Gödel–Rosser incompleteness and reflection phenomena [10,27], and diagonal arguments. These results are not used to prove RH or $\neg \mathsf{RH}$. They calibrate which uniform terminal classifications cannot exist for broad selected-package families. Where a barrier requires hypotheses stronger than mere consistency, such as $T\supseteq I{\Sigma }_1$ and the Hilbert–Bernays–Löb derivability conditions in the reflection-collapse theorem, those hypotheses are stated explicitly. Where the selected-resolution nullity theorem uses proof-status labels ${\mathbf{Prov}}_T$ and ${\mathbf{Ref}}_T$, the consistency of T is also stated explicitly; without consistency those labels become common labels for all packages.

Partial semantics.

The initial discussion of the raw Dirichlet-clause formula uses strict atomic partial-term semantics and strong Kleene truth values. Related background can be found in the literature on partial logic and many-valued semantics, including Kleene [12], Bochvar [5], Priest [22], and Lambert [15]. The point is not to replace classical RH, but to identify the formation step: the Dirichlet-series clause alone does not give a bivalent strip zero predicate; analytic continuation selects the classical proposition.

Remark 1 (Two uses of Turing) 

Turing enters the paper in two distinct ways. The computability-theoretic sections use Turing’s 1936 model of effective computation, expressed through Kleene’s primitive recursive computation predicate. The analytic-number-theory discussion of verified zeta zeros refers separately to Turing’s 1953 method for checking zeros of the zeta function. These are historically connected but mathematically separate roles.

Euler-product and explicit-formula routes.

Euler-product and explicit-formula routes are covered here at the level of the general exact-criterion absorption schema. That is, once such a route is presented as a decidable finite negative channel or decidable all-stage positive channel exactly equivalent to $\neg \mathsf{RH}$ or $\mathsf{RH}$, it is absorbed into the enriched selected package and is governed by the same nullity theorem. Without such a finite or all-stage certificate predicate, the phrase “Euler-product route” is not yet a selected channel in the formal sense used in this paper.

Parametricity and representation independence.

The selected-parametric resolver layer is an abstraction theorem for the stipulated pure selected-logical language. The package variable is opaque, and the language has no operation that can inspect concrete fibres, package codes, truth values, proof statuses, or coordinate-bearing analytic data. Therefore every pure selected-parametric resolver is invariant under arbitrary replacement of exact selected packages. This is the selected analogue of representation independence or relational parametricity in programming-language semantics. The resulting nullity theorem is not merely a conditional on an arbitrary strong invariance hypothesis; in the pure selected-logical regime, invariance is derived. The theorem is an induction on the stipulated syntax and semantics of the pure resolver language; it is not a claim about arbitrary mathematical reasoning.

Why the paper is long.

The paper includes both the $\Xi$-specific analytic/certificate layer and the global logical architecture. The theta-atlas and winding-certificate details are included not for numerical optimization but for checkability: every finite certificate predicate used in the logical arguments must be supported by terminating rational verification. The organization above separates the analytic, numerical, arithmetical, criterion-absorption, parametricity, and global logical layers so that each can be checked independently.

2. Terminology and Notation Guide

This section collects the nonstandard terminology used throughout the paper. Formal definitions are repeated or refined in the sections where the terms are used technically.

Term or notation	Meaning
${\mathsf{RH}}_0$	The raw strip formula containing the symbol $\zeta (·)$, before analytic continuation has been selected.
${\mathcal{E}}_{\mathrm{Dir}}$	The partial Dirichlet evaluator, in which $\zeta \left(s\right)$ is interpreted only by the Dirichlet series on $Re\left(s\right)>1$.
$\mathsf{G}$	The undefined truth value in strict atomic strong Kleene semantics.
Strip completion	An assignment of values to the raw $\zeta$-symbol on the critical strip, agreeing with the Dirichlet series where the latter is defined.
Analytic selector	The operation restricting completions to the unique meromorphic continuation of the Dirichlet series to $\mathbb{C}\setminus \left\{1\right\}$.
$\mathsf{RH}$	The classical Riemann Hypothesis after analytic selection.
$\xi \left(s\right)$, $\Xi \left(z\right)$	The completed zeta function and the Riemann $\Xi$-function $$\Xi \left(z\right)=\xi (1/2+\mathrm{i}z).$$
$\mathcal{U}$	The RH-relevant $\Xi$-region $$\{x+\mathrm{i}y:x>0,0<y<1/2\}.$$
$K_N$	The compact theta-atlas window $$K_N=[1/N,N]\times [1/N,1/2-1/N]\subset \mathcal{U}(N\ge 5).$$
$\mathsf{ThetaAtlas}(N,c)$	A finite rational theta-kernel certificate proving zero-freeness of $\Xi$ on $K_N$.
${\mathfrak{M}}_j\left(R\right)$	An explicit theta-kernel majorant satisfying $$\underset{\left|z\right|\le R}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le {\mathfrak{M}}_j\left(R\right).$$ For rational R, or for a supplied rational upper radius, it admits finite rational upper certificates.
Winding certificate	A finite rational record proving, by validated enclosures and a homotopy to a rational polygon, the exact number of zeros of $\Xi$ in a rectangle.
Certified tube radius	A rational number ${\tau }_Q>0$ proving that the ${\tau }_Q$-neighborhood of a rectangle boundary lies inside the compact guard used for a winding certificate. It is needed in quantitative winding mesh bounds.
${\mathsf{Bad}}_{\Xi }\left(w\right)$	A decidable finite certificate predicate for an off-critical zero of $\Xi$; $$\exists w{\mathsf{Bad}}_{\Xi }\left(w\right)$$ is equivalent to $\neg \mathsf{RH}$.
$\mathsf{LiNeg}(n,c)$	A finite certificate proving that the n-th Li coefficient is negative. By Li’s criterion, $$\exists n,c\mathsf{LiNeg}(n,c)\iff \neg \mathsf{RH}.$$
$\mathsf{LagNeg}(n,c)$	A finite certificate proving a strict violation of Lagarias’s criterion. By Lagarias’s theorem, $$\exists n,c\mathsf{LagNeg}(n,c)\iff \neg \mathsf{RH}.$$
Criterion absorption	The process by which an exact number-theoretic criterion, once formulated as a decidable finite negative channel or decidable all-stage positive channel, becomes a fibre of the selected RH package rather than an external route around nullity.
Sign-fibre occupation	The event that an exact selected sign channel is actually occupied. For a negative channel this means $$\exists wB\left(w\right),$$ which is equivalent to $\neg \mathsf{RH}$. For a positive all-stage channel this means $$\forall n\exists cC(n,c),$$ which is equivalent to $\mathsf{RH}$. A non-null sign outcome from an exact criterion is sign-fibre occupation, not an escape from nullity.
Enriched selected RH package	The selected RH package obtained by adjoining several exact number-theoretic criteria as channels, such as the $\Xi$-winding, Li-negative, Lagarias-negative, theta-atlas, zero-count, and half-plane-cover channels. The universal nullity theorem applies to this enriched package for pure enriched selected-parametric resolvers, and conditionally to arbitrary enriched resolvers satisfying full enriched-package replacement invariance, always for consistent background T when proof-status labels are included.
$\widehat{\mathsf{RH}}$	The literal ${\Pi }_1$ arithmetic representative obtained by forbidding accepting finite bad-witness computation histories.
Exact selected package	A structure $$(P,B,C,D,\widehat{P})$$ in which a proposition P has an exact finite negative channel, an all-stage positive channel, a trace checker, and an arithmetic representative.
Selected bridge architecture	The exact equivalence structure connecting P, its negative witness predicate, its positive stagewise certificate predicate, its accepting trace relation, and its arithmetic representative.
Selected-logical theorem	A theorem invariant under replacement of one exact selected package by another. It is a theorem of package form, not a theorem depending on accidental features of one package.
Pure selected-parametric resolver	A resolver generated in the stipulated pure resolver language where the selected package argument is opaque. Such a resolver may use selected bridge forms valid for every exact selected package, but may not inspect concrete fibres, package codes, semantic truth values, proof statuses, or coordinate-bearing data. Selected parametricity proves by induction on the generation of pure resolver terms that every pure selected-parametric resolver is fully replacement invariant.
Full selected-package replacement invariance	The property $$\mathcal{R}\left(\mathcal{P}\right)=\mathcal{R}\left(\mathcal{Q}\right)(\mathcal{P},\mathcal{Q}\in \mathfrak{Pkg}).$$ For arbitrary resolvers this is a hypothesis; for pure selected-parametric resolvers it is derived by parametricity.
${\Gamma }_{\mathrm{sel}}^{max}$	The full invariant theory of exact selected packages: all selected-logical consequences true for every exact selected package. The resolver induced by ${\Gamma }_{\mathrm{sel}}^{max}$ is invariant by construction.
Fixed-instance label	One of $$\mathbf{True},\mathbf{False},{\mathbf{Prov}}_T,{\mathbf{Ref}}_T,{\mathbf{Unprov}}_T,{\mathbf{Unref}}_T,{\mathbf{Ind}}_T.$$
Resolver	An assignment $$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right)$$ which outputs fixed-instance labels for exact selected packages.
Sound resolver	A resolver whose output labels are all actually correct for the package to which they are assigned. When proof-status labels are included, the universal nullity theorem assumes the background theory T is consistent.
Universal selected-resolution nullity	The theorem that, for every consistent $T\supseteq \mathsf{Q}$, every sound fully selected-package invariant resolver is identically empty. Together with selected parametricity, this gives that every sound pure selected-parametric resolver is identically empty. The enriched version gives the same conclusion for enriched exact selected packages.
Nullity matching	The structural matching of four empty invariant resolvents: $${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,$$ $${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀,$$ and, for consistent $T\supseteq \mathsf{Q}$, $$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀,$$ $$\bigcap _{{\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}}{Act}_T\left({\mathcal{P}}^{*}\right)=⌀.$$ These are, respectively, raw bivalence nullity, completion-invariant bivalence nullity, selected-package common-label nullity, and enriched-package common-label nullity.
Coordinate-bearing criterion	A criterion using instance-specific $\zeta$- or $\Xi$-data, such as the theta kernel, winding numbers, Li coefficients, Lagarias inequalities, Euler-product data, or explicit-formula data. Once such a criterion is made exact as a finite negative channel or all-stage positive channel, it is absorbed as an exact sign fibre of the enriched selected RH package. It is not an escape from nullity; its non-null force occurs only by fibre occupation.
Observation effect	A way for a resolver to leave the pure selected-parametric regime. Examples include querying concrete negative fibres $B_P\left(w\right)$, using a complete positive stream $C_P(n,c_n)$, inspecting coordinate-bearing analytic data, or adding an oracle/sign axiom.
Legacy support-filter laundering	An inference from a filtered bad-witness class to an unrestricted bad-witness class whose missing coverage premise is itself equivalent to the target RH statement.

Definition 1 

(Exact selected package). An exact selected package is a tuple

$$\mathcal{P}=(P,B_P,C_P,D_P,\widehat{P})$$

where:

(i)

P is a proposition;

(ii)

$\widehat{P}$ is an arithmetic sentence when arithmetical representation is being used;

(iii)

$B_P\left(w\right)$ is a decidable finite negative-witness predicate;

(iv)

$C_P(n,c)$ is a decidable all-stage positive-certificate predicate;

(v)

$D_P(w,u)$ is an accepting-trace predicate for the negative verifier;

and in the intended standard semantics,

$$P\iff \forall w\neg B_P\left(w\right),$$

$$P\iff \forall n\exists c{C}_P(n,c),$$

$$B_P\left(w\right)\iff \exists u{D}_P(w,u),$$

and

$$P\iff \mathbb{N}\vDash \widehat{P}.$$

The predicate $B_P$ is the finite negative channel, $C_P$ is the all-stage positive channel, $D_P$ is the finite accepting-trace relation, and $\widehat{P}$ is the arithmetic representative.

Definition 2 

(Selected bridge architecture). The selected bridge architecture of an exact selected package

$$\mathcal{P}=(P,B_P,C_P,D_P,\widehat{P})$$

is the collection of exact equivalences

$$P\iff \forall w\neg B_P\left(w\right),P\iff \forall n\exists c{C}_P(n,c),$$

$$B_P\left(w\right)\iff \exists u{D}_P(w,u),P\iff \mathbb{N}\vDash \widehat{P}.$$

A selected bridge theorem is a theorem whose proof uses only this package-form architecture and is invariant under replacement of one exact selected package by another.

Definition 3 

(Exact number-theoretic criterion channel). An exact number-theoretic criterion channel for RH is either:

(i)

a decidable finite negative-witness predicate $B\left(w\right)$ satisfying

$$\exists wB\left(w\right)\iff \neg \mathsf{RH};$$

or

(ii)

a decidable all-stage positive-certificate predicate $C(n,c)$ satisfying

$$\forall n\exists cC(n,c)\iff \mathsf{RH}.$$

Examples in this paper include the Ξ-winding negative channel, Li-negative channel, Lagarias-negative channel, theta-atlas positive channel, zero-count positive channel, and half-plane-cover positive channel.

Definition 4 

(Criterion absorption). Criterion absorption is the passage from an exact number-theoretic RH criterion to a sign fibre of an enriched selected RH package. If $B\left(w\right)$ is a decidable finite negative channel with

$$\exists wB\left(w\right)\iff \neg \mathsf{RH},$$

then B is absorbed as a negative fibre. If $C(n,c)$ is a decidable all-stage positive channel with

$$\forall n\exists cC(n,c)\iff \mathsf{RH},$$

then C is absorbed as a positive fibre. Exact criteria do not bypass selected-logical nullity; they become the fibres to which the nullity theorem applies.

Definition 5 

(Coordinate-bearing $\Xi$-data). Coordinate-bearing Ξ-data means data that refer to the actual analytic identity of the Riemann Ξ-function. Examples include:

(i)

the theta kernel ${\Phi }_{+}$;

(ii)

theta-majorant certificates for ${\Xi }^{\left(j\right)}$;

(iii)

point enclosures for $\Xi \left(z_0\right)$;

(iv)

theta-atlas certificates $\mathsf{ThetaAtlas}(N,c)$;

(v)

winding-number certificates for Ξ;

(vi)

Li coefficients and their certified signs;

(vii)

Lagarias or Robin inequalities tied to the divisor function.

Coordinate-bearing data are the RH-specific analytic and arithmetical content of the framework. They are not selected-logical invariants of arbitrary exact selected packages. Once such data are organized into decidable finite negative or all-stage positive criterion predicates, they are absorbed as exact sign fibres of the enriched selected RH package.

Definition 6 

(Pure selected-parametric resolver). A resolver is pure selected-parametric if it is generated in the stipulated pure resolver language whose package variable is opaque and whose formation rules are stable under arbitrary replacement of exact selected packages. The pure language may use only selected bridge schemes and theorem forms valid uniformly for every exact selected package. It has no eliminator for concrete fibre facts

$$B_P\left(w\right),C_P(n,c),D_P(w,u),$$

no package-code equality or reflection operation, no semantic truth oracle, no proof-status oracle, and no coordinate-bearing analytic or arithmetic data of a particular package. The selected-parametricity theorem below is the induction principle for this stipulated language.

Definition 7 

(Observation effects). A resolver construction has a negative-fibre observation effect if it may query concrete facts of the form $B_P\left(w\right)$ or accepting traces for such facts. It has a positive-fibre observation effect if it may use a complete all-stage stream of facts $C_P(n,c_n)$. It has a coordinate-bearing observation effect if it may use package-specific analytic or arithmetic data, such as the actual theta kernel of Ξ, actual winding certificates, Li coefficients, Lagarias inequalities, Euler-product data, or explicit-formula data.

Definition 8 

(Nullity matching). Nullity matching is the structural identification of four empty invariant resolvents:

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,$$

$${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀,$$

and, for every consistent $T\supseteq \mathsf{Q}$,

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀,$$

$$\bigcap _{{\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}}{Act}_T\left({\mathcal{P}}^{*}\right)=⌀.$$

The first two are formula-level and completion-level nullities. The latter two are selected-package and enriched-selected-package common-label nullities. Pure selected-parametric resolvers factor through the corresponding common-label intersection and are therefore null if sound.

Definition 9 

(Terminality predicate). A terminality predicate for a class of coded objects is an arithmetic predicate

$$\mathsf{Term}\left(x\right)$$

intended to mark that the object coded by x has reached a successful terminal state. Exactness of $\mathsf{Term}$ on a coded image means that, on that image, $\mathsf{Term}$ agrees with the corresponding semantic success predicate.

Definition 10 

(Uniform terminal chart). A uniform terminal chart for a class of selected packages is a single rule, predicate, or resolver which assigns terminal labels or terminal success values uniformly across the class. In this paper, the formal impossibility results for terminal charts are always stated as precise theorems about computable triage, truth-definitional terminality, same-theory reflection, diagonal universality, finite-prefix and finite-interface nonclosure, or selected-label resolvers.

Definition 11 

(Laundering). An inference from a restricted certificate class $B_{\mathrm{filt}}$ to an unrestricted class B is called laundering when the missing coverage premise

$$\forall w(B\left(w\right)\to B_{\mathrm{filt}}\left(w\right))$$

is itself equivalent to the target proposition being inferred.

3. Computability, Coding, and Certificate Conventions

Convention 2 

(Natural numbers, Turing computation, and Kleene coding). Throughout,

$$\mathbb{N}=\{0,1,2,\dots \}.$$

We fix:

(i)

an acceptable numbering ${\left({\phi }_e\right)}_{e\in \mathbb{N}}$ of the partial computable functions, equivalently a Gödel numbering of Turing machines up to the usual effective translations;

(ii)

a primitive recursive Kleene computation predicate ${\mathsf{T}}_{\mathrm{K}}(e,n,t)$, where t codes a finite halting computation history of the machine with index e on input n;

(iii)

a primitive recursive output map ${\mathsf{U}}_{\mathrm{K}}\left(t\right)$;

(iv)

primitive recursive codings of finite tuples, finite lists, finite computation histories, arithmetic formulas, arithmetic sentences, rational numbers, Gaussian rationals, rational intervals, rational disks, rational rectangles, finite verifier traces, theta-atlas records, theta-majorant records, winding-certificate records, Li-certificate records, and Lagarias-certificate records.

We count bounded formulas as both ${\Sigma }_0$ and ${\Pi }_0$, and allow bounded sentences to be viewed as degenerate ${\Pi }_1$ or ${\Sigma }_1$ sentences when convenient.

Convention 3 

(Fixed finite rational certificate grammar). Every certificate predicate in this paper is interpreted relative to a fixed finite syntactic grammar. This grammar is fixed before any certificate predicate is used and includes:

• encodings of integers, rationals, Gaussian rationals, rational intervals, rational disks, rational rectangles, finite lists, finite partitions, and finite polygonal loops;
• encodings of elementary-function enclosure certificates for

  $$\pi ,exp,log,Log,sin,cos;$$
• encodings of theta-majorant records, theta-point records, quadrature records, tail-bound records, winding records, half-plane records, Li-negative records, and Lagarias-negative records;
• primitive recursive parsing rules for every certificate type;
• finite rational containment and inequality checks, always carried out by integer arithmetic after clearing denominators;
• finite records for every backend enclosure call, including all truncation orders, cutoffs, partitions, elementary enclosures, derivative-bound subrecords, and rational output disks.

The verifier never accepts a high-level assertion such as “the backend computes this enclosure” as an oracle. The certificate contains the finite data from which the verifier recomputes and checks the enclosure using rational arithmetic and the fixed grammar. This convention is the common finite interface for $\mathsf{ThetaMajCert}$, $\mathsf{ThetaPointCert}$, $\mathsf{ThetaAtlas}$, ${\mathsf{WindCertRect}}_{\mathrm{AC}}$, ${\mathsf{WindCertRect}}_{\Theta }$, $\mathsf{LiNeg}$, $\mathsf{LagNeg}$, and the other certificate predicates defined below.

Remark 4 (Turing machines and Kleene normal form) 

The acceptable numbering ${\left({\phi }_e\right)}_{e\in \mathbb{N}}$ may be taken to be a Gödel numbering of Turing machines. The predicate

$${\mathsf{T}}_{\mathrm{K}}(e,n,t)$$

is written in Kleene normal form: it is a primitive recursive relation coding that the computation with index e on input n halts with finite computation history coded by t. Thus

$$\mathsf{TOT}=\{e:\forall n\exists t{\mathsf{T}}_{\mathrm{K}}(e,n,t)\}$$

is the standard totality index set for Turing-computable partial functions, and

$${\mathsf{K}}_0=\{e:\exists t{\mathsf{T}}_{\mathrm{K}}(e,0,t)\}$$

is the standard one-input halting index set. The use of Kleene notation is a coding convention, not a change of computational model. Any two acceptable Turing/Kleene/Gödel numberings give many-one equivalent versions of these index sets by the usual effective translation theorems.

Definition 12 

(Turing totality and one-input halting in Kleene form). Define

$$\mathsf{TOT}:=\{e:\forall n\exists t{\mathsf{T}}_{\mathrm{K}}(e,n,t)\},$$

the index set of total Turing-computable partial functions in the fixed acceptable numbering. Define also

$${\mathsf{K}}_0:=\{e:\exists t{\mathsf{T}}_{\mathrm{K}}(e,0,t)\},$$

the one-input halting set. The notation ${\mathsf{T}}_{\mathrm{K}}$ emphasizes the primitive recursive Kleene normal-form coding of finite Turing-machine computations.

Definition 13 

(Many-one reducibility). For $A,B\subseteq \mathbb{N}$, write $A{\le }_{m}B$ if there is a total computable function $f:\mathbb{N}\to \mathbb{N}$ such that

$$x\in A\iff f\left(x\right)\in B.$$

When primitive recursive reductions are needed, this is stated explicitly.

Definition 14 

(Theory-relative independence). For a theory T and an arithmetic sentence σ, write

$${Ind}_T\left(\sigma \right):⟺T⊬\sigma \mathrm{and}T⊬\neg \sigma .$$

Proposition 1 

(Standard completeness bases). $\mathsf{TOT}$ is ${\Pi }_2^0$-complete and ${\mathsf{K}}_0$ is ${\Sigma }_1^0$-complete.

Proof. 

Membership is immediate:

$$e\in \mathsf{TOT}\iff \forall n\exists t{\mathsf{T}}_{\mathrm{K}}(e,n,t),$$

and

$$e\in {\mathsf{K}}_0\iff \exists t{\mathsf{T}}_{\mathrm{K}}(e,0,t).$$

For ${\Pi }_2^0$-hardness, let

$$A=\{x:\forall n\exists mR(x,n,m\left)\right\}$$

with R decidable. By the s-m-n theorem, equivalently by the effective construction of Turing-machine indices, map x to an index $f\left(x\right)$ of the machine which, on input n, searches for an m satisfying $R(x,n,m)$. This machine halts on every input n iff

$$\forall n\exists mR(x,n,m),$$

so

$$x\in A\iff f\left(x\right)\in \mathsf{TOT}.$$

Thus $\mathsf{TOT}$ is ${\Pi }_2^0$-hard.

For ${\Sigma }_1^0$-hardness, let

$$B=\{x:\exists mS(x,m\left)\right\}$$

with S decidable. Effectively map x to an index $g\left(x\right)$ of the machine which, on input 0, searches for an m satisfying $S(x,m)$. Then

$$x\in B\iff g\left(x\right)\in {\mathsf{K}}_0.$$

Thus ${\mathsf{K}}_0$ is ${\Sigma }_1^0$-hard. □

4. Raw Zeta Clauses, Completions, and Analytic Selection

The first layer prevents a conflation. The Dirichlet series

$${\displaystyle \sum _{n=1}^{\infty }}{n}^{-s}$$

does not define $\zeta \left(s\right)$ in the critical strip. Therefore the raw formula

$$\forall s\left((0<Re(s)<1\wedge \zeta (s)=0)\Rightarrow Re\left(s\right)=1/2\right)$$

has no bivalent interpretation under the Dirichlet-series clause alone. The classical RH proposition is obtained only after the analytic selection of the meromorphic continuation. This section formalizes that distinction using partial-term semantics.

4.1. Partial-Term Semantics

Definition 15 

(Critical strip and critical line). Define

$$\mathcal{S}:=\{s\in \mathbb{C}:0<Re(s)<1\},\mathcal{L}:=\{s\in \mathcal{S}:Re(s)=1/2\}.$$

Definition 16 

(Raw and guarded RH formulas). In a language with a function symbol $\zeta (·)$, define

$${\mathsf{RH}}_0:\forall s\left((0<Re\left(s\right)<1\wedge \zeta \left(s\right)=0)\Rightarrow Re\left(s\right)=\frac{1}{2}\right),$$

and define the guarded variant

$${\mathsf{RH}}_{\mathrm{Def}}:\forall s\left((0<Re\left(s\right)<1\wedge \mathsf{Def}\left(\zeta \left(s\right)\right)\wedge \zeta \left(s\right)=0)\Rightarrow Re\left(s\right)=\frac{1}{2}\right).$$

Definition 17 

(Dirichlet evaluator). The Dirichlet evaluator ${\mathcal{E}}_{\mathrm{Dir}}$ interprets $\zeta \left(t\right)$ only by the rule: if t evaluates to s and $Re\left(s\right)>1$, then

$$\zeta \left(t\right)\Downarrow {\displaystyle \sum _{n=1}^{\infty }}{n}^{-s}.$$

No value judgment for $\zeta \left(t\right)$ is generated by this rule when $Re\left(s\right)\le 1$.

Definition 18 

(Strict atomic strong Kleene semantics). Terms may fail to denote. Equality and order atoms containing a non-denoting term receive the gap value $\mathsf{G}$. Definedness atoms are two-valued:

$$〚\mathsf{Def}\left(t\right){〛}_{\mathcal{E},\rho }=\mathsf{T}\mathrm{iff}t\mathrm{denotes}\mathrm{in}\mathcal{E}\mathrm{under}\rho .$$

Connectives and quantifiers use the strong Kleene order

$$\mathsf{F}<\mathsf{G}<\mathsf{T}.$$

Conjunction is minimum, disjunction is maximum, and negation is

$$\neg \mathsf{T}=\mathsf{F},\neg \mathsf{F}=\mathsf{T},\neg \mathsf{G}=\mathsf{G}.$$

Implication is defined by

$$\alpha \Rightarrow \beta :=\neg \alpha \vee \beta .$$

Universal quantification is interpreted as the infimum of the instance values in the order $\mathsf{F}<\mathsf{G}<\mathsf{T}$, and existential quantification as the corresponding supremum. For a closed sentence σ, write

$$〚\sigma {〛}_{\mathcal{E}}$$

for its semantic value.

We write

$$\mathcal{E}\vDash \sigma$$

for a closed sentence σ iff

$$〚\sigma {〛}_{\mathcal{E}}=\mathsf{T}.$$

Thus satisfaction means designated truth, not merely non-falsity.

Definition 19 

(Semantic spectrum). For a class $\mathcal{E}$ of evaluators and a sentence σ, define

$${Spec}_{\mathcal{E}}\left(\sigma \right):=\{〚\sigma {〛}_{\mathcal{E}}:\mathcal{E}\in \mathcal{E}\}.$$

Proposition 2 

(Base gap).

$$〚{\mathsf{RH}}_0{〛}_{{\mathcal{E}}_{\mathrm{Dir}}}=\mathsf{G}.$$

Proof. 

If $a\in \mathcal{S}\setminus \mathcal{L}$, then $\zeta \left(a\right)$ is undefined in ${\mathcal{E}}_{\mathrm{Dir}}$, so $\zeta \left(a\right)=0$ has value $\mathsf{G}$. The strip hypothesis is true and $Re\left(a\right)=1/2$ is false. Hence the instance has value

$$\mathsf{G}\Rightarrow \mathsf{F}=\mathsf{G}.$$

No instance is false: outside the strip the antecedent is false, and on the critical line the consequent is true. Thus the universal infimum is $\mathsf{G}$. □

Proposition 3 

(Guarded base truth).

$$〚{\mathsf{RH}}_{\mathrm{Def}}{〛}_{{\mathcal{E}}_{\mathrm{Dir}}}=\mathsf{T}.$$

Proof. 

If $a\in \mathcal{S}$, then $\mathsf{Def}\left(\zeta \right(a\left)\right)$ is false in ${\mathcal{E}}_{\mathrm{Dir}}$, so the antecedent of the implication in ${\mathsf{RH}}_{\mathrm{Def}}$ is false. If $a\notin \mathcal{S}$, the strip hypothesis is false. Hence every instance is true. □

Remark 5. 

The truth of ${\mathsf{RH}}_{\mathrm{Def}}$ in ${\mathcal{E}}_{\mathrm{Dir}}$ is definedness-vacuity, not classical RH.

4.2. Strip Completions

Definition 20 

(Dirichlet meaning). Define the partial function

$${\zeta }_{\mathrm{Dir}}\left(s\right)={\displaystyle \sum _{n=1}^{\infty }}{n}^{-s}(Re\left(s\right)>1),$$

undefined otherwise.

Definition 21 

(Strip completions). For any function

$$\kappa :\mathcal{S}\to \mathbb{C},$$

define the partial meaning

$$g_{\kappa }\left(s\right)=\left\{\begin{array}{cc}{\zeta }_{\mathrm{Dir}}\left(s\right),\hfill & Re\left(s\right)>1,\hfill \\ \kappa \left(s\right),\hfill & s\in \mathcal{S},\hfill \\ \mathrm{undefined},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Let ${\mathcal{E}}_{\kappa }$ be the deterministic evaluator induced by $g_{\kappa }$.

Proposition 4 

(Truth classification for strip completions). For every $\kappa :\mathcal{S}\to \mathbb{C}$,

$${\mathcal{E}}_{\kappa }\vDash {\mathsf{RH}}_0\iff \{a\in \mathcal{S}:\kappa \left(a\right)=0\}\subseteq \mathcal{L}.$$

Also,

$${\mathcal{E}}_{\kappa }\vDash {\mathsf{RH}}_{\mathrm{Def}}\iff {\mathcal{E}}_{\kappa }\vDash {\mathsf{RH}}_0.$$

Proof. 

In the strip, $\zeta \left(a\right)$ is defined and equals $\kappa \left(a\right)$. The raw implication fails exactly at off-line strip zeros. The guarded formula agrees with the raw formula because the definedness guard is true throughout the strip under ${\mathcal{E}}_{\kappa }$. □

Definition 22 

(Truth set on the strip). For a formula $A\left(s\right)$, define

$$TruthSet(A;\mathcal{S},\mathcal{E}):=\{a\in \mathcal{S}:〚A\left(s\right){〛}_{\mathcal{E},\rho [s\mapsto a]}=\mathsf{T}\}.$$

Theorem 6 

(Arbitrary strip zero-set realization). For every $Z\subseteq \mathcal{S}$, there is a strip completion ${\mathcal{E}}_Z$ such that

$$TruthSet(\zeta \left(s\right)=0;\mathcal{S},{\mathcal{E}}_Z)=Z.$$

Consequently

$${Spec}_{\left\{{\mathcal{E}}_{\mathrm{Dir}}\right\}\cup \{{\mathcal{E}}_{\kappa }:\kappa :\mathcal{S}\to \mathbb{C}\}}\left({\mathsf{RH}}_0\right)=\{\mathsf{G},\mathsf{T},\mathsf{F}\}.$$

Proof. 

Let

$${\kappa }_Z\left(a\right)=\left\{\begin{array}{cc}0,\hfill & a\in Z,\hfill \\ 1,\hfill & a\notin Z.\hfill \end{array}\right.$$

Then $\zeta \left(a\right)=0$ holds exactly on Z. Taking $Z=⌀$ makes the raw RH sentence true in the completion; taking Z to contain an off-line strip point makes it false; the base evaluator gives $\mathsf{G}$ by 2. □

4.3. A Canonical Completion Family

Definition 23 

(Off-line coding points). For $n\in \mathbb{N}$, set

$${\alpha }_n:={\displaystyle \frac{1}{4}}+\mathrm{i}n.$$

Then ${\alpha }_n\in \mathcal{S}\setminus \mathcal{L}$.

Definition 24 

(Divergence-coded strip completions). For each $e\in \mathbb{N}$, define ${\kappa }_e:\mathcal{S}\to \mathbb{C}$ by

$${\kappa }_e\left(s\right)=\left\{\begin{array}{cc}0,\hfill & s={\alpha }_n\mathrm{for}\mathrm{some}n\mathrm{with}{\phi }_e\left(n\right)\uparrow ,\hfill \\ 1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Let ${\mathcal{E}}_e:={\mathcal{E}}_{{\kappa }_e}$, and define

$$I_{{\mathsf{RH}}_0}:=\{e:{\mathcal{E}}_e\vDash {\mathsf{RH}}_0\},I_{{\mathsf{RH}}_{\mathrm{Def}}}:=\{e:{\mathcal{E}}_e\vDash {\mathsf{RH}}_{\mathrm{Def}}\}.$$

No effectiveness of the completion ${\kappa }_e$ is asserted; this is a completion-sensitivity coding family.

Theorem 7 

(Canonical completion-family complexity).

$$I_{{\mathsf{RH}}_0}=I_{{\mathsf{RH}}_{\mathrm{Def}}}=\mathsf{TOT}.$$

Hence these true-fibres are ${\Pi }_2^0$-complete.

Proof. 

If $e\in \mathsf{TOT}$, no ${\alpha }_n$ is assigned value 0, so there are no off-line strip zeros. Thus ${\mathcal{E}}_e\vDash {\mathsf{RH}}_0$. If $e\notin \mathsf{TOT}$, then for some n, ${\phi }_e\left(n\right)\uparrow$, so ${\kappa }_e\left({\alpha }_n\right)=0$, creating an off-line strip zero and falsifying ${\mathsf{RH}}_0$. The guarded formula agrees with the raw formula on strip-total completions by 4. Completeness follows from 1. □

4.4. Analytic Selection

The completion-sensitivity results above do not question the classical definition of $\zeta$. They identify the point at which the classical proposition is formed. Once analytic continuation has selected ${\zeta }_{\mathrm{AC}}$, all subsequent certificate constructions are ordinary classical constructions for $\xi$ and $\Xi$.

Definition 25 

(Analytic continuation). Let ${\zeta }_{\mathrm{AC}}$ denote the meromorphic continuation of the Riemann zeta function to $\mathbb{C}\setminus \left\{1\right\}$.

Definition 26 

(Selected classical RH). Let ${\mathcal{E}}_{\mathrm{AC}}$ be the exact evaluator in which ζ denotes ${\zeta }_{\mathrm{AC}}$ on $\mathbb{C}\setminus \left\{1\right\}$. Define

$$\mathsf{RH}:⟺{\mathcal{E}}_{\mathrm{AC}}\vDash {\mathsf{RH}}_0.$$

Theorem 8 

(Analytic selection). The strip assignments compatible with meromorphic continuation to $\mathbb{C}\setminus \left\{1\right\}$ and agreement with the Dirichlet series on $Re\left(s\right)>1$ form the singleton

$$\left\{{\zeta }_{\mathrm{AC}}{|}_{\mathcal{S}}\right\}.$$

Thus classical RH is the completion-level statement selected by analytic coherence.

Proof. 

If two meromorphic functions on $\mathbb{C}\setminus \left\{1\right\}$ agree with the Dirichlet series on the nonempty open half-plane $Re\left(s\right)>1$, they agree on all of $\mathbb{C}\setminus \left\{1\right\}$ by the identity theorem, since the domain is connected and the functions are holomorphic away from the common possible pole at $s=1$. □

5. The Selected $\Xi$-Plane Formulation

Definition 27 

(Completed $\xi$ and $\Xi$). Let $\xi \left(s\right)$ be the entire function obtained by filling in the removable singularities at $s=0$ and $s=1$ of

$${\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2){\zeta }_{\mathrm{AC}}\left(s\right).$$

Thus

$$\xi \left(0\right)=\xi \left(1\right)={\displaystyle \frac{1}{2}}.$$

Define

$$\Xi \left(z\right)=\xi \left({\displaystyle \frac{1}{2}}+\mathrm{i}z\right).$$

Define the RH-relevant Ξ-region:

$$\mathcal{U}:=\{z=x+\mathrm{i}y\in \mathbb{C}:x>0,0<y<1/2\}.$$

Lemma 1 

($\xi$-symmetries). For all $s\in \mathbb{C}$,

$$\xi \left(s\right)=\xi (1-s),\xi \left(\overline{s}\right)=\overline{\xi \left(s\right)}.$$

Consequently,

$$\Xi (-z)=\Xi \left(z\right),\Xi \left(\overline{z}\right)=\overline{\Xi \left(z\right)}.$$

If $\xi \left(\rho \right)=0$, then

$$\xi (1-\rho )=0,\xi \left(\overline{\rho }\right)=0,\xi (1-\overline{\rho })=0.$$

Proof. 

The functional equation for the completed zeta function gives

$$\xi \left(s\right)=\xi (1-s).$$

For $Re\left(s\right)>1$, the Dirichlet series has real coefficients, so

$$\zeta \left(\overline{s}\right)=\overline{\zeta \left(s\right)}.$$

By analytic continuation this identity holds on $\mathbb{C}\setminus \left\{1\right\}$. The remaining factors in the completed expression for $\xi$ also respect complex conjugation, and the removable values at 0 and 1 are real. Hence

$$\xi \left(\overline{s}\right)=\overline{\xi \left(s\right)}$$

for all s.

The evenness of $\Xi$ follows from

$$\Xi (-z)=\xi (1/2-\mathrm{i}z)=\xi (1-(1/2-\mathrm{i}z\left)\right)=\xi (1/2+\mathrm{i}z)=\Xi \left(z\right).$$

For the conjugation symmetry of $\Xi$, note that

$$\overline{\Xi \left(z\right)}=\overline{\xi (1/2+\mathrm{i}z)}=\xi (1/2-\mathrm{i}\overline{z})=\xi (1-(1/2-\mathrm{i}\overline{z}))=\xi (1/2+\mathrm{i}\overline{z})=\Xi \left(\overline{z}\right),$$

where the third equality uses the functional equation. The zero-symmetry consequences are immediate. □

Definition 28 

(Zero sets and zero counts). If F is holomorphic on an open set containing $E\subseteq \mathbb{C}$, define

$$Z(F;E):=\{z\in E:F(z)=0\}.$$

If $D\subseteq \mathbb{C}$ is a bounded open set whose closure lies in the domain of holomorphy of F, define $N(F;D)$ to be the number of zeros of F in D, counted with multiplicity.

Lemma 2 

(No real zeros of $\xi$ in the critical strip). For every real $s\in (0,1)$,

$$\zeta \left(s\right)<0,\xi \left(s\right)>0.$$

Proof. 

For $0<s<1$, the alternating eta series converges and satisfies

$$\eta \left(s\right)={\displaystyle \sum _{n=1}^{\infty }}{(-1)}^{n-1}{n}^{-s}>0.$$

The identity

$$\eta \left(s\right)=(1-2^{1-s})\zeta \left(s\right)$$

holds on this interval by analytic continuation from its initial domain of validity. Since $1-2^{1-s}<0$, it follows that $\zeta \left(s\right)<0$. In the definition of $\xi \left(s\right)$, the factors $s>0$, $s-1<0$, ${\pi }^{-s/2}>0$, and $\Gamma (s/2)>0$ combine with $\zeta \left(s\right)<0$ to give $\xi \left(s\right)>0$. □

Lemma 3 

(Classical RH in the $\Xi$-plane).

$$\mathsf{RH}\iff Z(\Xi ;\mathcal{U})=⌀.$$

Proof. 

For $z=x+\mathrm{i}y$,

$${\displaystyle \frac{1}{2}}+\mathrm{i}z={\displaystyle \frac{1}{2}}-y+\mathrm{i}x.$$

Thus points $z\in \mathcal{U}$ correspond to points s with

$$0<Re\left(s\right)<1/2,Im\left(s\right)>0.$$

In the critical strip, the factor

$$A\left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2)$$

has no zeros and no poles. Hence, in $0<Re\left(s\right)<1$,

$$\xi \left(s\right)=0\iff \zeta \left(s\right)=0.$$

If RH holds, no nontrivial zero has real part different from $1/2$, hence $Z(\Xi ;\mathcal{U})=⌀$.

Conversely, suppose RH fails. Then there is a zero

$$\rho =\sigma +\mathrm{i}t$$

of $\zeta$, equivalently of $\xi$, with

$$0<\sigma <1,\sigma \ne 1/2.$$

By 2, $t\ne 0$. By the conjugation symmetry in 1, replace $\rho$ by $\overline{\rho }$ if necessary, so $t>0$. If $\sigma <1/2$, set

$$z=t+\mathrm{i}(1/2-\sigma )\in \mathcal{U}.$$

Then

$${\displaystyle \frac{1}{2}}+\mathrm{i}z=\sigma +\mathrm{i}t=\rho ,$$

so $\Xi \left(z\right)=0$. If $\sigma >1/2$, use the combined symmetry

$$\rho \mapsto 1-\overline{\rho }$$

from 1. This replaces $\rho =\sigma +\mathrm{i}t$ by

$$1-\overline{\rho }=1-\sigma +\mathrm{i}t,$$

whose real part is $<1/2$ and whose imaginary part is still positive. The previous case then gives a zero of $\Xi$ in $\mathcal{U}$. Thus $Z(\Xi ;\mathcal{U})\ne ⌀$. □

6. The Quantitative Theta-Atlas Theorem

This section constructs a finite rational positive certificate predicate

$$\mathsf{ThetaAtlas}(N,c)$$

whose accepted certificates prove zero-freeness of $\Xi$ on the compact window

$$K_N=\left[{\displaystyle \frac{1}{N}},N\right]\times \left[{\displaystyle \frac{1}{N}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{N}}\right]\subset \mathcal{U}.$$

The construction is based on the classical theta-kernel representation of $\Xi$. The contribution here is the conversion of that representation into a finite rational certificate system with explicit theta-majorants, point certificates, atlas certificates, and quantitative cell-count and precision bounds.

All finite certificate predicates in this section use the fixed finite rational certificate grammar of 3. Thus every “certificate” below is a finite rational record parsed and checked by a terminating verifier using only the fixed syntactic rules, rational arithmetic, finite elementary-function enclosure certificates, finite truncation data, finite quadrature data, and finite tail inequalities.

6.1. The Theta-Kernel Representation

Definition 29 

(Theta functions). For $x>0$, define

$$\theta \left(x\right)={\displaystyle \sum _{n=-\infty }^{\infty }}{e}^{-\pi n^{2}x},\psi \left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{e}^{-\pi n^{2}x}.$$

Thus

$$\theta \left(x\right)=1+2\psi \left(x\right).$$

Lemma 4 

(Jacobi theta transformation). For every $x>0$,

$$\theta \left(x\right)=x^{-1/2}\theta (1/x).$$

Proof. 

Let

$$g_x\left(t\right)=e^{-\pi x{t}^2}.$$

With the Fourier transform convention

$$\widehat{g}\left(\xi \right)={\int }_{-\infty }^{\infty }g\left(t\right)e^{-2\pi \mathrm{i}t\xi }\mathrm{d}t,$$

one has

$${\widehat{g}}_x\left(\xi \right)=x^{-1/2}{e}^{-\pi {\xi }^2/x}.$$

Poisson summation gives

$$\sum _{n\in \mathbb{Z}}{g}_x\left(n\right)=\sum _{n\in \mathbb{Z}}{\widehat{g}}_x\left(n\right),$$

which is precisely

$$\sum _{n\in \mathbb{Z}}{e}^{-\pi n^{2}x}=x^{-1/2}\sum _{n\in \mathbb{Z}}{e}^{-\pi n^2/x}.$$

□

Lemma 5 

(Mellin representation). For $Re\left(s\right)>1$,

$${\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)={\int }_0^{\infty }\psi \left(x\right)x^{s/2-1}\mathrm{d}x.$$

Furthermore, after splitting at 1 and using 4, this continues to

$${\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)={\displaystyle \frac{1}{s(s-1)}}+{\int }_1^{\infty }\psi \left(x\right)\left(x^{s/2-1}+x^{(1-s)/2-1}\right)\mathrm{d}x$$

away from $s=0,1$.

Proof. 

For $Re\left(s\right)>1$, absolute convergence gives

$${\int }_0^{\infty }\psi \left(x\right)x^{s/2-1}\mathrm{d}x={\displaystyle \sum _{n=1}^{\infty }}{\int }_0^{\infty }{e}^{-\pi n^{2}x}{x}^{s/2-1}\mathrm{d}x.$$

With $u=\pi n^{2}x$, the inner integral equals

$${\left(\pi n^2\right)}^{-s/2}\Gamma (s/2).$$

Summing over n gives

$${\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right).$$

Split the integral at 1. In the part from 0 to 1, set $x=1/t$. Then

$${\int }_0^1\psi \left(x\right)x^{s/2-1}\mathrm{d}x={\int }_1^{\infty }\psi (1/t)t^{-s/2-1}\mathrm{d}t.$$

From the theta transformation,

$$1+2\psi \left(t\right)=t^{-1/2}(1+2\psi (1/t)).$$

Equivalently,

$$\psi (1/t)=t^{1/2}\psi \left(t\right)+{\displaystyle \frac{t^{1/2}-1}{2}}.$$

Therefore

$${\int }_0^1\psi \left(x\right)x^{s/2-1}\mathrm{d}x={\int }_1^{\infty }\psi \left(t\right)t^{(1-s)/2-1}\mathrm{d}t+{\displaystyle \frac{1}{2}}{\int }_1^{\infty }\left(t^{-(s+1)/2}-t^{-s/2-1}\right)\mathrm{d}t.$$

For $Re\left(s\right)>1$, the elementary integral equals

$${\displaystyle \frac{1}{s-1}}-{\displaystyle \frac{1}{s}}={\displaystyle \frac{1}{s(s-1)}}.$$

The integral on the right over $[1,\infty )$ converges locally uniformly for all $s\in \mathbb{C}$, because $\psi \left(x\right)$ decays exponentially as $x\to \infty$. Thus the right-hand side is meromorphic with only the displayed simple poles and gives the continuation away from $s=0,1$. □

Definition 30 

(Theta kernel). For $u\ge 0$, define

$$F\left(u\right):=e^{u/2}\psi \left(e^{2u}\right).$$

Define

$${\Phi }_{+}\left(u\right)={\displaystyle \sum _{n=1}^{\infty }}\left(2{\pi }^2{n}^4{e}^{9u/2}-3\pi n^2{e}^{5u/2}\right)e^{-\pi n^2{e}^{2u}}.$$

Lemma 6 

(Kernel differential identity).

$$F^{\prime \prime }\left(u\right)-{\displaystyle \frac{1}{4}}F\left(u\right)=2{\Phi }_{+}\left(u\right).$$

Proof. 

For one summand

$$f_n\left(u\right)=e^{u/2}{e}^{-\pi n^2{e}^{2u}},$$

put $a=\pi n^2$. Then

$$f_n^{\prime }\left(u\right)=\left({\displaystyle \frac{1}{2}}-2a{e}^{2u}\right)f_n\left(u\right),$$

and

$$f_n^{\prime \prime }\left(u\right)=\left({\displaystyle \frac{1}{4}}-6a{e}^{2u}+4a^2{e}^{4u}\right)f_n\left(u\right).$$

Hence

$$f_n^{\prime \prime }\left(u\right)-{\displaystyle \frac{1}{4}}{f}_n\left(u\right)=\left(4{\pi }^2{n}^4{e}^{9u/2}-6\pi n^2{e}^{5u/2}\right)e^{-\pi n^2{e}^{2u}}.$$

Summing over n gives

$$F^{\prime \prime }-{\displaystyle \frac{1}{4}}F=2{\Phi }_{+}.$$

Termwise differentiation is justified by superexponential convergence on compact u-intervals. □

Lemma 7 

(Boundary identity at 0).

$$F^{\prime }\left(0\right)=-{\displaystyle \frac{1}{4}}.$$

Proof. 

Using the theta transformation in the form

$$\psi \left(e^{-2u}\right)=e^u\psi \left(e^{2u}\right)+{\displaystyle \frac{e^u-1}{2}},$$

we get

$$F(-u)=e^{-u/2}\psi \left(e^{-2u}\right)=e^{u/2}\psi \left(e^{2u}\right)+{\displaystyle \frac{e^{u/2}-e^{-u/2}}{2}}=F\left(u\right)+sinh(u/2).$$

Differentiate at $u=0$. The left side gives $-F^{\prime }\left(0\right)$, the first term on the right gives $F^{\prime }\left(0\right)$, and ${(sinh(u/2))}^{\prime }{|}_{u=0}=1/2$. Hence

$$-F^{\prime }\left(0\right)=F^{\prime }\left(0\right)+{\displaystyle \frac{1}{2}},$$

so

$$F^{\prime }\left(0\right)=-{\displaystyle \frac{1}{4}}.$$

□

Theorem 9 

(Theta-kernel representation). With

$$\Xi \left(z\right)=\xi (1/2+\mathrm{i}z),$$

one has

$$\Xi \left(z\right)=4{\int }_0^{\infty }{\Phi }_{+}\left(u\right)cos\left(zu\right)\mathrm{d}u.$$

The integral converges absolutely and locally uniformly in $z\in \mathbb{C}$, and hence defines an entire function.

Proof. 

From 5,

$$\xi \left(s\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}s(s-1){\int }_1^{\infty }\psi \left(x\right)\left(x^{s/2-1}+x^{(1-s)/2-1}\right)\mathrm{d}x.$$

Set $x=e^{2u}$. Since $\mathrm{d}x=2e^{2u}\mathrm{d}u$,

$$x^{s/2-1}\mathrm{d}x=2e^{su}\mathrm{d}u,x^{(1-s)/2-1}\mathrm{d}x=2e^{(1-s)u}\mathrm{d}u.$$

Thus

$$\xi \left(s\right)={\displaystyle \frac{1}{2}}+s(s-1){\int }_0^{\infty }\psi \left(e^{2u}\right)\left(e^{su}+e^{(1-s)u}\right)\mathrm{d}u.$$

Put

$$s={\displaystyle \frac{1}{2}}+\mathrm{i}z.$$

Then

$$e^{su}+e^{(1-s)u}=2e^{u/2}cos\left(zu\right),$$

and

$$s(s-1)=-\left(z^2+{\displaystyle \frac{1}{4}}\right).$$

Therefore

$$\Xi \left(z\right)={\displaystyle \frac{1}{2}}-2\left(z^2+{\displaystyle \frac{1}{4}}\right){\int }_0^{\infty }F\left(u\right)cos\left(zu\right)\mathrm{d}u.$$

Let

$$I\left(z\right)={\int }_0^{\infty }F\left(u\right)cos\left(zu\right)\mathrm{d}u.$$

Since F and its derivatives decay superexponentially at $+\infty$, two integrations by parts give

$${\int }_0^{\infty }{F}^{\prime \prime }\left(u\right)cos\left(zu\right)\mathrm{d}u=-F^{\prime }\left(0\right)-z^{2}I\left(z\right).$$

Hence

$${\int }_0^{\infty }\left(F^{\prime \prime }\left(u\right)-{\displaystyle \frac{1}{4}}F\left(u\right)\right)cos\left(zu\right)\mathrm{d}u=-F^{\prime }\left(0\right)-\left(z^2+{\displaystyle \frac{1}{4}}\right)I\left(z\right).$$

By 7, $-F^{\prime }\left(0\right)=1/4$. Therefore

$$2{\int }_0^{\infty }\left(F^{\prime \prime }\left(u\right)-{\displaystyle \frac{1}{4}}F\left(u\right)\right)cos\left(zu\right)\mathrm{d}u={\displaystyle \frac{1}{2}}-2\left(z^2+{\displaystyle \frac{1}{4}}\right)I\left(z\right)=\Xi \left(z\right).$$

Since

$$F^{\prime \prime }-{\displaystyle \frac{1}{4}}F=2{\Phi }_{+}$$

by 6, we obtain

$$\Xi \left(z\right)=4{\int }_0^{\infty }{\Phi }_{+}\left(u\right)cos\left(zu\right)\mathrm{d}u.$$

For local uniform convergence, fix $R>0$. If $\left|z\right|\le R$, then

$$|cos\left(zu\right)|\le e^{Ru}.$$

Each summand in ${\Phi }_{+}$ is bounded by a finite sum of terms of the form

$$C{n}^m{e}^{\alpha u}{e}^{-\pi n^2{e}^{2u}},$$

whose n-sum and u-tail decay superexponentially. This gives absolute and locally uniform convergence. □

Differentiating under the integral is justified by the same locally uniform absolute convergence. Thus, for $j\ge 0$,

$${\Xi }^{\left(j\right)}\left(z\right)=4{\int }_0^{\infty }{\Phi }_{+}\left(u\right){\displaystyle \frac{{\partial }^j}{\partial z^j}}cos\left(zu\right)\mathrm{d}u.$$

6.2. Explicit Theta-Majorants

Definition 31 

(Gaussian theta-tail constants). For $m\ge 0$, define

$$G_m:={\displaystyle \sum _{n=1}^{\infty }}{n}^m{e}^{-\pi (n^2-1)}.$$

Definition 32 

(Superexponential tail integrals). For $j\ge 0$ and $\alpha \ge 0$, define

$$I_j\left(\alpha \right):={\int }_0^{\infty }{u}^j{e}^{\alpha u}{e}^{-\pi e^{2u}}\mathrm{d}u.$$

Definition 33 

(Theta majorants). For $R\ge 0$ and $j\ge 0$, define

$${\mathfrak{M}}_j\left(R\right)=4\left(2{\pi }^2{G}_4{I}_j(R+9/2)+3\pi G_2{I}_j(R+5/2)\right).$$

Theorem 10 

(Explicit theta-majorants). For all $R\ge 0$ and $j\ge 0$,

$$\underset{\left|z\right|\le R}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le {\mathfrak{M}}_j\left(R\right).$$

Proof. 

For $\left|z\right|\le R$,

$$\left|{\displaystyle \frac{{\partial }^j}{\partial z^j}}cos\left(zu\right)\right|\le u^j{e}^{Ru}.$$

Therefore

$$|{\Xi }^{\left(j\right)}\left(z\right)|\le 4{\int }_0^{\infty }{u}^j\left|{\Phi }_{+}\left(u\right)\right|e^{Ru}\mathrm{d}u.$$

By the definition of ${\Phi }_{+}$,

$$|{\Phi }_{+}\left(u\right)|\le {\displaystyle \sum _{n=1}^{\infty }}\left(2{\pi }^2{n}^4{e}^{9u/2}+3\pi n^2{e}^{5u/2}\right)e^{-\pi n^2{e}^{2u}}.$$

Since $e^{2u}\ge 1$,

$$e^{-\pi n^2{e}^{2u}}=e^{-\pi e^{2u}}{e}^{-\pi (n^2-1)e^{2u}}\le e^{-\pi e^{2u}}{e}^{-\pi (n^2-1)}.$$

Thus

$${\displaystyle \sum _{n=1}^{\infty }}{n}^m{e}^{-\pi n^2{e}^{2u}}\le G_m{e}^{-\pi e^{2u}}.$$

Substitution gives

$$|{\Phi }_{+}\left(u\right)|\le 2{\pi }^2{G}_4{e}^{9u/2}{e}^{-\pi e^{2u}}+3\pi G_2{e}^{5u/2}{e}^{-\pi e^{2u}}.$$

Hence

$$|{\Xi }^{\left(j\right)}\left(z\right)|\le 4\left(2{\pi }^2{G}_4{\int }_0^{\infty }{u}^j{e}^{(R+9/2)u}{e}^{-\pi e^{2u}}\mathrm{d}u+3\pi G_2{\int }_0^{\infty }{u}^j{e}^{(R+5/2)u}{e}^{-\pi e^{2u}}\mathrm{d}u\right),$$

which is precisely ${\mathfrak{M}}_j\left(R\right)$. □

6.3. Rational Upper Certificates for the Theta Majorants

The formula for ${\mathfrak{M}}_j\left(R\right)$ involves $\pi$, $G_2$, $G_4$, and $I_j\left(\alpha \right)$. We now record explicit rational enclosure procedures for these quantities. These procedures are encoded as finite certificate predicates using the fixed grammar of 3.

Lemma 8 

(Rational upper bounds for $G_m$). For each $m\ge 0$ and every prescribed rational tolerance $\eta >0$, there is a finite rational computation producing $G_m^{+}\in \mathbb{Q}$ such that

$$G_m\le G_m^{+}<G_m+\eta .$$

Moreover, the assertion $G_m\le G_m^{+}$ has a finite rational certificate.

Proof. 

Fix a rational lower bound ${\pi }_{-}$ for $\pi$. Then

$$e^{-\pi (n^2-1)}\le e^{-{\pi }_{-}(n^2-1)}.$$

For a cutoff L,

$$G_m={\displaystyle \sum _{n=1}^{L-1}}{n}^m{e}^{-\pi (n^2-1)}+{\displaystyle \sum _{n=L}^{\infty }}{n}^m{e}^{-\pi (n^2-1)}.$$

The finite sum is enclosed using rational interval enclosures for $\pi$ and exp.

For the tail, put $b={\pi }_{-}>0$. Then

$${\displaystyle \sum _{n=L}^{\infty }}{n}^m{e}^{-\pi (n^2-1)}\le e^b{\displaystyle \sum _{n=L}^{\infty }}{n}^m{e}^{-b{n}^2}.$$

For $m=0$, set $C_{0,b}^{+}=1$. For $m>0$, calculus gives

$$n^m{e}^{-b{n}^2/2}\le {\left({\displaystyle \frac{m}{be}}\right)}^{m/2}(n\ge 0),$$

and therefore

$$n^m\le {\left({\displaystyle \frac{m}{be}}\right)}^{m/2}{e}^{b{n}^2/2}.$$

In a finite rational certificate one does not use the generally nonrational constant

$$C_{m,b}:={\left({\displaystyle \frac{m}{be}}\right)}^{m/2}$$

itself. Instead, the record supplies a rational number

$$C_{m,b}^{+}\in \mathbb{Q}$$

together with a finite elementary-function certificate proving

$$C_{m,b}\le C_{m,b}^{+}.$$

Hence

$${\displaystyle \sum _{n=L}^{\infty }}{n}^m{e}^{-b{n}^2}\le C_{m,b}^{+}{\displaystyle \sum _{n=L}^{\infty }}{e}^{-b{n}^2/2}.$$

Since

$${(n+1)}^2-n^2=2n+1\ge 2L+1(n\ge L),$$

we have

$${\displaystyle \sum _{n=L}^{\infty }}{e}^{-b{n}^2/2}\le {\displaystyle \frac{e^{-b{L}^2/2}}{1-e^{-b(2L+1)/2}}}.$$

The finite certificate supplies rational upper bounds

$$E_b^{+}\ge e^b,E_L^{+}\ge e^{-b{L}^2/2},q_L^{+}\ge e^{-b(2L+1)/2},$$

with

$$q_L^{+}<1.$$

The certified rational tail bound is then

$$e^b{\displaystyle \sum _{n=L}^{\infty }}{n}^m{e}^{-b{n}^2}\le E_b^{+}{C}_{m,b}^{+}{\displaystyle \frac{E_L^{+}}{1-q_L^{+}}}.$$

As $L\to \infty$, this bound tends effectively to 0, after refining the elementary exponential enclosures if necessary. Thus an arbitrarily sharp rational upper bound $G_m^{+}$ is obtained by finite rational computation. □

Lemma 9 

(Rational upper bounds for $I_j\left(\alpha \right)$). Let $j\ge 0$ and let $\alpha \in {\mathbb{Q}}_{\ge 0}$. For every rational tolerance $\eta >0$, there is a finite rational computation producing $I_j^{+}\left(\alpha \right)\in \mathbb{Q}$ such that

$$I_j\left(\alpha \right)\le I_j^{+}\left(\alpha \right)<I_j\left(\alpha \right)+\eta .$$

Moreover, the assertion $I_j\left(\alpha \right)\le I_j^{+}\left(\alpha \right)$ has a finite rational certificate.

Proof. 

Write

$$I_j\left(\alpha \right)={\int }_0^U{u}^j{e}^{\alpha u}{e}^{-\pi e^{2u}}\mathrm{d}u+{\int }_U^{\infty }{u}^j{e}^{\alpha u}{e}^{-\pi e^{2u}}\mathrm{d}u.$$

On the compact interval $[0,U]$, interval arithmetic on a rational partition gives an upper Riemann-sum enclosure for the first integral. Refining the partition and the elementary exponential enclosures makes this upper enclosure arbitrarily sharp.

For the tail, use a rational lower bound ${\pi }_{-}$ for $\pi$. Then

$${\int }_U^{\infty }{u}^j{e}^{\alpha u}{e}^{-\pi e^{2u}}\mathrm{d}u\le {\int }_U^{\infty }{u}^j{e}^{\alpha u}{e}^{-{\pi }_{-}{e}^{2u}}\mathrm{d}u.$$

Put $t=e^{2u}$, so $u={\displaystyle \frac{1}{2}}logt$ and

$$\mathrm{d}u={\displaystyle \frac{\mathrm{d}t}{2t}}.$$

Thus the right-hand side equals

$$2^{-(j+1)}{\int }_{e^{2U}}^{\infty }{(logt)}^j{t}^{\alpha /2-1}{e}^{-{\pi }_{-}t}\mathrm{d}t.$$

Choose an integer $L_0\ge 1$. For $t\ge 1$,

$$logt\le {\displaystyle \frac{L_0}{e}}{t}^{1/L_0}.$$

The finite certificate supplies a rational number $C_{log}^{+}$ proving

$${\left({\displaystyle \frac{L_0}{e}}\right)}^j\le C_{log}^{+}.$$

Set

$$\nu ={\displaystyle \frac{\alpha }{2}}+{\displaystyle \frac{j}{L_0}},c=max(\nu -1,0).$$

Choose U sufficiently large, and then choose a rational number

$$A_0\ge 1$$

such that the finite elementary exponential enclosure certifies

$$A_0\le e^{2U}$$

and such that the rational inequality

$$A_0>{\displaystyle \frac{c}{{\pi }_{-}}}$$

holds. This implies the denominator positivity condition

$${\pi }_{-}-{\displaystyle \frac{c}{A_0}}>0.$$

Since the integrand is nonnegative and $A_0\le e^{2U}$,

$${\int }_{e^{2U}}^{\infty }{(logt)}^j{t}^{\alpha /2-1}{e}^{-{\pi }_{-}t}\mathrm{d}t\le {\int }_{A_0}^{\infty }{(logt)}^j{t}^{\alpha /2-1}{e}^{-{\pi }_{-}t}\mathrm{d}t.$$

Using the logarithmic bound and the definition of $\nu$, this is at most

$$C_{log}^{+}{\int }_{A_0}^{\infty }{t}^{\nu -1}{e}^{-{\pi }_{-}t}\mathrm{d}t.$$

For $t\ge A_0$,

$$t^{\nu -1}\le A_0^{\nu -1}exp\left({\displaystyle \frac{c}{A_0}}(t-A_0)\right).$$

Indeed, if $\nu \le 1$, then $c=0$ and $t^{\nu -1}\le A_0^{\nu -1}$. If $\nu >1$, the inequality follows from

$$log(t/A_0)\le {\displaystyle \frac{t-A_0}{A_0}}.$$

Therefore

$${\int }_{A_0}^{\infty }{t}^{\nu -1}{e}^{-{\pi }_{-}t}\mathrm{d}t\le {\displaystyle \frac{A_0^{\nu -1}{e}^{-{\pi }_{-}{A}_0}}{{\pi }_{-}-c/A_0}}.$$

The finite certificate encloses

$$A_0^{\nu -1}=exp((\nu -1)log{A}_0)$$

by outward rational interval arithmetic, allowing negative rational exponents through the displayed exponential-logarithmic representation, encloses $e^{-{\pi }_{-}{A}_0}$ from above, and checks the rational denominator lower bound

$${\pi }_{-}-c/A_0>0.$$

Thus it obtains the rational tail certificate

$${\int }_U^{\infty }{u}^j{e}^{\alpha u}{e}^{-\pi e^{2u}}\mathrm{d}u\le 2^{-(j+1)}{C}_{log}^{+}{\displaystyle \frac{{\left(A_0^{\nu -1}\right)}^{+}{E}_A^{+}}{{\pi }_{-}-c/A_0}},$$

where

$${\left(A_0^{\nu -1}\right)}^{+}\ge A_0^{\nu -1},E_A^{+}\ge e^{-{\pi }_{-}{A}_0}$$

are certified rational upper bounds. As U, and hence the allowable rational cutoff $A_0$, tends to infinity, this upper bound tends effectively to 0. Combining the compact integral upper enclosure with this rational tail bound gives the desired rational upper certificate. □

Definition 34 

(Majorant certificate predicate). For rational $R\ge 0$, integer $j\ge 0$, rational $M>0$, and a finite code μ, write

$$\mathsf{ThetaMajCert}(j,R,M,\mu )$$

if μ contains rational numbers

$${\pi }^{+},{\left({\pi }^2\right)}^{+},G_2^{+},G_4^{+},I_{j,5}^{+},I_{j,9}^{+}$$

together with finite rational certificates proving

$$\pi \le {\pi }^{+},{\pi }^2\le {\left({\pi }^2\right)}^{+},$$

$$G_2\le G_2^{+},G_4\le G_4^{+},$$

and

$$I_j(R+5/2)\le I_{j,5}^{+},I_j(R+9/2)\le I_{j,9}^{+}.$$

The $G_m^{+}$ and $I_j^{+}$ subrecords include their own rational lower bounds for π, elementary-function enclosure records, cutoff parameters, finite sum data, quadrature data where needed, and tail inequalities. These subrecords are checked by finite rational arithmetic and not treated as oracle calls.

The verifier then checks the purely rational inequality

$$M\ge 4\left(2{\left({\pi }^2\right)}^{+}{G}_4^{+}{I}_{j,9}^{+}+3{\pi }^{+}{G}_2^{+}{I}_{j,5}^{+}\right).$$

If all these finite rational checks pass, then

$$\mathsf{ThetaMajCert}(j,R,M,\mu )$$

holds.

If an analytic estimate is needed on a non-rational radius $R_0$, the finite certificate uses instead a supplied rational upper radius $R^{+}\ge R_0$. The certified estimate is then

$$\underset{\left|z\right|\le R_0}{sup}|{\Xi }^{\left(j\right)}\left(z\right)|\le \underset{\left|z\right|\le R^{+}}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le {\mathfrak{M}}_j\left(R^{+}\right)\le M.$$

When $R^{+}\ge R_0$ is checked by squaring in a finite record, the verifier also checks $R^{+}\ge 0$.

Proposition 5 

(Soundness and completeness of majorant certificates). The predicate

$$\mathsf{ThetaMajCert}(j,R,M,\mu )$$

is decidable for rational $R\ge 0$. If it holds, then

$$\underset{\left|z\right|\le R}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le M.$$

Conversely, for every rational $R\ge 0$, every $j\ge 0$, and every rational $M>{\mathfrak{M}}_j\left(R\right)$, there exists μ such that

$$\mathsf{ThetaMajCert}(j,R,M,\mu ).$$

More generally, if an analytic radius $R_0$ is bounded by a supplied rational $R^{+}\ge R_0$, then every rational $M>{\mathfrak{M}}_j\left(R^{+}\right)$ admits a certificate which is sound on $\left|z\right|\le R_0$.

Proof. 

Decidability follows because $\mu$ contains only finite rational data: constant enclosures, finite sums, tail bounds, elementary-function certificates, quadrature or integration certificates where used, and rational inequalities.

For soundness, the finite record proves rational upper bounds for every non-rational ingredient in ${\mathfrak{M}}_j\left(R\right)$, and the verifier checks

$$M\ge 4\left(2{\left({\pi }^2\right)}^{+}{G}_4^{+}{I}_{j,9}^{+}+3{\pi }^{+}{G}_2^{+}{I}_{j,5}^{+}\right).$$

Thus

$$M\ge 4\left(2{\pi }^2{G}_4{I}_j(R+9/2)+3\pi G_2{I}_j(R+5/2)\right)={\mathfrak{M}}_j\left(R\right).$$

The analytic majorant theorem then gives

$$\underset{\left|z\right|\le R}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le M.$$

Completeness follows from and the elementary rational enclosure procedures for $\pi$ and ${\pi }^2$: the upper bounds for all ingredients of ${\mathfrak{M}}_j\left(R\right)$ can be made arbitrarily sharp. Thus any rational $M>{\mathfrak{M}}_j\left(R\right)$ eventually admits a finite certificate. The final assertion follows by applying this statement at $R=R^{+}$. □

Remark 11 (Rational radii in finite certificates) 

The analytic estimate

$$\underset{\left|z\right|\le R}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le {\mathfrak{M}}_j\left(R\right)$$

holds for every real $R\ge 0$. The finite predicate $\mathsf{ThetaMajCert}(j,R,M,\mu )$, however, is a rational certificate predicate and is therefore stated for rational R. In all later certificate applications the radius appearing in the finite record is rational. If a real radius occurs in a mathematical existence proof, one first chooses any rational upper radius.

6.4. Theta Point Certificates

The atlas certificates require point enclosures for $\Xi \left(z_0\right)$ at rational points $z_0$. The point certificate predicate is finite and rational. It is constructed from theta-kernel truncation, validated quadrature, and explicit tail bounds.

Lemma 10 

(Elementary rational interval primitives). There are finite rational certificate systems for:

$$\pi ,expx,logx,sinx,cosx,$$

on rational compact input intervals separated from singularities where relevant. In particular, every asserted interval enclosure used below can be checked by finite rational arithmetic.

Proof. 

For $\pi$, use Machin’s identity

$${\displaystyle \frac{\pi }{4}}=4arctan{\displaystyle \frac{1}{5}}-arctan{\displaystyle \frac{1}{239}}$$

and the alternating Taylor expansion for $arctanx$ on $\left|x\right|\le 1$, with a rational alternating-series remainder.

For exp, use the Taylor series with the Lagrange remainder on bounded rational intervals. For log, reduce by powers of 2 and use

$$logx=2\left(y+{\displaystyle \frac{y^3}{3}}+{\displaystyle \frac{y^5}{5}}+\dots \right),y={\displaystyle \frac{x-1}{x+1}},$$

with $\left|y\right|<1$ and an explicit geometric tail bound. For sine and cosine, use Taylor series after range reduction using a rational enclosure of $\pi$, or use the exponential formulas. All remainders are bounded by rational quantities and therefore give finite certificates. □

Remark 12 (Elementary-function certificate grammar) 

The formal certificate grammar fixes once and for all the real and complex interval/disk schemes used for

$$\pi ,exp,log,Log,sin,cos.$$

For real enclosures the record contains range-reduction data, Taylor/Machin truncation orders, and rational remainder bounds. For complex enclosures the record contains rational disks or rectangles for the input, the chosen branch data where Log is used, the Taylor or exponential-formula truncation data, and rational disk/rectangle remainder bounds. The verifier does not accept an asserted elementary-function enclosure as an oracle: it recomputes the enclosure from the displayed finite data using rational arithmetic and outward containment checks.

Definition 35 

(Truncated theta kernel). For an integer $L\ge 1$, define

$${\Phi }_L\left(u\right)={\displaystyle \sum _{n=1}^{L-1}}\left(2{\pi }^2{n}^4{e}^{9u/2}-3\pi n^2{e}^{5u/2}\right)e^{-\pi n^2{e}^{2u}}.$$

For fixed rational $z_0$, define

$$g_L(u;z_0)=4{\Phi }_L\left(u\right)cos\left(z_{0}u\right).$$

Lemma 11 

(Finite n-tail on compact u-intervals). Let $R_0\ge \left|z_0\right|$, $U>0$, and $L\ge 1$. Then

$$\left|4{\int }_0^U\left({\Phi }_{+}\left(u\right)-{\Phi }_L\left(u\right)\right)cos\left(z_{0}u\right)\mathrm{d}u\right|$$

is bounded above by

$$4U{e}^{R_{0}U}\left(2{\pi }^2{e}^{9U/2}{\displaystyle \sum _{n=L}^{\infty }}{n}^4{e}^{-\pi n^2}+3\pi e^{5U/2}{\displaystyle \sum _{n=L}^{\infty }}{n}^2{e}^{-\pi n^2}\right).$$

The right side admits finite rational upper certificates tending effectively to 0 as $L\to \infty$.

Proof. 

For $0\le u\le U$,

$$|cos\left(z_{0}u\right)|\le e^{|z_0|u}\le e^{R_{0}U},$$

and

$$e^{-\pi n^2{e}^{2u}}\le e^{-\pi n^2}.$$

Thus

$$|{\Phi }_{+}\left(u\right)-{\Phi }_L\left(u\right)|\le 2{\pi }^2{e}^{9U/2}{\displaystyle \sum _{n=L}^{\infty }}{n}^4{e}^{-\pi n^2}+3\pi e^{5U/2}{\displaystyle \sum _{n=L}^{\infty }}{n}^2{e}^{-\pi n^2}.$$

Integrating over $[0,U]$ gives the bound. The Gaussian tails are bounded as in 8. □

Lemma 12 

(Infinite u-tail for point evaluation). Let $R_0\ge \left|z_0\right|$. Then

$$\left|4{\int }_U^{\infty }{\Phi }_{+}\left(u\right)cos\left(z_{0}u\right)\mathrm{d}u\right|\le 4\left(2{\pi }^2{G}_4{I}_0^{\left(U\right)}(R_0+9/2)+3\pi G_2{I}_0^{\left(U\right)}(R_0+5/2)\right),$$

where

$$I_0^{\left(U\right)}\left(\alpha \right)={\int }_U^{\infty }{e}^{\alpha u}{e}^{-\pi e^{2u}}\mathrm{d}u.$$

The right side admits finite rational upper certificates tending effectively to 0 as $U\to \infty$.

Proof. 

The proof is the same as the proof of 10, but with the u-integration restricted to $[U,\infty )$. The rational tail certificates are obtained by the change of variables $t=e^{2u}$ as in 9. □

Lemma 13 

(Validated midpoint quadrature). Let g be continuously differentiable on $[a,b]$, and let

$$m={\displaystyle \frac{a+b}{2}}.$$

If

$$g\left(m\right)\in \overline{B}(C,\rho )$$

and

$$\underset{u\in [a,b]}{sup}\left|g^{\prime }\left(u\right)\right|\le B,$$

then

$${\int }_a^{b}g\left(u\right)\mathrm{d}u\in \overline{B}\left((b-a)C,(b-a)\rho +{\displaystyle \frac{{(b-a)}^2}{4}}B\right).$$

Proof. 

For $u\in [a,b]$,

$$g\left(u\right)=g\left(m\right)+{\int }_m^u{g}^{\prime }\left(v\right)\mathrm{d}v.$$

Thus

$$\left|{\int }_a^b(g\left(u\right)-g\left(m\right))\mathrm{d}u\right|\le B{\int }_a^b|u-m|\mathrm{d}u={\displaystyle \frac{{(b-a)}^2}{4}}B.$$

The point-enclosure error contributes $(b-a)\rho$. □

Remark 13 (Validated quadrature certificate grammar) 

A validated quadrature record is purely finite and syntactic. It lists the rational partition, the rational midpoint of each subinterval, the rational disk or rectangle enclosing the integrand at that midpoint, the finite elementary enclosure subrecords used to obtain that midpoint enclosure, a rational upper bound for the derivative on the subinterval, the finite rational interval subrecord proving that derivative bound, and the rational arithmetic showing that the accumulated midpoint and derivative errors are bounded by the declared radius. The verifier recomputes the quadrature enclosure from these displayed finite data; it does not call an external numerical oracle.

Definition 36 

(Theta point certificate). Let $z_0\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$, $A\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$, and $\epsilon \in {\mathbb{Q}}_{>0}$. The predicate

$$\mathsf{ThetaPointCert}(z_0,A,\epsilon ,\omega )$$

means that ω is a finite rational record containing:

• rational enclosures for the elementary constants and functions used;
• a rational $R_0\in {\mathbb{Q}}_{\ge 0}$ with $R_0\ge \left|z_0\right|$;
• integers $L,m\ge 1$ and a rational cutoff $U>0$;
• a rational partition

  $$0=u_0<u_1<\dots <u_m=U;$$
• for each interval $[u_{\ell },u_{\ell +1}]$, a rational disk enclosure for

  $$g_L\left({\displaystyle \frac{u_{\ell }+u_{\ell +1}}{2}};z_0\right);$$
• for each interval, a rational upper bound, with a finite rational subcertificate, for

  $$\underset{u\in [u_{\ell },u_{\ell +1}]}{sup}\left|{\displaystyle \frac{\mathrm{d}}{\mathrm{d}u}}{g}_L(u;z_0)\right|;$$
• rational upper bounds for the compact n-tail from 11;
• rational upper bounds for the infinite u-tail from 12;

such that the resulting rational disk enclosure for

$$4{\int }_0^{\infty }{\Phi }_{+}\left(u\right)cos\left(z_{0}u\right)\mathrm{d}u$$

is contained in

$$\overline{B}(A,\epsilon ).$$

Since $|z_0|$ need not be rational, the verifier checks

$$R_0\ge \left|z_0\right|$$

by checking both

$$R_0\ge 0$$

and the rational inequality

$$R_0^2\ge {(Re{z}_0)}^2+{(Im{z}_0)}^2.$$

Proposition 6 

(Point-certificate decidability and soundness). The predicate

$$\mathsf{ThetaPointCert}(z_0,A,\epsilon ,\omega )$$

is decidable. If it holds, then

$$|\Xi (z_0)-A|\le \epsilon .$$

Proof. 

Decidability follows because the verifier checks only finitely many rational enclosures and rational inequalities, including the nonnegative squared-radius check for $R_0\ge \left|z_0\right|$.

Soundness follows from 9: the integral equals $\Xi \left(z_0\right)$. The finite quadrature enclosures are sound by 13. The omitted n-tail and u-tail are bounded by . Hence the final disk contains $\Xi \left(z_0\right)$. Since the verifier checks that this disk is contained in $\overline{B}(A,\epsilon )$, the result follows. □

Proposition 7 

(Point-certificate completeness). For every rational point $z_0\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ and every rational $\epsilon >0$, there exist $A\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ and a finite code ω such that

$$\mathsf{ThetaPointCert}(z_0,A,\epsilon ,\omega ).$$

Proof. 

The theta integral converges absolutely. Choose U large enough that the u-tail is less than $\epsilon /4$. Choose L large enough that the compact n-tail over $[0,U]$ is less than $\epsilon /4$. On the compact interval $[0,U]$, the truncated integrand $g_L(u;z_0)$ is continuously differentiable. Refine the rational partition until the sum of midpoint quadrature errors is less than $\epsilon /4$. Finally refine elementary constant and function enclosures so that all point-evaluation interval errors contribute less than $\epsilon /4$. Choose the truncation parameters, quadrature partition, and elementary function enclosures so that the final rational disk enclosure has radius strictly less than $\epsilon$. Let $A\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ be the rational center of this final disk. Then the final enclosure is contained in $\overline{B}(A,\epsilon )$, and the resulting finite data form $\omega$. □

Corollary 1 

(Theta jet point certificates). For every integer $j\ge 0$, every rational point $z_0\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$, and every rational $\epsilon >0$, there is a finite rational certificate enclosing ${\Xi }^{\left(j\right)}\left(z_0\right)$ in a rational disk of radius at most ε. The corresponding predicate is decidable.

Proof. 

Differentiate the theta-kernel representation under the integral:

$${\Xi }^{\left(j\right)}\left(z\right)=4{\int }_0^{\infty }{\Phi }_{+}\left(u\right){\displaystyle \frac{{\partial }^j}{\partial z^j}}cos\left(zu\right)\mathrm{d}u.$$

The differentiated integrand differs from the point-evaluation integrand only by a factor $u^j$ and by replacing cos with $\pm cos$ or $\pm sin$. The same finite truncation, validated quadrature, and tail-bound argument used in applies, with the tail integrals $I_j\left(\alpha \right)$. All checks remain finite and rational. □

6.5. Theta-Atlas Windows and Certificates

Definition 37 

(Theta-atlas windows). For $N\ge 5$, define

$$K_N=\left[{\displaystyle \frac{1}{N}},N\right]\times \left[{\displaystyle \frac{1}{N}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{N}}\right]\subset \mathcal{U}.$$

For $x+\mathrm{i}y\in K_N$, this means

$${\displaystyle \frac{1}{N}}\le x\le N,{\displaystyle \frac{1}{N}}\le y\le {\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{N}}.$$

Lemma 14 

(Window exhaustion).

$$\mathcal{U}=\bigcup _{N\ge 5}{K}_N^{\circ }.$$

Proof. 

Let $z=x+\mathrm{i}y\in \mathcal{U}$. Then $x>0$ and $0<y<1/2$. Choose $N\ge 5$ large enough that

$${\displaystyle \frac{1}{N}}<x<N$$

and

$${\displaystyle \frac{1}{N}}<y<{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{N}}.$$

Then $z\in K_N^{\circ }$. □

Definition 38 

(Rational disk). A rational disk is a closed disk

$$D=\overline{B}(a,r)=\{z\in \mathbb{C}:|z-a|\le r\},a\in \mathbb{Q}+\mathrm{i}\mathbb{Q},r\in {\mathbb{Q}}_{\ge 0}.$$

Remark 14 (Finite rational containment checks) 

All disk and rectangle containment checks used below are made by rational inequalities. For example,

$$\overline{B}(a,r)\subseteq \overline{B}(b,s)$$

is checked by

$$s-r\ge {0\mathrm{and}|a-b|}^2\le {(s-r)}^2.$$

The condition

$$\left|a\right|+r\le R$$

is checked by

$$R-r\ge {0\mathrm{and}\left|a\right|}^2\le {(R-r)}^2.$$

Containment of a rational rectangle Q in a rational disk $\overline{B}(a,r)$ is checked by testing the finitely many vertices of Q, since the squared distance from a is convex on Q and attains its maximum at a vertex. Inclusion of one rational rectangle in a finite union of rational rectangles is decidable by rational cell decomposition.

Definition 39 

(Theta-atlas certificate for a rectangle). Let $K\subset \mathbb{C}$ be a rational closed rectangle. A theta-atlas certificate for K is a finite rational record consisting of:

• a rational radius $R>0$;
• a rational number $M>0$ and a code μ such that

  $$\mathsf{ThetaMajCert}(1,R,M,\mu );$$
• finitely many rational closed rectangles $Q_{\nu }$;
• rational disks

  $$D_{\nu }=\overline{B}(a_{\nu },r_{\nu });$$
• rational points $A_{\nu }\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$;
• rational radii ${\epsilon }_{\nu }>0$;
• point-certificate codes ${\omega }_{\nu }$;

such that the verifier checks:

(a)

$$K\subseteq \bigcup _{\nu }{Q}_{\nu };$$

(a)

$$Q_{\nu }\subseteq D_{\nu };$$

(a)

$$|a_{\nu }|+r_{\nu }\le R;$$

in the finite rational verifier, this is checked by

$$R-r_{\nu }\ge 0\mathrm{and}{\left|a_{\nu }\right|}^2\le {(R-r_{\nu })}^2;$$

(a)

$$\mathsf{ThetaPointCert}(a_{\nu },A_{\nu },{\epsilon }_{\nu },{\omega }_{\nu });$$

(a)

$${\epsilon }_{\nu }+M{r}_{\nu }<\left|A_{\nu }\right|.$$

Write

$$\mathsf{ThetaAtlasRect}(K,c)$$

if the finite record c satisfies these conditions.

Definition 40 

(Window atlas predicate). For $N\ge 5$, define

$$\mathsf{ThetaAtlas}(N,c)$$

to mean

$$\mathsf{ThetaAtlasRect}(K_N,c).$$

For $N<5$, define $\mathsf{ThetaAtlas}(N,c)$ to be false.

Proposition 8 

(Decidability of theta-atlas predicates). The predicates

$$\mathsf{ThetaAtlasRect}(K,c)$$

and

$$\mathsf{ThetaAtlas}(N,c)$$

are decidable.

Proof. 

The verifier performs only finitely many rational checks:

• parsing of the finite record;
• rational rectangle and disk containment;
• finite rectangle-cover inclusion, decidable by rational cell decomposition;
• the decidable majorant predicate $\mathsf{ThetaMajCert}$;
• the decidable point predicate $\mathsf{ThetaPointCert}$;
• the rational inequalities

  $${\epsilon }_{\nu }+M{r}_{\nu }<\left|A_{\nu }\right|.$$

Since $|A_{\nu }{|}^2\in \mathbb{Q}$, the last condition is checked as

$${({\epsilon }_{\nu }+M{r}_{\nu })}^2<{\left|A_{\nu }\right|}^2.$$

□

Theorem 15 

(Theta-atlas soundness). If

$$\mathsf{ThetaAtlasRect}(K,c),$$

then

$$\Xi \left(z\right)\ne 0(z\in K).$$

In particular,

$$\mathsf{ThetaAtlas}(N,c)⟹\Xi \left(z\right)\ne 0(z\in K_N).$$

Proof. 

Let $z\in K$. By the cover condition, choose $\nu$ such that

$$z\in Q_{\nu }.$$

Since $Q_{\nu }\subseteq D_{\nu }=\overline{B}(a_{\nu },r_{\nu })$,

$$|z-a_{\nu }|\le r_{\nu }.$$

The certificate proves

$$|a_{\nu }|+r_{\nu }\le R.$$

Therefore the line segment from $a_{\nu }$ to z lies in the disk

$$\left\{\right|w|\le R\}.$$

By $\mathsf{ThetaMajCert}(1,R,M,\mu )$ and 5,

$$|{\Xi }^{\prime }\left(w\right)|\le M(\left|w\right|\le R).$$

Thus

$$|\Xi \left(z\right)-\Xi \left(a_{\nu }\right)|\le M|z-a_{\nu }|\le M{r}_{\nu }.$$

By the point certificate,

$$|\Xi \left(a_{\nu }\right)-A_{\nu }|\le {\epsilon }_{\nu }.$$

Therefore

$$|\Xi \left(z\right)-A_{\nu }|\le {\epsilon }_{\nu }+M{r}_{\nu }.$$

The certificate checks

$${\epsilon }_{\nu }+M{r}_{\nu }<\left|A_{\nu }\right|.$$

Hence $\Xi \left(z\right)\ne 0$. Since $z\in K$ was arbitrary, K is zero-free. □

6.6. Quantitative Completeness of Theta Atlases

Theorem 16 

(Quantitative atlas completeness for rectangles). Let

$$K=[a,b]\times [c,d]\subset \mathbb{C}$$

be a rational closed rectangle. Suppose

$$\Xi \left(z\right)\ne 0(z\in K).$$

Put

$$W=b-a,H=d-c,$$

and

$${\delta }_K:=\underset{z\in K}{min}|\Xi \left(z\right)|>0.$$

Let $R>0$ be rational, and suppose $K\subset \left\{\right|z|<R-1\}$. Let $M>0$ be a rational number with a majorant certificate

$$\mathsf{ThetaMajCert}(1,R,M,\mu ).$$

Set

$$M^{+}:=max(1,M),{\Delta }_K:=min(1,{\delta }_K).$$

Then there exists a certificate c with

$$\mathsf{ThetaAtlasRect}(K,c)$$

using at most

$$\left(1+⌈{\displaystyle \frac{2\sqrt{2}{M}^{+}W}{{\Delta }_K}}⌉\right)\left(1+⌈{\displaystyle \frac{2\sqrt{2}{M}^{+}H}{{\Delta }_K}}⌉\right)$$

local rectangles. The point-enclosure radii may be taken to satisfy

$${\epsilon }_{\nu }\le {\Delta }_K/8.$$

In an actual rational certificate construction, $\sqrt{2}$ is replaced by any certified rational upper bound $q_2>0$ satisfying $q_2^2>2$. The theorem is an existential completeness statement in terms of the real number ${\delta }_K$; the later lower-bound-to-atlas theorem gives an effective construction from a supplied rational lower bound for ${\delta }_K$.

Proof. 

Let

$${\ell }_0:={\displaystyle \frac{{\Delta }_K}{2\sqrt{2}{M}^{+}}}.$$

Choose integers

$$m_x=1+⌈{\displaystyle \frac{W}{{\ell }_0}}⌉,m_y=1+⌈{\displaystyle \frac{H}{{\ell }_0}}⌉.$$

Subdivide K into $m_x{m}_y$ rational rectangles $Q_{\nu }$ whose side lengths are at most

$${\ell }_x={\displaystyle \frac{W}{m_x}}<{\ell }_0,{\ell }_y={\displaystyle \frac{H}{m_y}}<{\ell }_0.$$

The center $a_{\nu }$ of each $Q_{\nu }$ is rational. Let $h_{\nu }$ be the half-diagonal of $Q_{\nu }$. Since both side lengths are strictly less than ${\ell }_0$,

$$h_{\nu }<{\displaystyle \frac{\sqrt{2}}{2}}{\ell }_0={\displaystyle \frac{{\Delta }_K}{4M^{+}}}.$$

Choose a rational number $r_{\nu }$ satisfying

$$h_{\nu }\le r_{\nu }<{\displaystyle \frac{{\Delta }_K}{4M^{+}}}.$$

This is possible by density of $\mathbb{Q}$ in $\mathbb{R}$. Then

$$D_{\nu }=\overline{B}(a_{\nu },r_{\nu })$$

is a rational disk containing $Q_{\nu }$. Moreover,

$$M{r}_{\nu }\le M^{+}{r}_{\nu }<{\displaystyle \frac{{\Delta }_K}{4}}.$$

Also,

$$r_{\nu }<{\displaystyle \frac{{\Delta }_K}{4M^{+}}}\le {\displaystyle \frac{1}{4}}.$$

Since $a_{\nu }\in K$ and $K\subset \left\{\right|z|<R-1\}$, we have

$$|a_{\nu }|<R-1.$$

Consequently

$$|a_{\nu }|+r_{\nu }<(R-1)+{\displaystyle \frac{1}{4}}<R.$$

Thus the verifier’s containment condition

$$|a_{\nu }|+r_{\nu }\le R$$

holds for every cell.

By 7, choose rational point enclosures

$$|\Xi \left(a_{\nu }\right)-A_{\nu }|\le {\epsilon }_{\nu }$$

with

$${\epsilon }_{\nu }\le {\Delta }_K/8.$$

Then

$${\epsilon }_{\nu }+M{r}_{\nu }<{\displaystyle \frac{{\Delta }_K}{8}}+{\displaystyle \frac{{\Delta }_K}{4}}={\displaystyle \frac{3{\Delta }_K}{8}}.$$

On the other hand,

$$|A_{\nu }|\ge |\Xi \left(a_{\nu }\right)|-{\epsilon }_{\nu }\ge {\delta }_K-{\displaystyle \frac{{\Delta }_K}{8}}.$$

If ${\Delta }_K={\delta }_K\le 1$, then

$$|A_{\nu }|\ge {\displaystyle \frac{7{\delta }_K}{8}}={\displaystyle \frac{7{\Delta }_K}{8}}.$$

If ${\Delta }_K=1<{\delta }_K$, then

$$|A_{\nu }|>{\delta }_K-{\displaystyle \frac{1}{8}}>{\displaystyle \frac{7}{8}}={\displaystyle \frac{7{\Delta }_K}{8}}.$$

In either case,

$$|A_{\nu }|\ge {\displaystyle \frac{7{\Delta }_K}{8}}.$$

Therefore

$${\epsilon }_{\nu }+M{r}_{\nu }<{\displaystyle \frac{3{\Delta }_K}{8}}<{\displaystyle \frac{7{\Delta }_K}{8}}\le \left|A_{\nu }\right|.$$

Thus every local inequality required by the atlas verifier holds. The resulting finite record is accepted and uses $m_x{m}_y$ local rectangles, which gives the claimed bound.

For literal rational certificate records, choose any rational $q_2>0$ with $q_2^2>2$ and replace $\sqrt{2}$ by $q_2$ in the mesh choice. This only refines the mesh and preserves all inequalities. □

Corollary 2 

(Quantitative atlas completeness for $K_N$). Let $N\ge 5$, and suppose

$$\Xi \left(z\right)\ne 0(z\in K_N).$$

Put

$${\delta }_N:=\underset{z\in K_N}{min}|\Xi \left(z\right)|>0,{\Delta }_N:=min(1,{\delta }_N).$$

Let $M_N>0$ be rational and suppose there is a certificate

$$\mathsf{ThetaMajCert}(1,N+2,M_N,{\mu }_N).$$

Put

$$M_N^{+}=max(1,M_N).$$

Then there exists c with

$$\mathsf{ThetaAtlas}(N,c),$$

using at most

$$\left(1+⌈{\displaystyle \frac{2\sqrt{2}{M}_N^{+}}{{\Delta }_N}}\left(N-{\displaystyle \frac{1}{N}}\right)⌉\right)\left(1+⌈{\displaystyle \frac{2\sqrt{2}{M}_N^{+}}{{\Delta }_N}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{2}{N}}\right)⌉\right)$$

local rectangles. The point-enclosure radii may be taken to satisfy

$${\epsilon }_{\nu }\le {\Delta }_N/8.$$

In finite rational certificate records, $\sqrt{2}$ is replaced by any certified rational upper bound $q_2>0$ with $q_2^2>2$. This is an existential bound in terms of the actual real lower bound ${\delta }_N$; effective construction from supplied rational lower bounds is isolated below.

Proof. 

For $K_N$,

$$W_N=N-{\displaystyle \frac{1}{N}},H_N=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{N}}\right)-{\displaystyle \frac{1}{N}}={\displaystyle \frac{1}{2}}-{\displaystyle \frac{2}{N}}.$$

Also, for $z\in K_N$,

$$\left|z\right|\le \sqrt{N^2+{(1/2)}^2}<N+1.$$

The local disk radii constructed in 16 are strictly less than $1/4$, hence all local disks lie in

$$\left\{\right|z|\le N+2\}.$$

Equivalently,

$$K_N\subset \left\{\right|z|<N+1\}=\left\{\right|z|<(N+2)-1\}.$$

Apply 16 with $R=N+2$. The rationalization of $\sqrt{2}$ is exactly as in that theorem. □

Remark 17 (Rationalizing 2 in finite records) 

The displayed cell-count bounds use $\sqrt{2}$ as a mathematical shorthand for the diagonal of a square. The finite certificate verifier never needs an irrational constant. In an actual rational certificate record one supplies any rational number

$$q_2\in \mathbb{Q},q_2>0,q_2^2>2,$$

and replaces $\sqrt{2}$ by $q_2$ in all mesh inequalities. This only enlarges the finite mesh and leaves the soundness and completeness arguments unchanged.

Theorem 18 

(Explicit quantitative theta-atlas criterion for RH).

$$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

Moreover, if RH holds, then for each $N\ge 5$, after choosing any rational $M_N>{\mathfrak{M}}_1(N+2)$ and a corresponding majorant certificate

$$\mathsf{ThetaMajCert}(1,N+2,M_N,{\mu }_N),$$

an accepted certificate exists with the quantitative cell-count bound of 2. This is an existential completeness statement in terms of the actual positive minimum

$${\delta }_N=\underset{K_N}{min}|\Xi |.$$

An effective construction of the certificate requires supplied rational lower-bound data as in 19.

Proof. 

Assume RH. By 3,

$$\Xi \left(z\right)\ne 0(z\in \mathcal{U}).$$

Since

$$K_N\subset \mathcal{U},$$

each $K_N$ is zero-free. By 5, there are rational majorant certificates for any rational $M_N>{\mathfrak{M}}_1(N+2)$. Applying 2 gives a finite accepted atlas certificate for $K_N$. Thus

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

Conversely, suppose

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

By 15, each $K_N$ is zero-free. By 14,

$$\mathcal{U}=\bigcup _{N\ge 5}{K}_N^{\circ }.$$

Therefore $\Xi$ has no zeros in $\mathcal{U}$. By 3, RH follows.

The quantitative assertion is exactly 2, with any certified rational $M_N>{\mathfrak{M}}_1(N+2)$. □

6.7. Effective Lower Bounds Imply Explicit Certificates

The preceding theorem is qualitative in the sense that, under RH, it uses the positive real number

$${\delta }_N=\underset{K_N}{min}|\Xi |$$

whose value is not supplied by RH itself. The next theorem isolates the exact sign-fibre occupation data needed to construct a stage certificate.

Definition 41 

(Certified stage lower bound). For $N\ge 5$, a certified stage lower bound is a rational number

$${\lambda }_N>0$$

together with a finite verification record proving

$${\lambda }_N\le |\Xi \left(z\right)|(z\in K_N).$$

In practice such a verification record may itself be a theta-atlas certificate, a collection of smaller local enclosures, a winding/argument-principle exclusion, or any other finite rational certificate whose soundness implies the displayed inequality.

Theorem 19 

(Lower-bound-to-atlas conversion). Fix $N\ge 5$. Suppose the following data are supplied:

• a rational lower bound

  $$0<{\lambda }_N\le \underset{z\in K_N}{min}|\Xi \left(z\right)|;$$
• a rational majorant certificate

  $$\mathsf{ThetaMajCert}(1,N+2,M_N,{\mu }_N).$$

Put

$${\Delta }_N^{\lambda }:=min(1,{\lambda }_N),M_N^{+}:=max(1,M_N).$$

Then there is an effectively constructible certificate c with

$$\mathsf{ThetaAtlas}(N,c),$$

using at most

$$\left(1+⌈{\displaystyle \frac{2\sqrt{2}{M}_N^{+}}{{\Delta }_N^{\lambda }}}\left(N-{\displaystyle \frac{1}{N}}\right)⌉\right)\left(1+⌈{\displaystyle \frac{2\sqrt{2}{M}_N^{+}}{{\Delta }_N^{\lambda }}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{2}{N}}\right)⌉\right)$$

local rectangles and point precision

$${\epsilon }_{\nu }\le {\Delta }_N^{\lambda }/8.$$

In finite rational records, $\sqrt{2}$ is replaced by a certified rational upper bound $q_2>0$ satisfying $q_2^2>2$.

Proof. 

This is the construction in 2, with the unknown ${\delta }_N$ replaced by the supplied lower bound ${\lambda }_N$. Since

$${\lambda }_N\le {\delta }_N,$$

all inequalities used in the proof remain valid with

$${\Delta }_N^{\lambda }=min(1,{\lambda }_N)$$

in place of ${\Delta }_N=min(1,{\delta }_N)$. The subdivision, point precision, and finite certificate construction are effective because all parameters are rational and the point-certificate predicate is complete by 7. Operationally, one constructs the prescribed mesh and then computes point certificates to the prescribed precision; the final atlas certificate is accepted by the independent verifier. As before, an actual rational mesh uses a rational upper bound for $\sqrt{2}$. □

Corollary 3 

(Computable lower-bound stream gives a total atlas selector). Suppose there is a total computable procedure which, on input $N\ge 5$, outputs:

• a rational ${\lambda }_N>0$ with

  $${\lambda }_N\le \underset{K_N}{min}|\Xi |;$$
• a majorant certificate

  $$\mathsf{ThetaMajCert}(1,N+2,M_N,{\mu }_N).$$

Then there is a total computable function $a_{\Theta }\left(N\right)$ such that

$$\mathsf{ThetaAtlas}(N,a_{\Theta }\left(N\right))(N\ge 5).$$

Consequently RH holds.

Proof. 

On input N, use the supplied data and the effective construction of 19 to build an accepted atlas certificate $a_{\Theta }\left(N\right)$. Since every $K_N$ has an accepted atlas certificate, 18 gives RH. □

Remark 20 (Lower bounds and sign-fibre occupation) 

A stream of positive lower bounds

$${\lambda }_N\le \underset{K_N}{min}|\Xi |$$

for all N supplies enough information to occupy the theta-atlas positive fibre. This is not an escape from selected-logical nullity. It is occupation of the all-stage positive sign fibre

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c),$$

which is exactly RH by 18.

Remark 21 (Lower bounds versus finite prefixes) 

A certified lower bound for a fixed window $K_N$ is information about $\Xi$ on that window. It is not the same as a finite prefix of the all-stage assertion. A finite prefix proves zero-freeness only on the windows it covers, whereas a computable stream of positive lower bounds for all N yields a total atlas selector and hence RH by 3.

7. Criterion Absorption and Selected-Logical Nullity

This section clarifies how the number-theoretic criteria used later interact with selected-logical nullity. The framework is a theory of RH-specific analytic and number-theoretic reasoning with an effect discipline. The correct relation between RH-specific criteria and nullity is absorption, not escape. A theta-atlas certificate, a winding certificate, a Li-negative certificate, or a Lagarias-negative certificate is not an external route around the nullity theorem. Once made exact, it is a coordinate-bearing channel of the selected RH package.

The theta-atlas theorem gives a positive all-stage fibre:

$$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

The winding theorem proved later gives a finite negative fibre:

$$\neg \mathsf{RH}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right).$$

The later Li and Lagarias comparisons give replay-equivalent finite negative fibres. The universal nullity theorem applies to the unoccupied enriched criterion architecture for pure selected-parametric resolvers, and conditionally to arbitrary resolvers satisfying full replacement invariance. When proof-status labels are included, the selected-resolution nullity theorem is stated for every consistent $T\supseteq \mathsf{Q}$. This is an unconditional metatheorem about the class of consistent theories; the consistency antecedent is essential because inconsistent theories make proof and refutation labels common under the label semantics used here. A non-null sign outcome occurs when one of the sign fibres is occupied, and such occupation is itself equivalent to $\mathsf{RH}$ or $\neg \mathsf{RH}$.

Definition 42 

(Selected-logical invariant inference). An inference about exact selected packages is selected-logical invariant if its validity uses only the selected bridge architecture

$$P\iff \forall w\neg B_P\left(w\right),P\iff \forall n\exists c{C}_P(n,c),B_P\left(w\right)\iff \exists u{D}_P(w,u),P\iff \mathbb{N}\vDash \widehat{P},$$

and remains valid under arbitrary replacement of one exact selected package by another. Such an inference does not distinguish the RH package from an arbitrary exact selected package.

Definition 43 

(Pure selected-logical inference). A selected-logical invariant inference is pure if it is formulated in the opaque selected-parametric language: it may use only bridge forms and rules valid uniformly for all exact selected packages, and it may not inspect concrete fibre facts, package codes, truth values, proof statuses, or coordinate-bearing instance data. Pure selected-logical inferences are replacement invariant by selected parametricity.

Proposition 9 

(Theta-atlas criterion as positive fibre). The predicate

$$C_{\Theta }(n,c):=\mathsf{ThetaAtlas}(n+5,c)$$

is an exact all-stage positive criterion channel for RH:

$$\forall n\exists c{C}_{\Theta }(n,c)\iff \mathsf{RH}.$$

Thus the theta-atlas theorem supplies a positive sign fibre of the selected RH package, not an external route around selected-logical nullity.

Proof. 

This is exactly 18, with the shift $N=n+5$. □

Principle 22 

(Criterion absorption principle). Any exact number-theoretic RH criterion, once formulated as a decidable finite negative channel or decidable all-stage positive channel, is absorbed into the selected RH package as a sign fibre. Its non-null force occurs only by occupation of that fibre:

$$\exists wB\left(w\right)\iff \neg \mathsf{RH}$$

for a negative channel, or

$$\forall n\exists cC(n,c)\iff \mathsf{RH}$$

for a positive channel. Thus exact criteria do not bypass selected-logical nullity; they become the fibres to which it applies.

Theorem 23 

(Theta-atlas occupation is sign-fibre occupation). A verification of

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c)$$

is not a selected-logical escape from universal nullity. It is occupation of the theta-atlas positive fibre and hence is exactly a proof of $\mathsf{RH}$. Before this all-stage occupation is supplied, the availability of the theta-atlas criterion as an exact channel does not yield any pure selected-parametric invariant truth or proof-status label.

Proof. 

The equivalence

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c)\iff \mathsf{RH}$$

is 18. Therefore a complete verification of the left-hand side is precisely occupation of an exact positive sign fibre. The universal selected-resolution nullity theorem concerns invariant label output from the package architecture itself, for consistent T when proof-status labels are included. It is not bypassed by fibre occupation; rather, fibre occupation is the exact sign condition. In the pure selected-parametric regime, invariant label output is null if sound. A complete theta-atlas stream is coordinate-bearing RH-specific all-stage sign data. It is not pure package-form output; it is positive sign-fibre occupation inside the enriched RH-specific calculus. □

Remark 24 (How this addresses conditional RH information) 

The quantitative atlas theorem gives conditional information of the form: if one supplies positive lower bounds

$$0<{\lambda }_N\le \underset{K_N}{min}|\Xi |,$$

then explicit certificates exist with controlled cell count and precision. This does not stand outside selected-logical nullity. It is a route to occupying the theta-atlas positive fibre. If all stages are occupied, the result is exactly RH; before all-stage occupation is supplied, the criterion remains part of the unoccupied enriched criterion architecture.

Remark 25 (Coordinate-bearing data and pure invariance) 

The theta kernel, theta-majorants, point enclosures, and atlas certificates are coordinate-bearing data of the actual $\Xi$-function. A resolver that uses them is not pure selected-parametric and need not be invariant under arbitrary replacement of exact selected packages. This is not a defect. It is the effect discipline of the framework: coordinate-bearing information may occupy a sign fibre, whereas pure selected-logical form alone has null label output.

8. Validated $\Xi$-Certificate Backend

The theta-atlas theorem in Section 6 gives a positive zero-free atlas channel using the theta kernel. The negative channel and the exact zero-count calculus use winding-number certificates. These winding certificates can be verified using either:

• an analytic-continuation/Euler–Maclaurin backend for $\Xi$;
• the entire theta-kernel backend;
• or any other backend satisfying the same finite rational enclosure interface.

This section states the finite interface needed by the winding and local zero-exclusion certificate calculus. The theta-atlas section already supplied a concrete theta-kernel construction for point enclosures and derivative majorants. Here we also record the general shrinking-domain enclosure theorem, because it is the backend principle behind both winding certificates and local image-disk certificates.

The key point is the shrinking-domain form. For a fixed positive-radius disk D, increasing arithmetic precision alone cannot make an enclosure for $\Xi \left(D\right)$ shrink to radius 0; the image itself usually has positive diameter. The correct certificate completeness mechanism refines the domain mesh and increases precision simultaneously.

All certificate predicates in this section use the finite rational certificate grammar fixed in 3. Thus an accepted backend record is a finite rational object whose enclosure claims are checked from displayed truncation, quadrature, elementary-function, containment, and tail data; no high-level numerical oracle is part of the verifier.

Definition 44 

(Rational compact guards). A rational compact guard $K⋐\mathcal{U}$ is a finite union of rational closed rectangles contained in $\mathcal{U}$, together with explicit rational inequalities certifying positive distance from $\partial \mathcal{U}$. In certificate records, each domain disk is supplied with finite rational containment data proving that it lies inside K.

Definition 45 

(Backend parameter data). A backend parameter datum, sometimes called a precision package, is a finite code specifying all numerical choices made by a validated enclosure call. Depending on the backend, it may include:

• the rational compact guard K;
• the rational point or domain disk D;
• the requested precision p;
• Euler–Maclaurin truncation parameters;
• gamma-product or polygamma truncation parameters;
• theta-kernel truncation parameters;
• quadrature partitions;
• rational interval enclosures for elementary constants;
• rational output disks;
• all rational inequalities used to verify the claimed containment.

A backend parameter datum is not an oracle. It is a finite rational object checked by the certificate verifier using the terminating backend algorithms fixed in the paper.

Theorem 26 

(Computable analytic $\Xi$-jet backend). For $j=0,1,2,3$, the functions ${\Xi }^{\left(j\right)}$ are computable holomorphic functions on $\mathcal{U}$, uniformly on rational compact guards. More explicitly, for every rational compact guard $K⋐\mathcal{U}$ and every $j=0,1,2,3$, there is an algorithm which computes a rational number

$$M_{j,K}>0$$

with

$$\underset{z\in K}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le M_{j,K}.$$

There is also an algorithm which, given a rational point $z_0\in K$, $j\le 3$, and precision p, outputs a rational disk of radius at most $2^{-p}$ containing ${\Xi }^{\left(j\right)}\left(z_0\right)$.

Proof. 

There are two constructive routes.

The first is the analytic-continuation route. On rational compact guards in $\mathcal{U}$, the corresponding s-region

$$s=1/2+\mathrm{i}z$$

is compactly separated from $s=1$, and $s/2$ is compactly contained in the right half-plane. Euler–Maclaurin formulas give uniformly computable enclosures for

$${\zeta }^{\left(q\right)}\left(s\right),0\le q\le 3,$$

on such compact sets, with explicit rational tail bounds. Weierstrass-product, finite-subdivision inversion, logarithmic gamma, or polygamma series give enclosures for

$$\Gamma (s/2),\psi (s/2),{\psi }^{\prime }(s/2),{\psi }^{\prime \prime }(s/2).$$

The detailed appendix uses a corrected compact-subdivision argument whenever an inverse disk enclosure is needed on a positive-radius domain; it does not assume that a nonvanishing compact image can always be enclosed by one disk avoiding 0. Combining these with the identities

$$\xi \left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right),\Xi \left(z\right)=\xi (1/2+\mathrm{i}z),$$

and differentiating through order 3, gives point enclosures and compact sup bounds.

The second route is the theta-kernel route from Section 6. The identity

$$\Xi \left(z\right)=4{\int }_0^{\infty }{\Phi }_{+}\left(u\right)cos\left(zu\right)\mathrm{d}u$$

and its differentiated forms give point enclosures by truncating the n-sum, truncating the u-integral, and using validated quadrature on a finite rational partition. The explicit majorants of 10 give compact sup bounds on Euclidean disks. A rational compact guard is covered by finitely many Euclidean disks, so finite maxima give compact guard bounds.

Both routes use only terminating rational interval computations with outward rounding. The detailed constructive algorithms are included in the appendices. □

Theorem 27 

(Shrinking-domain enclosure theorem). Let $K⋐\mathcal{U}$ be a rational compact guard. For $j=0,1,2$, there is a terminating rational-disk enclosure routine which, given a rational disk

$$D=\overline{B}(z_0,r)\subseteq K$$

and a precision p, outputs a rational disk $E_{j,p}\left(D\right)$ satisfying

$${\Xi }^{\left(j\right)}\left(D\right)\subseteq E_{j,p}\left(D\right)$$

and

$$rad\left(E_{j,p}\left(D\right)\right)\le 2^{-p}+M_{j+1,K}r,$$

where $M_{j+1,K}$ is a validated rational upper bound for

$$\underset{z\in K}{sup}\left|{\Xi }^{(j+1)}\left(z\right)\right|.$$

In particular, uniformly for $D\subseteq K$,

$$r\to 0,p\to \infty ⟹rad\left(E_{j,p}\left(D\right)\right)\to 0.$$

Proof. 

Compute a point enclosure

$${\Xi }^{\left(j\right)}\left(z_0\right)\in \overline{B}(a,2^{-p})$$

using 26. For $z\in D$, the segment from $z_0$ to z lies in $D\subseteq K$. The fundamental theorem of calculus gives

$$|{\Xi }^{\left(j\right)}\left(z\right)-{\Xi }^{\left(j\right)}\left(z_0\right)|\le \underset{\zeta \in K}{sup}|{\Xi }^{(j+1)}\left(\zeta \right)\left|\right|z-z_0|\le M_{j+1,K}r.$$

Thus

$$E_{j,p}\left(D\right):=\overline{B}(a,2^{-p}+M_{j+1,K}r)$$

is sound. The convergence assertion follows immediately. □

Definition 46 

(Finite backend image-enclosure certificate). Let $K⋐\mathcal{U}$ be a rational compact guard, let

$$D=\overline{B}(z_0,r)\subseteq K$$

be a rational disk, and let E be a rational disk. Write

$$\mathsf{XiImageCert}(K,D,E,\beta )$$

if β is a finite rational backend record, in the grammar fixed in 3, whose displayed subrecords prove

$$\Xi \left(D\right)\subseteq E.$$

The record specifies the chosen backend, AC or theta, the point-enclosure data, the derivative-majorant data, the shrinking-domain radius computation, and every rational containment check. The predicate

$$\mathsf{XiImageCert}(K,D,E,\beta )$$

is decidable. If it holds, then

$$\Xi \left(D\right)\subseteq E.$$

Conversely, whenever the shrinking-domain theorem supplies such an enclosure, the corresponding finite backend data can be written as a certificate β.

Remark 28. 

The theta-atlas certificates of Section 6 use the special case $j=0$, together with an explicit global majorant for $j=1$. The winding certificates below use the same shrinking-domain principle to enclose $\Xi$ on boundary subarc disks.

Remark 29 (Two roles of boundary refinement) 

In quantitative winding certificates, domain refinement has two independent roles: it controls the variation of $\Xi$ through a derivative bound, and it keeps each boundary-subarc disk inside the compact guard on which that derivative bound has been certified. This is why the quantitative winding theorem includes both a value-separation term and a certified tube-radius term.

9. Exact-Count, Zero-Free, and Winding Certificates

This section develops exact zero-count certificates by the argument principle. The theta-atlas predicate gives a direct zero-free positive stage certificate. The winding certificates give exact local zero counts and therefore finite negative witnesses when an off-critical zero exists.

9.1. Admissible Rectangles and Stage Windows

Definition 47 

(Admissible rectangle). An admissible rectangle is a nondegenerate rational closed rectangle

$$Q=[a,b]\times [c,d]\subset \mathcal{U},a<b,c<d.$$

Here Q is identified with the set of points $x+\mathrm{i}y$ with $a\le x\le b$ and $c\le y\le d$.

Definition 48 

(Enumeration of admissible rectangles). Fix a primitive recursive enumeration

$${\left(Q_k\right)}_{k\in \mathbb{N}}$$

of all admissible rational rectangles in $\mathcal{U}$, with repetitions allowed.

Definition 49 

(Stage windows for zero-count certificates). Fix a primitive recursive enumeration

$${\left(R_n\right)}_{n\in \mathbb{N}}$$

of all admissible rational rectangles in $\mathcal{U}$, with repetitions allowed, such that the interiors of the $R_n$’s cover $\mathcal{U}$.

9.2. Parameterized Tube-Homotopy Winding Certificates

The winding certificate is a finite rational version of the argument principle. It does not require the verifier to know the exact curve $\Xi (\partial Q)$. Instead, the certificate supplies rational image disks avoiding 0, each containing both the true image of a boundary subarc and a rational polygonal segment. Convexity gives a homotopy in ${\mathbb{C}}^{\times }$, so the rational polygon’s winding number is the true winding number.

Definition 50 

(Boundary parametrization). For an admissible rectangle Q, let

$${\gamma }_Q:[0,1]\to \partial Q$$

be the standard positively oriented piecewise-linear rational parametrization, starting and ending at the lower-left corner and moving counterclockwise with constant speed on each side.

Definition 51 

(Tube winding certificate record). A tube winding certificate record for an admissible rectangle Q consists of:

• a rational compact guard $K⋐\mathcal{U}$;
• rational containment data proving $Q\subset K$;
• a rational partition

  $$0=t_0<t_1<\dots <t_m=1$$
  which contains the four side-break parameters of ${\gamma }_Q$. Thus each interval $[t_i,t_{i+1}]$ lies on a single side of Q;
• rational disks $D_i\subset K$ such that

  $${\gamma }_Q\left([t_i,t_{i+1}]\right)\subset D_i(0\le i<m);$$
• rational image disks $E_i=\overline{B}(a_i,{\rho }_i)\subset \mathbb{C}$;
• finite backend image-enclosure records ${\beta }_i$ and rational disks $E_i^{\prime }$ such that

  $$\mathsf{XiImageCert}(K,D_i,E_i^{\prime },{\beta }_i)$$
  holds, together with rational containment checks

  $$E_i^{\prime }\subseteq E_i;$$
• a rational polygonal loop

  $$P:[0,1]\to \mathbb{C},P\left(0\right)=P\left(1\right),$$
  linear on each $[t_i,t_{i+1}]$, with rational vertices $p_i=P\left(t_i\right)$;
• rational containment checks proving

  $$P\left([t_i,t_{i+1}]\right)\subseteq E_i(0\le i<m);$$
• zero-avoidance checks

  $$0\notin E_i(0\le i<m);$$
  in rational form this means

  $$|a_i|>{\rho }_i,$$
  checked as

  $$|a_i{|}^2>{\rho }_i^2;$$
• a rational ray direction $v\in {\mathbb{Q}}^2\setminus \left\{0\right\}$, representing the ray

  $${\ell }_v=\{\lambda v:\lambda \ge 0\},$$
  together with finite rational general-position checks proving:

  (a)

  no polygon vertex $p_i$ lies on ${\ell }_v$;

  (b)

  no polygon segment $[p_i,p_{i+1}]$ is collinear with ${\ell }_v$;

  (c)

  no polygon segment contains 0;

  (d)

  every intersection of a polygon segment with ${\ell }_v$, if it occurs, is transverse and occurs in the relative interior of the segment and at positive ray parameter.

  The winding number is computed by the fixed signed ray-crossing convention: for each segment, solve the two-by-two rational linear system

  $$p_i+\lambda (p_{i+1}-p_i)=\mu v,$$
  check whether $0<\lambda <1$ and $\mu >0$, and if so add the sign determined by the fixed orientation convention for the crossing. The sum is the declared integer winding number.

All data are finite and rational.

Definition 52 

(Winding certificate predicate). ${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q,r,c)$ means that c encodes a finite tube winding certificate record for Q, all finite rational checks and backend enclosure checks accept, $r\in \mathbb{Z}$, and the rational ray-crossing winding calculation for P has value r.

Proposition 10 

(Decidability of winding certificates). ${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q,r,c)$ is decidable.

Proof. 

The verifier parses finite rational data, checks rational containment conditions, checks finitely many backend image-enclosure records

$$\mathsf{XiImageCert},$$

checks finitely many disk-containment and zero-avoidance inequalities, checks containment of rational segments in rational disks, checks the finite ray general-position conditions, and computes a rational polygon winding number by the fixed signed ray-crossing rule. □

Lemma 15 

(Parameterized tube homotopy). If the winding verifier accepts all geometric and enclosure data before the final winding comparison, then the rational polygonal loop P is homotopic to $\Xi \circ {\gamma }_Q$ in ${\mathbb{C}}^{\times }$.

Proof. 

For $t\in [t_i,t_{i+1}]$, the certificate proves

$$\Xi \left({\gamma }_Q\left(t\right)\right)\in E_i,P\left(t\right)\in E_i,E_i\subset {\mathbb{C}}^{\times }.$$

Since each $E_i$ is convex, the straight-line homotopy

$$H(\lambda ,t)=(1-\lambda )\Xi \left({\gamma }_Q\left(t\right)\right)+\lambda P\left(t\right)$$

lies in $E_i\subset {\mathbb{C}}^{\times }$ for $t\in [t_i,t_{i+1}]$. At partition endpoints the adjacent interval verifications agree because both $\Xi \circ {\gamma }_Q$ and P are continuous closed loops. Thus H is a homotopy in ${\mathbb{C}}^{\times }$. □

Theorem 30 

(Exact-count certificate theorem). The predicate ${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q,r,c)$ is decidable and satisfies:

• Soundness.If ${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q,r,c)$, then Ξ has no zero on $\partial Q$ and

  $$N(\Xi ;Q^{\circ })=r.$$
• Completeness.If $Q\subset \mathcal{U}$ is admissible and Ξ has no zero on $\partial Q$, then there exist $r,c$ with

  $${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q,r,c),$$
  and necessarily

  $$r=N(\Xi ;Q^{\circ }).$$

Proof. 

Soundness follows from 15. The accepted enclosure data prove that

$$\Xi \left({\gamma }_Q\left(t\right)\right)\in E_i,P\left(t\right)\in E_i,0\notin E_i$$

for $t\in [t_i,t_{i+1}]$. Since the disks $E_i$ are convex, the straight-line homotopy from $\Xi \circ {\gamma }_Q$ to P stays in ${\mathbb{C}}^{\times }$. Therefore the two loops have the same winding number around 0. The rational ray-crossing calculation gives

$$wind(P,0)=r.$$

Hence

$$wind(\Xi \circ {\gamma }_Q,0)=r.$$

The accepted zero-avoidance disks also exclude boundary zeros. The argument principle then gives

$$N(\Xi ;Q^{\circ })=r.$$

For completeness, assume $\Xi \ne 0$ on $\partial Q$. Then

$$\delta :=\underset{z\in \partial Q}{min}|\Xi \left(z\right)|>0.$$

Choose a compact tubular neighborhood $N_Q⋐\mathcal{U}$ of $\partial Q$ such that

$$|\Xi \left(z\right)|>{\displaystyle \frac{3\delta }{4}}(z\in N_Q).$$

Since $Q⋐\mathcal{U}$, choose a rational compact guard

$$K⋐\mathcal{U}$$

containing

$$Q\cup N_Q.$$

Choose

$$0<\epsilon <{\displaystyle \frac{\delta }{100}}.$$

By uniform continuity of $\Xi$ on $N_Q$, choose the rational boundary partition fine enough that each subarc ${\gamma }_Q\left([t_i,t_{i+1}]\right)$ is contained in a rational disk

$$D_i⋐N_Q$$

and, writing

$$z_i:={\gamma }_Q\left(t_i\right),$$

one has

$$\Xi \left(D_i\right)\subset B(\Xi \left(z_i\right),\epsilon ).$$

Refine the partition so that all four side-break parameters of ${\gamma }_Q$ occur among the $t_i$’s. Since $z_{i+1}\in D_i$, this also gives

$$|\Xi \left(z_{i+1}\right)-\Xi \left(z_i\right)|<\epsilon .$$

Choose rational vertices $p_i$ satisfying

$$|p_i-\Xi \left(z_i\right)|<\epsilon (0\le i<m),$$

and put $p_m=p_0$. Then

$$|p_{i+1}-p_i|\le |p_{i+1}-\Xi \left(z_{i+1}\right)|+|\Xi \left(z_{i+1}\right)-\Xi \left(z_i\right)|+|\Xi \left(z_i\right)-p_i|<3\epsilon .$$

Therefore every point of the polygon segment $[p_i,p_{i+1}]$ lies within $3\epsilon$ of $p_i$, and hence within $4\epsilon$ of $\Xi \left(z_i\right)$. At the same time,

$$\Xi \left(D_i\right)\subset B(\Xi \left(z_i\right),\epsilon )\subset B(\Xi \left(z_i\right),4\epsilon ).$$

Since

$$|\Xi (z_i\left)\right|\ge \delta$$

and $4\epsilon <\delta /2$, the disk

$$B(\Xi \left(z_i\right),4\epsilon )$$

is disjoint from 0. Choose a rational image disk $E_i$ such that

$$\Xi \left(D_i\right)\cup [p_i,p_{i+1}]\subseteq E_i\subset {\mathbb{C}}^{\times }.$$

This is possible by density of rational centers and radii and the strict positive distance from 0.

By the shrinking-domain backend, having made the domain disks $D_i$ sufficiently small, we may increase precision until there are rational image disks $E_i^{\prime }$ and finite backend records ${\beta }_i$ with

$$\mathsf{XiImageCert}(K,D_i,E_i^{\prime },{\beta }_i)$$

and

$$E_i^{\prime }\subseteq E_i.$$

Thus the certificate contains explicit finite data proving

$$\Xi \left(D_i\right)\subseteq E_i^{\prime }\subseteq E_i.$$

Finally choose a rational ray direction satisfying the finite general-position conditions. Such a direction exists because the polygon has only finitely many vertices and finitely many segments. The forbidden directions form a finite subset of the projective line of directions: directions through polygon vertices, directions parallel to polygon segments, and directions that would make a ray intersection non-transverse or occur at an endpoint. Nonzero rational vectors in ${\mathbb{Q}}^2$ determine a dense set of directions, so one can choose

$$v\in {\mathbb{Q}}^2\setminus \left\{0\right\}$$

outside this finite exceptional set. The rational ray-crossing winding computation then returns the true winding number. The resulting finite rational data form an accepted tube winding certificate. □

Definition 53 

(Certified tube radius). Let $Q⋐\mathcal{U}$ be an admissible rational rectangle and let $N_Q⋐\mathcal{U}$ be a rational compact tubular neighborhood of $\partial Q$. A certified tube radius for $N_Q$ around $\partial Q$ is a rational number

$${\tau }_Q>0$$

together with finite rational containment data proving

$$\{z\in \mathbb{C}:dist(z,\partial Q)\le {\tau }_Q\}\subseteq N_Q.$$

For rational rectangles and rational compact guards this is a decidable rational containment condition, using rational cell decomposition and distance estimates to finitely many line segments.

Theorem 31 

(Quantitative winding certificates). Let $Q⋐\mathcal{U}$ be an admissible rational rectangle and suppose

$$\Xi \left(z\right)\ne 0(z\in \partial Q).$$

Put

$$d_Q:=\underset{z\in \partial Q}{min}|\Xi \left(z\right)|>0.$$

Let $N_Q⋐\mathcal{U}$ be a rational compact tubular neighborhood of $\partial Q$, let

$$M_Q\ge \underset{z\in N_Q}{sup}\left|{\Xi }^{\prime }\left(z\right)\right|$$

be a certified rational upper bound, and let ${\tau }_Q>0$ be a certified tube radius for $N_Q$ around $\partial Q$. Put

$$M_Q^{+}:=max(1,M_Q),{\Delta }_Q:=min(1,d_Q),$$

and let $L_Q$ be the perimeter of Q. Define

$$h_Q:=min\left({\displaystyle \frac{{\Delta }_Q}{64M_Q^{+}}},{\displaystyle \frac{{\tau }_Q}{2}}\right).$$

Then Q has an exact winding-number zero-count certificate using at most

$$4+⌈{\displaystyle \frac{L_Q}{h_Q}}⌉$$

boundary arcs and point precision

$$2^{-p}\le {\Delta }_Q/256.$$

Proof. 

Parametrize each of the four sides of $\partial Q$ by arclength. On each side choose an integer subdivision fine enough that every resulting subarc has rational length ${\ell }_i\le h_Q$. If the four side lengths are $L_1,L_2,L_3,L_4$, this produces at most

$${\displaystyle \sum _{j=1}^4}⌈{\displaystyle \frac{L_j}{h_Q}}⌉$$

subarcs. Since for nonnegative real numbers $x_1,\dots ,x_4$,

$${\displaystyle \sum _{j=1}^4}\lceil x_j\rceil \le ⌈{\displaystyle \sum _{j=1}^4}{x}_j⌉+3\le 4+⌈{\displaystyle \sum _{j=1}^4}{x}_j⌉,$$

the number of subarcs is at most

$$4+⌈{\displaystyle \frac{L_Q}{h_Q}}⌉.$$

The four side-break parameters are included as partition points.

Let $z_i$ be the initial point of the i-th subarc, and let

$$D_i=\overline{B}(z_i,{\ell }_i).$$

Then $D_i$ is a rational disk and the corresponding subarc is contained in $D_i$. Since ${\ell }_i\le h_Q\le {\tau }_Q/2<{\tau }_Q$, every point of $D_i$ has distance at most ${\ell }_i\le {\tau }_Q$ from $\partial Q$, and therefore

$$D_i\subseteq N_Q.$$

For every $z\in D_i$, the derivative bound gives

$$|\Xi \left(z\right)-\Xi \left(z_i\right)|\le M_Q{\ell }_i\le M_Q{h}_Q\le {\displaystyle \frac{{\Delta }_Q}{64}}.$$

Use the backend to compute a rational point enclosure centered at a rational point $p_i$ such that

$$|\Xi \left(z_i\right)-p_i|\le {\Delta }_Q/256.$$

Then for every $z\in D_i$,

$$|\Xi \left(z\right)-p_i|\le {\displaystyle \frac{{\Delta }_Q}{64}}+{\displaystyle \frac{{\Delta }_Q}{256}}={\displaystyle \frac{5{\Delta }_Q}{256}}.$$

Also,

$$|p_i|\ge |\Xi \left(z_i\right)|-|\Xi \left(z_i\right)-p_i|\ge d_Q-{\displaystyle \frac{{\Delta }_Q}{256}}>{\displaystyle \frac{{\Delta }_Q}{2}}.$$

For consecutive vertices,

$$|p_{i+1}-p_i|\le |p_{i+1}-\Xi \left(z_{i+1}\right)|+|\Xi \left(z_{i+1}\right)-\Xi \left(z_i\right)|+|\Xi \left(z_i\right)-p_i|\le {\displaystyle \frac{{\Delta }_Q}{256}}+{\displaystyle \frac{{\Delta }_Q}{64}}+{\displaystyle \frac{{\Delta }_Q}{256}}={\displaystyle \frac{3{\Delta }_Q}{128}}.$$

Hence the polygon segment $[p_i,p_{i+1}]$ lies in

$$\overline{B}\left(p_i,{\displaystyle \frac{3{\Delta }_Q}{128}}\right).$$

The true image of the domain disk $D_i$ lies in

$$\overline{B}\left(p_i,{\displaystyle \frac{5{\Delta }_Q}{256}}\right).$$

Choose a rational radius ${\rho }_i$ satisfying

$$max\left({\displaystyle \frac{3{\Delta }_Q}{128}},{\displaystyle \frac{5{\Delta }_Q}{256}}\right)<{\rho }_i<\left|p_i\right|.$$

Such a rational ${\rho }_i$ exists because

$$max\left({\displaystyle \frac{3{\Delta }_Q}{128}},{\displaystyle \frac{5{\Delta }_Q}{256}}\right)<{\displaystyle \frac{{\Delta }_Q}{32}}<{\displaystyle \frac{{\Delta }_Q}{2}}<\left|p_i\right|.$$

Then

$$E_i=\overline{B}(p_i,{\rho }_i)$$

is a rational image disk containing both the true image of $D_i$ and the polygon segment, and it avoids 0.

The shrinking-domain backend supplies rational image disks $E_i^{\prime }$ and finite backend records ${\beta }_i$ satisfying

$$\mathsf{XiImageCert}(N_Q,D_i,E_i^{\prime },{\beta }_i)$$

and

$$E_i^{\prime }\subseteq E_i$$

once the point precision satisfies $2^{-p}\le {\Delta }_Q/256$. Hence the finite record explicitly proves

$$\Xi \left(D_i\right)\subseteq E_i^{\prime }\subseteq E_i.$$

The rational polygon with vertices $p_i$ is contained segmentwise in the disks $E_i$, and each $E_i$ avoids 0. Choose a rational ray direction satisfying the finite general-position checks from the winding-certificate definition. The resulting tube-homotopy winding certificate is accepted, and by the argument principle it has the exact zero count. □

Theorem 32 

(Lower-bound-to-winding conversion). Let $Q⋐\mathcal{U}$ be an admissible rational rectangle. Suppose the following data are supplied:

• a rational lower bound

  $$0<{\lambda }_Q\le \underset{z\in \partial Q}{min}|\Xi \left(z\right)|,$$
  together with whatever finite verification record is being used to certify this boundary separation;
• a rational compact tubular neighborhood $N_Q⋐\mathcal{U}$ of $\partial Q$;
• a certified rational upper bound

  $$M_Q\ge \underset{z\in N_Q}{sup}\left|{\Xi }^{\prime }\left(z\right)\right|;$$
• a certified tube radius ${\tau }_Q>0$ proving

  $$\{z:dist(z,\partial Q)\le {\tau }_Q\}\subseteq N_Q.$$

Put

$$M_Q^{+}:=max(1,M_Q),{\Delta }_Q^{\lambda }:=min(1,{\lambda }_Q),$$

and

$$h_Q^{\lambda }:=min\left({\displaystyle \frac{{\Delta }_Q^{\lambda }}{64M_Q^{+}}},{\displaystyle \frac{{\tau }_Q}{2}}\right).$$

Let $L_Q$ be the perimeter of Q. Then there are effectively constructible $r\in \mathbb{Z}$ and a finite rational certificate c such that

$${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q,r,c),$$

using at most

$$4+⌈{\displaystyle \frac{L_Q}{h_Q^{\lambda }}}⌉$$

boundary arcs and point precision

$$2^{-p}\le {\Delta }_Q^{\lambda }/256.$$

The integer r is the exact zero count

$$r=N(\Xi ;Q^{\circ }).$$

The same construction gives a theta-backend certificate

$${\mathsf{WindCertRect}}_{\Theta }(Q,r,c_{\Theta })$$

when the image enclosures in the finite record are required to be certified by the theta-kernel backend.

Proof. 

Run the construction in 31 with ${\Delta }_Q^{\lambda }=min(1,{\lambda }_Q)$ in place of ${\Delta }_Q=min(1,d_Q)$, where

$$d_Q=\underset{z\in \partial Q}{min}|\Xi \left(z\right)|.$$

Since

$$0<{\lambda }_Q\le d_Q,$$

all separation estimates used in that proof remain valid after replacing ${\Delta }_Q$ by ${\Delta }_Q^{\lambda }$. The supplied derivative majorant and certified tube radius give the same variation control and guard-containment control. All mesh parameters are now rationally specified, and the backend point enclosures can be computed to the prescribed rational precision. The resulting rational polygon has a computable ray-crossing winding number r, and the finite tube-homotopy record is accepted by the winding verifier. Soundness of the winding certificate gives

$$r=N(\Xi ;Q^{\circ }).$$

If the theta backend is required, use the theta point and majorant enclosure routines instead of the AC backend in the same finite construction. □

9.3. Zero-Count Predicates and Zero-Free Rectangle Certificates

Definition 54 

(Zero-count predicates). Define

$$\mathsf{CountCertRect}(Q,r,c):={\mathsf{WindCertRect}}_{\mathrm{AC}}(Q,r,c),$$

and

$$\mathsf{ZeroCertRect}(Q,c):={\mathsf{WindCertRect}}_{\mathrm{AC}}(Q,0,c).$$

Define

$$\mathsf{ZeroCert}(n,c):=\mathsf{ZeroCertRect}(R_n,c).$$

Corollary 4 

(Zero-certificate form of RH).

$$\mathsf{RH}\iff \forall n\exists c\mathsf{ZeroCert}(n,c).$$

Proof. 

If RH holds, then $Z(\Xi ;\mathcal{U})=⌀$ by 3. Hence every $R_n$ is zero-free and has a zero-count 0 certificate by 30. Conversely, if every $R_n$ has a zero-count 0 certificate, the interiors of the $R_n$’s cover $\mathcal{U}$, so $Z(\Xi ;\mathcal{U})=⌀$, hence RH by 3. □

Remark 33. 

The theta-atlas positive channel

$$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c)$$

is the primary positive channel used in the final selected RH package. The zero-count channel

$$\mathsf{RH}\iff \forall n\exists c\mathsf{ZeroCert}(n,c)$$

is a companion all-stage positive channel based on the argument principle. In the enriched package, both are positive sign fibres.

9.4. Half-Plane Certificates

The paper also retains a local half-plane zero-free certificate. It is useful because it expresses local zero-freeness without computing a winding number and without requiring a global lower bound.

Definition 55 

(Half-plane rectangle certificate). $\mathsf{HalfPlaneCertRect}(Q,\alpha ,\beta ,\delta ,c)$ means that c is a finite guarded boundary certificate, checked using rational boundary disk covers and 27, proving

$$\alpha Re(\Xi (z\left)\right)+\beta Im(\Xi (z\left)\right)\ge \delta >0(z\in \partial Q),$$

where $\alpha ,\beta ,\delta \in \mathbb{Q}$ and $(\alpha ,\beta )\ne (0,0)$. Operationally, the verifier covers $\partial Q$ by small rational disks inside a rational compact guard, encloses Ξ on those disks, and checks that the linear functional has rational lower bound at least δ on every image enclosure.

Proposition 11 

(Decidability and completeness of half-plane certificates). $\mathsf{HalfPlaneCertRect}(Q,\alpha ,\beta ,\delta ,c)$ is decidable. Moreover, if for some rational $\alpha ,\beta$ and real ${\delta }_0>0$,

$$\alpha Re(\Xi \left(z\right))+\beta Im(\Xi \left(z\right))\ge {\delta }_0(z\in \partial Q),$$

then for every rational $0<\delta <{\delta }_0$ there exists c such that

$$\mathsf{HalfPlaneCertRect}(Q,\alpha ,\beta ,\delta ,c).$$

Proof. 

Decidability is finite rational verification. The verifier parses finite rational data, checks rational rectangle and guard containment, runs finitely many terminating backend enclosure routines, and checks rational linear-functional lower bounds.

For completeness, put

$$L\left(w\right)=\alpha Re\left(w\right)+\beta Im\left(w\right).$$

Assume

$$L(\Xi \left(z\right))\ge {\delta }_0(z\in \partial Q).$$

Let $0<\delta <{\delta }_0$ be rational. Choose

$$\delta <{\delta }_1<{\delta }_0.$$

By compactness of $\partial Q$ and continuity of $L\circ \Xi$, there is a compact tubular neighborhood $N_Q⋐\mathcal{U}$ of $\partial Q$ such that

$$L(\Xi \left(z\right))\ge {\delta }_1(z\in N_Q).$$

Choose a rational compact guard $K⋐\mathcal{U}$ containing $N_Q$, and choose a finite rational disk cover of $\partial Q$ by disks $D_{\ell }⋐N_Q$. The real-linear functional L satisfies

$$|L\left(w\right)-L\left(w^{\prime }\right)|\le (\left|\alpha \right|+\left|\beta \right|\left)\right|w-w^{\prime }|.$$

Choose the disks small enough and precision high enough that the resulting shrinking-domain image enclosures $E_{\ell }$ satisfy

$$\underset{w\in E_{\ell }}{inf}L\left(w\right)\ge \delta .$$

The finite list of disks, enclosure parameters, and rational inequalities is an accepted certificate. □

Lemma 16 

(Half-plane certificate implies zero-free). If

$$\mathsf{HalfPlaneCertRect}(Q,\alpha ,\beta ,\delta ,c),$$

then Q is zero-free.

Proof. 

The boundary image lies in the convex open half-plane

$$\{w:\alpha Re\left(w\right)+\beta Im\left(w\right)>0\}\subset {\mathbb{C}}^{\times }.$$

Thus its winding number around 0 is 0. The strict boundary inequality excludes boundary zeros, and the argument principle excludes interior zeros. □

Definition 56 

(Half-plane cover certificate). $\mathsf{HalfPlaneCoverCert}(n,c)$ means that c decodes a finite family

$${(Q_j,{\alpha }_j,{\beta }_j,{\delta }_j,c_j)}_{j<m}$$

such that:

• each $Q_j\subset \mathcal{U}$ is admissible;
• $$R_n\subseteq \bigcup _{j<m}{Q}_j;$$
• each

  $$\mathsf{HalfPlaneCertRect}(Q_j,{\alpha }_j,{\beta }_j,{\delta }_j,c_j)$$
  holds.

Proposition 12 

(Decidability of half-plane covers). $\mathsf{HalfPlaneCoverCert}(n,c)$ is decidable.

Proof. 

Tuple decoding is primitive recursive. Inclusion of a rational rectangle in a finite union of rational rectangles is decidable by rational cell decomposition. Each component certificate is decidable by 11. □

Theorem 34 

(Half-plane cover form of RH).

$$\mathsf{RH}\iff \forall n\exists c\mathsf{HalfPlaneCoverCert}(n,c).$$

Proof. 

Assume RH. Then $Z(\Xi ;\mathcal{U})=⌀$. Fix n. The compact rectangle $R_n$ is zero-free. For each $p\in R_n$, define

$$L_p\left(w\right):=Re\left(\overline{\Xi \left(p\right)}w\right).$$

Then

$$L_p(\Xi \left(p\right))={|\Xi \left(p\right)|}^2>0.$$

By continuity there is a small neighborhood $V_p⋐\mathcal{U}$ of p and a real margin ${\eta }_p>0$ such that

$$L_p(\Xi \left(z\right))\ge 2{\eta }_p(z\in V_p).$$

Approximate the coefficients of $L_p$ by rationals closely enough and shrink $V_p$ if necessary. This yields a rational linear functional

$$L_p^{\mathbb{Q}}\left(w\right)={\alpha }_{p}Re\left(w\right)+{\beta }_{p}Im\left(w\right),{\alpha }_p,{\beta }_p\in \mathbb{Q},$$

not both zero, and a rational admissible rectangle $Q_p⋐V_p$ with

$$p\in Q_p^{\circ }$$

such that

$$L_p^{\mathbb{Q}}(\Xi \left(z\right))\ge {\eta }_p(z\in Q_p).$$

In particular the inequality holds on $\partial Q_p$. Choose rational

$$0<{\delta }_p<{\eta }_p.$$

By 11, there exists a finite certificate $c_p$ with

$$\mathsf{HalfPlaneCertRect}(Q_p,{\alpha }_p,{\beta }_p,{\delta }_p,c_p).$$

The interiors $Q_p^{\circ }$, $p\in R_n$, form an open cover of the compact set $R_n$. A finite subcover gives a half-plane cover certificate.

Conversely, suppose every $R_n$ has a half-plane cover certificate. Each component rectangle in such a cover is zero-free by 16. Since $R_n$ is covered by finitely many such closed zero-free rectangles, $R_n$ itself contains no zero. The interiors of the $R_n$’s cover $\mathcal{U}$, so $Z(\Xi ;\mathcal{U})=⌀$. By 3, RH follows. □

10. Bad Witnesses and Admissibility Filters

The selected negative channel is finite. A bad witness is an exact-count winding certificate for an admissible rectangle in $\mathcal{U}$ with positive zero count.

10.1. Selected Bad Witnesses

Definition 57 

(Selected bad-witness predicate). Define ${\mathsf{Bad}}_{\Xi }\left(w\right)$ to mean that w decodes as

$$w=\langle k,r,c\rangle ,$$

with $r\ge 1$ and

$${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q_k,r,c).$$

Proposition 13 

(Decidability of selected bad witnesses). ${\mathsf{Bad}}_{\Xi }\left(w\right)$ is decidable.

Proof. 

Tuple decoding, the inequality $r\ge 1$, and the winding-certificate predicate are all decidable. □

Theorem 35 

(Finite selected negative channel).

$$\neg \mathsf{RH}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right).$$

Consequently

$$\mathsf{RH}\iff \forall w\neg {\mathsf{Bad}}_{\Xi }\left(w\right).$$

Proof. 

If ${\mathsf{Bad}}_{\Xi }\left(w\right)$, some admissible rectangle $Q_k\subset \mathcal{U}$ has certified exact zero count $r\ge 1$. Hence $Z(\Xi ;\mathcal{U})\ne ⌀$, so $\neg \mathsf{RH}$.

Conversely, if $\neg \mathsf{RH}$, choose $z_0\in \mathcal{U}$ with $\Xi \left(z_0\right)=0$. Since $\Xi$ is not identically zero, its zeros are isolated. Choose an admissible rational rectangle $Q_k$ containing $z_0$ in its interior and with zero-free boundary. By completeness of 30, $Q_k$ has an exact-count certificate with count $r\ge 1$, yielding w with ${\mathsf{Bad}}_{\Xi }\left(w\right)$. □

Definition 58 

(Least bad-witness search). The partial computation

$$\mu w{\mathsf{Bad}}_{\Xi }\left(w\right)$$

tests ${\mathsf{Bad}}_{\Xi }\left(0\right),{\mathsf{Bad}}_{\Xi }\left(1\right),{\mathsf{Bad}}_{\Xi }\left(2\right),\dots$ in order and halts with the least bad witness if one exists.

Corollary 5 

(Negative semidecision).

$$\mu w{\mathsf{Bad}}_{\Xi }\left(w\right)\downarrow \iff \neg \mathsf{RH}.$$

Equivalently,

$$\mu w{\mathsf{Bad}}_{\Xi }\left(w\right)\uparrow \iff \mathsf{RH}.$$

Proof. 

Since ${\mathsf{Bad}}_{\Xi }$ is decidable, least search halts exactly when

$$\exists w{\mathsf{Bad}}_{\Xi }\left(w\right).$$

Now apply 35. □

10.2. Theta Winding Certificates

The theta-atlas predicate uses theta point enclosures and theta majorants for positive zero-free atlases. The same theta backend supports winding certificates without using the analytic-continuation backend.

Definition 59 

(Theta winding certificate predicate). Define

$${\mathsf{WindCertRect}}_{\Theta }(Q,r,c)$$

by the same tube-homotopy winding certificate format as ${\mathsf{WindCertRect}}_{\mathrm{AC}}$, including the requirement that the partition contain the four side-break parameters of ${\gamma }_Q$, but require that every image enclosure call be certified using theta-kernel point enclosures and theta-majorant/shrinking-domain bounds from Section 6. Equivalently, the record contains finite rational theta data proving

$$\Xi \left(D_i\right)\subseteq E_i^{\prime }\subseteq E_i$$

for every boundary subarc disk $D_i$.

Theorem 36 

(Entire theta-backend winding theorem). The predicate

$${\mathsf{WindCertRect}}_{\Theta }(Q,r,c)$$

is decidable and satisfies the same soundness and completeness clauses as ${\mathsf{WindCertRect}}_{\mathrm{AC}}$:

• If ${\mathsf{WindCertRect}}_{\Theta }(Q,r,c)$, then Ξ has no zero on $\partial Q$ and

  $$N(\Xi ;Q^{\circ })=r.$$
• If $Q\subset \mathcal{U}$ is admissible and Ξ has no zero on $\partial Q$, then there exist $r,c$ with

  $${\mathsf{WindCertRect}}_{\Theta }(Q,r,c),$$
  and necessarily

  $$r=N(\Xi ;Q^{\circ }).$$

Proof. 

Decidability is finite rational verification: tuple decoding, rational containment checks, finitely many theta point/majorant enclosure checks, disk-containment checks, zero-avoidance checks, polygon-segment containment checks, finite ray general-position checks, and rational ray-crossing winding computation.

Soundness is the same tube-homotopy argument as in 15. The accepted data prove that for each boundary subarc,

$$\Xi \left({\gamma }_Q\left([t_i,t_{i+1}]\right)\right)\subseteq E_i,P\left([t_i,t_{i+1}]\right)\subseteq E_i,0\notin E_i.$$

Convexity of $E_i$ gives a straight-line homotopy in ${\mathbb{C}}^{\times }$ from $\Xi \circ {\gamma }_Q$ to P. Hence their winding numbers are equal, and the argument principle gives the zero count.

For completeness, assume $\Xi \ne 0$ on $\partial Q$. Let

$$\delta =\underset{z\in \partial Q}{min}|\Xi \left(z\right)|>0.$$

Choose a compact tubular neighborhood $N_Q⋐\mathcal{U}$ of $\partial Q$ on which

$$|\Xi (z\left)\right|>3\delta /4.$$

Choose a rational compact guard $K⋐\mathcal{U}$ containing $Q\cup N_Q$, a sufficiently fine rational boundary partition containing the four side-break parameters, and rational boundary disks $D_i⋐N_Q$ covering the corresponding subarcs. Make the disks small enough that the theta-majorant shrinking-domain term is much smaller than $\delta$. Then increase theta point precision so the total theta-image enclosure radius is also much smaller than $\delta$. As in the AC completeness proof, choose a rational polygonal loop P, rational image disks $E_i\subset {\mathbb{C}}^{\times }$ containing both the true image and the polygon segment, and a rational ray direction satisfying the finite general-position checks. Such a rational ray direction exists because only finitely many directions are excluded and rational directions are dense. The verifier then accepts with the true winding number. □

Definition 60 

(Theta replay transform). Define a total computable function

$$\mathsf{ThetaReplay}:\mathbb{N}\to \mathbb{N}$$

as follows. On input w, if w does not decode as $\langle k,r,c\rangle$ with

$${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q_k,r,c),$$

output 0. If it does, search for the least $c_{\Theta }$ such that

$${\mathsf{WindCertRect}}_{\Theta }(Q_k,r,c_{\Theta }),$$

and output that code.

Lemma 17 

(Totality of theta replay). $\mathsf{ThetaReplay}$ is total computable.

Proof. 

The AC condition is decidable. If it fails, the output is the fixed default 0. If it holds, AC soundness gives that $Q_k$ has zero-free boundary and exact count r. By completeness of ${\mathsf{WindCertRect}}_{\Theta }$, a theta-backend certificate with the same count exists. Since ${\mathsf{WindCertRect}}_{\Theta }$ is decidable, least search terminates. Thus totality is a mathematical consequence of the soundness and completeness of the two backends. No efficiency claim is made, and no primitive-recursive or closed-form a priori bound on this search is asserted. □

Definition 61 

(Repaired entire admissibility). Define ${\mathsf{Adm}}_{\Xi }^{\Theta }\left(w\right)$ to mean that $w=\langle k,r,c\rangle$ with

$${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q_k,r,c)$$

and

$${\mathsf{WindCertRect}}_{\Theta }(Q_k,r,\mathsf{ThetaReplay}\left(w\right)).$$

Define

$${\mathsf{Bad}}_{\Xi }^{\Theta }\left(w\right):={\mathsf{Bad}}_{\Xi }\left(w\right)\wedge {\mathsf{Adm}}_{\Xi }^{\Theta }\left(w\right).$$

Theorem 37 

(Repaired entire filter equivalence). For every w,

$${\mathsf{Bad}}_{\Xi }^{\Theta }\left(w\right)\iff {\mathsf{Bad}}_{\Xi }\left(w\right).$$

Consequently,

$$\mathsf{RH}\iff \forall w\neg {\mathsf{Bad}}_{\Xi }^{\Theta }\left(w\right).$$

Proof. 

The forward implication is immediate. Conversely, if ${\mathsf{Bad}}_{\Xi }\left(w\right)$, then $w=\langle k,r,c\rangle$ with $r\ge 1$ and

$${\mathsf{WindCertRect}}_{\mathrm{AC}}(Q_k,r,c).$$

By 17, $\mathsf{ThetaReplay}\left(w\right)$ returns a genuine theta-certificate for the same rectangle and count. Thus ${\mathsf{Adm}}_{\Xi }^{\Theta }\left(w\right)$, so ${\mathsf{Bad}}_{\Xi }^{\Theta }\left(w\right)$. The final equivalence follows from 35. □

10.3. Legacy Dirichlet Support-Strict Filtering

Definition 62 

(Legacy Dirichlet support-strict admissibility). A selected witness code $w=\langle k,r,c\rangle$ is legacy Dirichlet support-strict admissible, written ${\mathsf{Adm}}_{\Xi }^{\mathrm{Dir}}\left(w\right)$, if every domain disk D appearing in the guarded boundary record encoded by c is certified to satisfy

$$Re\left({\displaystyle \frac{1}{2}}+\mathrm{i}z\right)>1(z\in D).$$

Equivalently, every associated s-disk under $s=1/2+\mathrm{i}z$ lies wholly inside the Dirichlet half-plane $Re\left(s\right)>1$. This is a finite rational support-domain condition and is decidable. For malformed codes the predicate is false by convention.

Proposition 14 

(Support mismatch).

$${\mathsf{Bad}}_{\Xi }\left(w\right)\Rightarrow \neg {\mathsf{Adm}}_{\Xi }^{\mathrm{Dir}}\left(w\right).$$

Proof. 

A bad witness rectangle lies in $\mathcal{U}$, and its accepted guarded boundary disks are contained in a rational compact guard $K⋐\mathcal{U}$. For any $z=x+\mathrm{i}y\in K$, the corresponding s-input is

$$s=1/2+\mathrm{i}z=1/2-y+\mathrm{i}x.$$

Since $K\subset \mathcal{U}$, one has $0<y<1/2$, and hence

$$0<Re\left(s\right)=1/2-y<1/2<1.$$

Thus the associated s-domains are not contained in the Dirichlet half-plane $Re\left(s\right)>1$. Therefore a selected bad witness cannot be legacy Dirichlet support-strict admissible. □

Definition 63 

(Legacy filtered bad class). Define

$${\mathsf{Bad}}_{\Xi }^{\mathrm{Dir}}\left(w\right):={\mathsf{Bad}}_{\Xi }\left(w\right)\wedge {\mathsf{Adm}}_{\Xi }^{\mathrm{Dir}}\left(w\right).$$

Corollary 6 

(Legacy filtered emptiness).

$$\forall w\neg {\mathsf{Bad}}_{\Xi }^{\mathrm{Dir}}\left(w\right).$$

Proof. 

Immediate from 14. □

Theorem 38 

(Legacy coverage laundering). Under the support mismatch theorem, the coverage premise

$$\forall w({\mathsf{Bad}}_{\Xi }\left(w\right)\to {\mathsf{Adm}}_{\Xi }^{\mathrm{Dir}}\left(w\right))$$

is equivalent to RH. Hence passing from

$$\forall w\neg {\mathsf{Bad}}_{\Xi }^{\mathrm{Dir}}\left(w\right)$$

to

$$\forall w\neg {\mathsf{Bad}}_{\Xi }\left(w\right)$$

by asserting legacy coverage launders RH.

Proof. 

If $\forall w\neg {\mathsf{Bad}}_{\Xi }\left(w\right)$, coverage holds vacuously. Conversely, if coverage holds and ${\mathsf{Bad}}_{\Xi }\left(w\right)$ held, then ${\mathsf{Adm}}_{\Xi }^{\mathrm{Dir}}\left(w\right)$ would hold by coverage, while $\neg {\mathsf{Adm}}_{\Xi }^{\mathrm{Dir}}\left(w\right)$ holds by support mismatch. Contradiction. Thus

$$\forall w({\mathsf{Bad}}_{\Xi }\left(w\right)\to {\mathsf{Adm}}_{\Xi }^{\mathrm{Dir}}\left(w\right))\iff \forall w\neg {\mathsf{Bad}}_{\Xi }\left(w\right)\iff \mathsf{RH}$$

by 35. □

11. Classical RH Criteria as Finite Negative Channels

This section connects the selected $\Xi$-bad-witness channel to classical RH criteria. The purpose is not to claim that Li’s criterion or Lagarias’s criterion is new. The purpose is to show that their strict finite failure witnesses fit the same exact negative-channel pattern as the $\Xi$-winding witnesses.

The result is a finite witness-fibre theorem: the $\Xi$-winding, Li-negative, and Lagarias-negative channels are extensionally equivalent finite negative channels for $\neg \mathsf{RH}$, and witnesses in one channel can be computably replayed into witnesses in the others. In the next section these channels are absorbed into one enriched selected RH criterion package.

11.1. Li-Negative Certificates

Recall Li’s criterion. Let

$${\lambda }_n={\displaystyle \frac{1}{(n-1)!}}{\displaystyle \frac{d^n}{d{s}^n}}{\left[s^{n-1}log\xi \left(s\right)\right]}_{s=1}(n\ge 1).$$

Li’s theorem says

$$\mathsf{RH}\iff {\lambda }_n\ge 0(n\ge 1).$$

The coefficients ${\lambda }_n$ are real. This follows from the real symmetry $\xi \left(\overline{s}\right)=\overline{\xi \left(s\right)}$, the fact that $\xi \left(1\right)=1/2>0$, and the local real branch of $log\xi \left(s\right)$ at $s=1$; it is also part of the standard formulation of Li’s criterion.

The certificate system below uses arbitrary-order theta-jet enclosures near $s=1$. This avoids evaluating the singular Euler-product expression for $\xi \left(s\right)$ at the removable point $s=1$. Instead, one uses

$$\xi \left(s\right)=\Xi (-\mathrm{i}(s-1/2\left)\right),$$

and computes the required finite jet of the entire function $\Xi$ at

$$z=-\mathrm{i}/2.$$

Lemma 18 

(Arbitrary-order theta-jet point certificates). For every $j\ge 0$, every rational point $z_0\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$, and every rational $\epsilon >0$, there are finite rational certificates producing a rational disk of radius at most ε containing ${\Xi }^{\left(j\right)}\left(z_0\right)$.

Proof. 

This is 1. □

Lemma 19 

(Computability of Li coefficients by finite theta jets). The real numbers ${\lambda }_n$ are uniformly computable, with finite rational certificates for rigorous interval enclosures.

Proof. 

Fix $n\ge 1$. Put $w=s-1$, and write

$$A\left(w\right):=\xi (1+w)={\displaystyle \sum _{k=0}^{\infty }}{a}_k{w}^k,a_0=\xi \left(1\right)={\displaystyle \frac{1}{2}}.$$

Since

$$\xi \left(s\right)=\Xi (-\mathrm{i}(s-1/2\left)\right),$$

we have

$${\xi }^{\left(k\right)}\left(1\right)={(-\mathrm{i})}^k{\Xi }^{\left(k\right)}(-\mathrm{i}/2),a_k={\displaystyle \frac{{(-\mathrm{i})}^k}{k!}}{\Xi }^{\left(k\right)}(-\mathrm{i}/2).$$

By 18, the coefficients $a_1,\dots ,a_n$ can be enclosed to arbitrary rational precision.

Let

$$B\left(w\right):=logA\left(w\right)={\displaystyle \sum _{k=0}^{\infty }}{b}_k{w}^k$$

be the local logarithmic branch determined by $A\left(0\right)=1/2>0$. This branch is well-defined in a neighborhood of $w=0$ because $A\left(0\right)\ne 0$. The coefficients $b_1,\dots ,b_n$ are determined algebraically from $a_0,\dots ,a_n$, without needing a global branch, by the identity

$$A\left(w\right)B^{\prime }\left(w\right)=A^{\prime }\left(w\right).$$

Comparing coefficients of $w^m$, for $0\le m\le n-1$, gives

$${\displaystyle \sum _{j=0}^m}{a}_j(m-j+1)b_{m-j+1}=(m+1)a_{m+1}.$$

Since $a_0=1/2$, this recursion uniquely determines $b_{m+1}$:

$$b_{m+1}={\displaystyle \frac{(m+1)a_{m+1}-{\sum }_{j=1}^m{a}_j(m-j+1)b_{m-j+1}}{a_0(m+1)}}.$$

Rational interval arithmetic applied to this finite recursion gives arbitrarily sharp enclosures for $b_1,\dots ,b_n$.

Now

$$s^{n-1}={(1+w)}^{n-1},$$

and

$${\displaystyle \frac{1}{(n-1)!}}{\displaystyle \frac{d^n}{d{s}^n}}{\left[s^{n-1}log\xi \left(s\right)\right]}_{s=1}=n\left[w^n\right]\left({(1+w)}^{n-1}B\left(w\right)\right).$$

The constant term $b_0$ cannot contribute to the coefficient of $w^n$, since ${(1+w)}^{n-1}$ has degree $n-1$. Therefore

$${\lambda }_n=n{\displaystyle \sum _{k=1}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k-1}}\right)b_k.$$

This is a finite rational combination of $b_1,\dots ,b_n$. The interval computation may be performed in complex rectangular or disk arithmetic throughout the recursion. The exact number ${\lambda }_n$ is real. Thus, if the final complex enclosure for the displayed finite sum is a disk

$$\overline{B}(A,\rho ),A\in \mathbb{Q}+\mathrm{i}\mathbb{Q},\rho \in {\mathbb{Q}}_{\ge 0},$$

then the real interval

$$[ReA-\rho ,ReA+\rho ]$$

contains ${\lambda }_n$. More generally, if the final enclosure is a complex rectangle

$$[X_{-},X_{+}]+\mathrm{i}[Y_{-},Y_{+}],$$

then the real interval $[X_{-},X_{+}]$ contains ${\lambda }_n$. Refining the theta-jet enclosures and the rational interval arithmetic makes this real interval arbitrarily small. Hence ${\lambda }_n$ is uniformly computable and admits finite rational interval certificates. □

Definition 64 

(Li-negative certificate). For $n\ge 1$, define $\mathsf{LiNeg}(n,c)$ to mean that c contains a finite validated rational computation consisting of:

• arbitrary-order theta-jet point certificates enclosing

  $${\Xi }^{\left(k\right)}(-\mathrm{i}/2)(1\le k\le n);$$
• the induced interval enclosures for

  $$a_k={\displaystyle \frac{{(-\mathrm{i})}^k}{k!}}{\Xi }^{\left(k\right)}(-\mathrm{i}/2)(1\le k\le n),$$
  with $a_0=1/2$ exact;
• a rational interval-arithmetic execution of the recursion

  $$b_{m+1}={\displaystyle \frac{(m+1)a_{m+1}-{\sum }_{j=1}^m{a}_j(m-j+1)b_{m-j+1}}{a_0(m+1)}}(0\le m<n);$$
• the resulting complex enclosure for

  $${\lambda }_n=n{\displaystyle \sum _{k=1}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k-1}}\right)b_k,$$
  computed by rational complex interval or disk arithmetic, together with its projected real interval

  $$I_c=[A_c,B_c]\subset \mathbb{Q}$$
  obtained by taking the real projection of the final complex enclosure;
• the strict rational inequality

  $$B_c<0.$$

The verifier checks all finite theta-jet certificates, all rational interval operations, the real projection step, and the final inequality.

Proposition 15 

(Decidability of Li-negative certificates). $\mathsf{LiNeg}(n,c)$ is decidable.

Proof. 

The certificate contains only finite rational data: finitely many theta-jet point certificates, finite rational interval operations for the logarithmic jet, a finite real projection of the final complex enclosure, and a final rational sign check. Each component verifier is terminating and decidable. Therefore $\mathsf{LiNeg}(n,c)$ is decidable. □

Theorem 39 

(Li negative channel).

$$\neg \mathsf{RH}\iff \exists n\exists c\mathsf{LiNeg}(n,c).$$

Proof. 

If $\mathsf{LiNeg}(n,c)$, then the projected real interval $I_c=[A_c,B_c]$ contains the real number ${\lambda }_n$, and the certificate verifies

$$B_c<0.$$

Hence

$${\lambda }_n<0.$$

By Li’s criterion, RH is false.

Conversely, if RH is false, Li’s criterion implies that not all ${\lambda }_n$ are nonnegative. Hence for some $n\ge 1$,

$${\lambda }_n<0.$$

By 19, ${\lambda }_n$ admits arbitrarily sharp rational interval enclosures. Choose one with upper endpoint $<0$. The corresponding finite rational computation is a certificate c with $\mathsf{LiNeg}(n,c)$. □

11.2. Lagarias-Negative Certificates

Lagarias’s criterion states that RH is equivalent to

$$\sigma \left(n\right)\le H_n+exp\left(H_n\right)log{H}_n(n\ge 1),$$

where

$$\sigma \left(n\right)=\sum _{d\mid n}d,H_n={\displaystyle \sum _{k=1}^n}{\displaystyle \frac{1}{k}}.$$

Definition 65 

(Lagarias-negative certificate). For $n\ge 1$, define $\mathsf{LagNeg}(n,c)$ to mean that c contains:

• an exact factorization of n and the exact integer $\sigma \left(n\right)$;
• the exact rational number $H_n$;
• rational intervals $E_c$ and $L_c$ with

  $$exp\left(H_n\right)\in E_c,log{H}_n\in L_c;$$
• rational upper endpoints $E_c^{+}$, $L_c^{+}$ satisfying

  $$E_c\subseteq [0,E_c^{+}],L_c\subseteq [0,L_c^{+}];$$
• the strict rational inequality

  $$\sigma \left(n\right)>H_n+E_c^{+}{L}_c^{+}.$$

For $n=1$, $H_1=1$ and $log{H}_1=0$, so the interval convention includes the endpoint 0.

Proposition 16 

(Decidability of Lagarias-negative certificates). $\mathsf{LagNeg}(n,c)$ is decidable.

Proof. 

Factoring n, computing $\sigma \left(n\right)$, and computing $H_n$ are finite exact integer/rational operations. The interval enclosures for $exp\left(H_n\right)$ and $log{H}_n$ are checked by rational interval algorithms. The final strict inequality is rational. □

Theorem 40 

(Lagarias negative channel).

$$\neg \mathsf{RH}\iff \exists n\exists c\mathsf{LagNeg}(n,c).$$

Proof. 

If $\mathsf{LagNeg}(n,c)$, then

$$\sigma \left(n\right)>H_n+exp\left(H_n\right)log{H}_n,$$

so Lagarias’s criterion implies $\neg \mathsf{RH}$.

Conversely, if $\neg \mathsf{RH}$, Lagarias’s criterion gives some n with the strict violation

$$\sigma \left(n\right)>H_n+exp\left(H_n\right)log{H}_n.$$

The margin is positive. Since $exp\left(H_n\right)$ and $log{H}_n$ are computable reals, sufficiently sharp rational intervals $E_c,L_c$ certify the strict inequality. Thus $\mathsf{LagNeg}(n,c)$ holds for some c. □

11.3. Replay Equivalence of Finite Negative Channels

Definition 66 

(Packaged negative predicates). Define the decidable predicates

$$B_{\Xi }\left(w\right):={\mathsf{Bad}}_{\Xi }\left(w\right),$$

$$B_{\mathrm{Li}}\left(v\right):=\exists n\exists c\left(v=\langle n,c\rangle \wedge \mathsf{LiNeg}(n,c)\right),$$

and

$$B_{\mathrm{Lag}}\left(v\right):=\exists n\exists c\left(v=\langle n,c\rangle \wedge \mathsf{LagNeg}(n,c)\right).$$

Here the existential quantifiers are bounded by decoding v, so $B_{\mathrm{Li}}$ and $B_{\mathrm{Lag}}$ are decidable.

Theorem 41 

(Finite negative channel equivalence).

$$\exists w{B}_{\Xi }\left(w\right)\iff \exists v{B}_{\mathrm{Li}}\left(v\right)\iff \exists v{B}_{\mathrm{Lag}}\left(v\right)\iff \neg \mathsf{RH}.$$

Proof. 

The first equivalence is 35. The second is 39. The third is 40. □

Theorem 42 

(Negative-channel replay). Let $B_1,B_2$ be decidable predicates such that

$$\exists w{B}_1\left(w\right)\iff \exists v{B}_2\left(v\right).$$

Then there is a total computable function

$${\tau }_{1\to 2}:\mathbb{N}\to \mathbb{N}$$

such that

$$B_1\left(w\right)\Rightarrow B_2\left({\tau }_{1\to 2}\left(w\right)\right).$$

Proof. 

On input w, first decide $B_1\left(w\right)$. If false, output 0. If true, then

$$\exists v{B}_2\left(v\right)$$

by the assumed equivalence. Since $B_2$ is decidable, search for the least v satisfying $B_2\left(v\right)$. The search terminates. Output that v. This is a total computable function, and if $B_1\left(w\right)$ holds, the output satisfies $B_2$. □

Corollary 7 

(Replay equivalence of $\Xi$, Li, and Lagarias witnesses). There are total computable functions

$${\tau }_{\Xi \to \mathrm{Li}},{\tau }_{\mathrm{Li}\to \Xi },{\tau }_{\Xi \to \mathrm{Lag}},{\tau }_{\mathrm{Lag}\to \Xi },{\tau }_{\mathrm{Li}\to \mathrm{Lag}},{\tau }_{\mathrm{Lag}\to \mathrm{Li}}$$

such that, for example,

$${\mathsf{Bad}}_{\Xi }\left(w\right)\Rightarrow B_{\mathrm{Li}}\left({\tau }_{\Xi \to \mathrm{Li}}\left(w\right)\right),$$

$$B_{\mathrm{Li}}\left(v\right)\Rightarrow {\mathsf{Bad}}_{\Xi }\left({\tau }_{\mathrm{Li}\to \Xi }\left(v\right)\right),$$

and similarly for the remaining pairs.

Proof. 

Apply 42 to the decidable predicates $B_{\Xi }$, $B_{\mathrm{Li}}$, and $B_{\mathrm{Lag}}$, using 41. □

Remark 43. 

The replay maps are not asserted to preserve numerical efficiency or witness size. They express exact finite-witness equivalence: a finite certificate of $\neg \mathsf{RH}$ in one channel can be transformed by a total computable procedure into a finite certificate in any other exact negative channel. In the enriched selected RH package, these are different presentations of the same negative sign fibre.

12. Number-Theoretic Criterion Absorption

The preceding sections produced several exact number-theoretic channels for the same selected RH sign fibres. The purpose of this section is to record that these channels do not stand outside the selected-logical nullity theorem. They are absorbed into an enriched selected RH package.

Definition 67 

(Enriched exact selected package). An enriched exact selected package is a tuple

$${\mathcal{P}}^{*}=\left(P,\widehat{P},{(B_i,D_i)}_{i<I},{\left(C_j\right)}_{j<J}\right),$$

where $I,J\ge 1$ are finite, P is a proposition, $\widehat{P}$ is an arithmetic sentence, each $B_i\left(w\right)$ is a decidable finite negative-witness predicate, each $D_i(w,u)$ is a decidable accepting-trace predicate for $B_i$, and each $C_j(n,c)$ is a decidable all-stage positive-certificate predicate, such that in the intended standard semantics:

$$P\iff \forall w\neg B_i\left(w\right)(i<I),$$

equivalently,

$$\neg P\iff \exists w{B}_i\left(w\right)(i<I),$$

$$B_i\left(w\right)\iff \exists u{D}_i(w,u)(i<I),$$

$$P\iff \forall n\exists c{C}_j(n,c)(j<J),$$

and

$$P\iff \mathbb{N}\vDash \widehat{P}.$$

The predicates $B_i$ are the negative sign fibres and the predicates $C_j$ are the positive sign fibres of the enriched package. When a theorem concerns a fixed enriched arity $(I,J)$, the benchmark packages used in the nullity arguments are taken with exactly that same arity.

Definition 68 

(Positive criterion fibres used below). Define

$$C_{\Theta }(n,c):=\mathsf{ThetaAtlas}(n+5,c),$$

$$C_0(n,c):=\mathsf{ZeroCert}(n,c),$$

and

$$C_{\mathrm{hp}}(n,c):=\mathsf{HalfPlaneCoverCert}(n,c).$$

Definition 69 

(Enriched selected RH criterion package). Let

$$B_{\Xi }\left(w\right):={\mathsf{Bad}}_{\Xi }\left(w\right).$$

Let $B_{\mathrm{Li}}$ and $B_{\mathrm{Lag}}$ be the packaged Li-negative and Lagarias-negative predicates:

$$B_{\mathrm{Li}}\left(v\right):=\exists n\exists c\left(v=\langle n,c\rangle \wedge \mathsf{LiNeg}(n,c)\right),$$

$$B_{\mathrm{Lag}}\left(v\right):=\exists n\exists c\left(v=\langle n,c\rangle \wedge \mathsf{LagNeg}(n,c)\right).$$

Here the existential quantifiers are bounded by decoding v, so these predicates are decidable.

Let $D_{\Xi },D_{\mathrm{Li}},D_{\mathrm{Lag}}$ denote the corresponding complete accepting low-level trace predicates for the finite verifiers for $B_{\Xi },B_{\mathrm{Li}},B_{\mathrm{Lag}}$.

The symbol $\widehat{\mathsf{RH}}$ in the following definition is introduced in Section 13 and proved equivalent to $\mathsf{RH}$ in 52. Its appearance here is a forward reference in the exposition only; equivalently, the definition of ${\mathcal{P}}_{\mathsf{RH}}^{*}$ may be read after Section 13. If desired, one may temporarily write $\widehat{\mathsf{RH}}$ for the arithmetic representative and identify $\widehat{\mathsf{RH}}=\widehat{\mathsf{RH}}$ after the arithmetization theorem.

Define the enriched selected RH criterion package

$${\mathcal{P}}_{\mathsf{RH}}^{*}=\left(\mathsf{RH},\widehat{\mathsf{RH}},(B_{\Xi },D_{\Xi }),(B_{\mathrm{Li}},D_{\mathrm{Li}}),(B_{\mathrm{Lag}},D_{\mathrm{Lag}});C_{\Theta },C_0,C_{\mathrm{hp}}\right).$$

Theorem 44 

(Number-theoretic criterion absorption). The enriched selected RH criterion package ${\mathcal{P}}_{\mathsf{RH}}^{*}$ is an enriched exact selected package. Explicitly,

$$\exists w{B}_{\Xi }\left(w\right)\iff \exists v{B}_{\mathrm{Li}}\left(v\right)\iff \exists v{B}_{\mathrm{Lag}}\left(v\right)\iff \neg \mathsf{RH},$$

and

$$\forall n\exists c{C}_{\Theta }(n,c)\iff \forall n\exists c{C}_0(n,c)\iff \forall n\exists c{C}_{\mathrm{hp}}(n,c)\iff \mathsf{RH}.$$

Thus the Ξ-winding, Li-negative, Lagarias-negative, theta-atlas, zero-count, and half-plane-cover criteria are absorbed as exact sign fibres of one enriched selected RH package.

Proof. 

The $\Xi$-winding negative equivalence is 35. The Li-negative equivalence is 39. The Lagarias-negative equivalence is 40. Hence all three negative channels are equivalent to $\neg \mathsf{RH}$.

The theta-atlas positive equivalence is 18, with the shift $N=n+5$. The zero-count positive equivalence is 4. The half-plane-cover positive equivalence is 34. Hence all three positive channels are equivalent to $\mathsf{RH}$.

The trace predicates $D_{\Xi },D_{\mathrm{Li}},D_{\mathrm{Lag}}$ are exact by the complete low-level accepting-history construction used throughout the arithmetization section. Finally,

$$\mathbb{N}\vDash \widehat{\mathsf{RH}}\iff \mathsf{RH}$$

by 52. Therefore ${\mathcal{P}}_{\mathsf{RH}}^{*}$ satisfies all clauses of 67. □

Theorem 45 

(Fibre-complexity absorption). For each positive criterion fibre

$$C_j\in \{C_{\Theta },C_0,C_{\mathrm{hp}}\},$$

define its program-realizer fibre by

$${\mathcal{F}}_{C_j}^{+}=\left\{e:\forall n\exists t\left({\mathsf{T}}_{\mathrm{K}}(e,n,t)\wedge C_j(n,{\mathsf{U}}_{\mathrm{K}}\left(t\right))\right)\right\}.$$

Then

$${\mathcal{F}}_{C_j}^{+}\ne ⌀\iff \mathsf{RH}.$$

If $\mathsf{RH}$, then ${\mathcal{F}}_{C_j}^{+}$ is ${\Pi }_2^0$-complete.

For the negative fibres

$$B_i\in \{B_{\Xi },B_{\mathrm{Li}},B_{\mathrm{Lag}}\},$$

one has

$$\exists w{B}_i\left(w\right)\iff \neg \mathsf{RH}.$$

Moreover the three finite negative witness fibres are pairwise computably replay-equivalent.

Proof. 

For each positive $C_j$, the equivalence

$$\forall n\exists c{C}_j(n,c)\iff \mathsf{RH}$$

is part of 44. Since each $C_j$ is decidable, least search gives a total selector if the all-stage statement holds. Conversely any program realizer gives the all-stage statement. Thus

$${\mathcal{F}}_{C_j}^{+}\ne ⌀\iff \mathsf{RH}.$$

If nonempty, ${\mathcal{F}}_{C_j}^{+}$ is ${\Pi }_2^0$-complete by 63.

The negative equivalences are also part of 44. Pairwise replay equivalence of the finite negative fibres is 7. □

Theorem 46 

(General exact-criterion absorption schema). Let

$$\mathcal{P}=(P,B,C,D,\widehat{P})$$

be an exact selected package. Suppose $B^{\prime }\left(w\right)$ is a decidable predicate with

$$\exists w{B}^{\prime }\left(w\right)\iff \neg P,$$

and suppose $D^{\prime }(w,u)$ is an accepting-trace predicate satisfying

$$B^{\prime }\left(w\right)\iff \exists u{D}^{\prime }(w,u).$$

Then adjoining $B^{\prime }$ and $D^{\prime }$ to $\mathcal{P}$ gives an enriched exact selected package.

Similarly, if $C^{\prime }(n,c)$ is a decidable predicate with

$$\forall n\exists c{C}^{\prime }(n,c)\iff P,$$

then adjoining $C^{\prime }$ to $\mathcal{P}$ gives an enriched exact selected package.

Consequently, any exact Euler-product, explicit-formula, prime-sum, theta, winding, Li, Lagarias, Robin, or other number-theoretic criterion becomes a selected sign fibre once it is formulated as such a decidable finite negative or all-stage positive channel.

Proof. 

The enriched package clauses are exactly the displayed equivalences together with the original exact package equivalences. Decidability and trace exactness give the required finite verifier clauses. No additional argument is needed: exact criteria are absorbed by definition into the enriched selected package. □

Corollary 8 

(Exact criteria do not bypass nullity). The exact Ξ-winding, Li-negative, Lagarias-negative, theta-atlas, zero-count, and half-plane-cover criteria are the RH-specific coordinate-bearing criteria of this framework. They do not stand outside selected-logical nullity. They are sign fibres of ${\mathcal{P}}_{\mathsf{RH}}^{*}$. A non-null sign outcome from one of these criteria is possible only by fibre occupation:

$$\exists w{B}_i\left(w\right)\iff \neg \mathsf{RH}$$

for a negative fibre, or

$$\forall n\exists c{C}_j(n,c)\iff \mathsf{RH}$$

for a positive fibre. Before such occupation is supplied, the availability, exactness, and replay equivalence of the criteria do not yield an invariant selected-logical label.

Proof. 

This is an immediate consequence of . □

Remark 47 (RH-specific reasoning as fibre occupation) 

The enriched selected RH package is not an abstraction away from RH-specific analytic reasoning. It is the formal location of such reasoning. The theta kernel, $\Xi$-point enclosures, winding numbers, Li coefficients, Lagarias inequalities, zero-count data, half-plane covers, Euler-product data, and explicit-formula data are coordinate-bearing RH-specific inputs. If organized as exact finite negative channels or exact all-stage positive channels, they become fibres of ${\mathcal{P}}_{\mathsf{RH}}^{*}$. Their successful use is not a bypass of the framework: it is precisely finite negative occupation

$$\exists w{B}_i\left(w\right)\iff \neg \mathsf{RH}$$

or all-stage positive occupation

$$\forall n\exists c{C}_j(n,c)\iff \mathsf{RH}.$$

Pure selected-logical nullity describes what remains when these RH-specific coordinates and fibres are not observed.

Remark 48 (Euler-product and explicit-formula criteria) 

Euler-product and explicit-formula methods are not treated here as informal exceptions to the selected calculus. To enter the formal framework they must be presented as finite negative channels, all-stage positive channels, formal proofs in a specified theory, or explicit oracle/sign axioms. In the first two cases they are absorbed by 46; in the proof case they are governed by the proof-status calibration; in the oracle case the assumption is explicitly oracular.

13. Literal Arithmetization

The arithmetization step is deliberately literal. We do not assert that the high-level certificate verifier eventually halts. Instead, the trace variable u contains the entire finite accepting computation history. The arithmetic formula checks only bounded local consistency of this displayed history. Thus $\neg \widehat{\mathsf{RH}}$ is witnessed by a concrete finite trace whenever RH is false.

Convention 49 

(Concrete bounded $\beta$-style access coding). We fix once and for all a Gödel-β-style bounded access convention for finite sequences and finite tableaux. One convenient primitive access relation is:

$$Beta(b,c,i,y)$$

meaning that y is the remainder of b modulo $1+(i+1)c$. In the base language of arithmetic this is represented by the bounded formula

$$y<1+(i+1)c\wedge \exists q\le b\left(b=q(1+(i+1\left)c\right)+y\right).$$

Finite sequences and finite tableaux are coded by iterated bounded β-access relations and explicit length bounds. Every access to a configuration entry, tape cell, register entry, object-code position, subtrace entry, tuple component, rational numerator, rational denominator, or finite-list entry is expanded into such bounded access formulas. All quantifiers in these access relations are bounded by the displayed trace code and its displayed length parameters.

Convention 50 

(Bounded low-level Turing computation-history coding). We fix once and for all concrete Turing-machine or register-machine implementations of the finite verifiers used in the paper, including the verifiers for ${\mathsf{Bad}}_{\Xi }$, $\mathsf{ThetaAtlas}$, $\mathsf{LiNeg}$, and $\mathsf{LagNeg}$. A purported accepting trace is not a high-level assertion that one of these verifiers halts. It is a complete low-level finite computation tableau.

We work in a finite notational trace extension ${\mathsf{Q}}_{\mathrm{pr}}$ of Robinson arithmetic $\mathsf{Q}$ containing symbols for the fixed coding operations used to display finite machine traces. The expansion is purely notational. Every occurrence of the added notation is eliminated by replacing it with the fixed base-language bounded relation that checks the corresponding component of the trace, using the bounded β-style access convention of 49.

The bounded sequence-access and tableau-access relations are fixed once and for all by this convention. Thus all quantifiers in the accepting-history checker range over positions, entries, configurations, tape cells, registers, or subobjects bounded by the displayed trace code.

A trace code u contains the full accepting computation history of the fixed machine on the displayed input. In particular, u contains:

• the time bound and every machine configuration;
• the input and all decoded tuple components;
• every decoded rational object inspected by the verifier;
• every intermediate integer and rational arithmetic result;
• every subtrace needed for coding, decoding, comparison, arithmetic, containment, ray-crossing, theta-jet verification, quadrature verification, and backend-output verification;
• the final accepting configuration.

The checker verifies only local consistency of adjacent configurations and finite relations among entries appearing inside u. All quantifiers in the checker are bounded by lengths, positions, entries, configurations, tape cells, registers, or subobjects explicitly coded inside u.

If a subroutine is not one of the fixed bounded-access primitives of 49, then its complete finite computation history is displayed as a subtrace inside u and is checked locally. Therefore no unbounded assertion that a primitive recursive or computable subroutine halts is hidden in the matrix.

With this low-level trace convention, for every fixed verifier V used in the paper, the relation

$$u\mathrm{is}\mathrm{an}\mathrm{accepting}\mathrm{computation}\mathrm{history}\mathrm{of}V\mathrm{on}\mathrm{input}x$$

is represented by a ${\Delta }_0$ formula of the expanded trace language, and after eliminating the finite trace notation by a ${\Delta }_0$ formula in the original language of arithmetic.

The only proof-theoretic property used below is closed-numeral exactness: for every closed standard numeral instance of such a bounded verifier formula, the formula evaluates exactly as the corresponding external finite computation; true closed instances are provable in $\mathsf{Q}$ after definitional elimination, and false closed instances have their negations provable in $\mathsf{Q}$ after definitional elimination.

Warning 51 (No hidden primitive recursion in the Π1 matrix) 

The assertion that a verifier uses a primitive recursive or computable subroutine is never used as an unbounded assertion inside ${\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}$. Every subroutine computation inspected by the verifier is either one of the fixed bounded access/arithmetic relations of the chosen tableau coding, or else its complete finite computation history is displayed as a subtrace inside the trace code u. In particular, sequence access, tuple projection, rational arithmetic, comparison, containment checking, elementary-function enclosure checking, quadrature checking, and backend-output verification are all checked locally against data bounded by the displayed trace. No unbounded exponentiation, halting assertion, totality assertion, or primitive-recursive computation is hidden in the matrix. After definitional trace elimination, the remaining formula contains only bounded quantifiers over positions, entries, configurations and subobjects explicitly coded in u.

Lemma 20 

(Bounded tableau coding). Fix a deterministic Turing-machine or register-machine program M. Fix also the concrete bounded β-style coding of finite sequences and finite tableaux from 49, with bounded access relations for entries, configurations, tape cells, registers, states, and subtraces. After definitional elimination of this fixed coding notation, there is a bounded arithmetic formula

$${Acc}_M(x,u)$$

whose standard interpretation is:

$$u\mathrm{codes}\mathrm{a}\mathrm{complete}\mathrm{accepting}\mathrm{computation}\mathrm{tableau}\mathrm{of}M\mathrm{on}\mathrm{input}x.$$

More precisely, u codes a time bound $T\le u$, a finite list of configurations of length $T+1$, and all displayed subcomputations used by M. The formula ${Acc}_M(x,u)$ asserts, using only quantifiers bounded by u, that:

• the first configuration is the initial configuration of M on input x;
• every configuration entry, tape-cell entry, register entry, state code, head position, decoded object, rational numerator and denominator, and subtrace entry is bounded by u;
• every adjacent pair of configurations satisfies one of the finitely many transition rules of the fixed machine M;
• every displayed subtrace is locally consistent and its initial and terminal configurations match the call and return data shown in the main trace;
• every arithmetic, comparison, tuple-decoding, rational-containment, interval-enclosure, ray-crossing, quadrature, theta-certification, or backend verification step inspected by M is certified by the corresponding displayed bounded local computation;
• the last configuration is an accepting configuration.

Therefore ${Acc}_M(x,u)$ is ${\Delta }_0$ after the fixed coding notation is eliminated.

Proof. 

All data inspected by the fixed machine M are required to occur explicitly inside the tableau coded by u. Sequence length, configuration number, tape cell number, register number, subtrace number, object-code position, and arithmetic-entry position are all bounded by u. The relation saying that a given entry occurs at a given bounded position in the tableau is represented by the fixed bounded access relation for the chosen coding, ultimately by the bounded Beta-style formula of 49.

The initial-configuration condition, the accepting-final-configuration condition, and each one-step transition condition are finite disjunctions of bounded relations, because M has only finitely many transition rules. Each subverifier call is checked by a displayed subtrace, again with all quantifiers bounded by the length and entries of that subtrace, which are themselves bounded by u. Hence no unbounded assertion of halting or totality is used. The formula checks only bounded local consistency of a displayed finite object. After replacing the fixed tableau notation by its chosen base-language bounded definition, the resulting formula is ${\Delta }_0$. □

Lemma 21 

(Base-language bounded trace representation). After eliminating the finite trace notation fixed in 50, the accepting-history relation for each fixed verifier used in the paper is represented in the original language of arithmetic by a bounded formula. In particular, there is a ${\Delta }_0$ formula

$${Acc}_{V_{{\mathsf{Bad}}_{\Xi }}}(w,u)$$

whose standard interpretation is exactly:

$$u\mathrm{is}\mathrm{a}\mathrm{complete}\mathrm{accepting}\mathrm{low}-\mathrm{level}\mathrm{computation}\mathrm{history}\mathrm{of}{V}_{{\mathsf{Bad}}_{\Xi }}\mathrm{on}\mathrm{input}w.$$

Proof. 

Apply 20 to the fixed verifier $V_{{\mathsf{Bad}}_{\Xi }}$. This gives a bounded formula

$${Acc}_{V_{{\mathsf{Bad}}_{\Xi }}}(w,u)$$

whose standard interpretation is exactly that u is a complete accepting low-level computation history of $V_{{\mathsf{Bad}}_{\Xi }}$ on input w. The same construction applies to the theta-atlas, Li-negative, and Lagarias-negative verifiers. Since every inspected output is accompanied by its own displayed low-level computation history unless it is one of the fixed bounded access primitives, no high-level halting assertion is hidden in the matrix. □

Definition 70 

(Bad-witness trace checker). Let ${\mathsf{BadCheck}}_{\Xi }(w,u)$ mean that u is a complete accepting computation history of the fixed finite verifier for ${\mathsf{Bad}}_{\Xi }\left(w\right)$.

Proposition 17 

(Trace representation).

$${\mathsf{Bad}}_{\Xi }\left(w\right)\iff \exists u{\mathsf{BadCheck}}_{\Xi }(w,u).$$

Moreover ${\mathsf{BadCheck}}_{\Xi }(w,u)$ is represented by a bounded arithmetic formula

$${\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u).$$

Proof. 

The verifier accepts exactly the valid finite rational certificate codes for ${\mathsf{Bad}}_{\Xi }\left(w\right)$. By the chosen finite computation-history coding and 21, accepting histories are checked by bounded local conditions. □

Definition 71 

(Li and Lagarias accepting traces). Let

$$\mathsf{LiNegCheck}(v,u)$$

mean that u is a complete accepting low-level computation history of the fixed verifier for the packaged predicate $B_{\mathrm{Li}}\left(v\right)$. Let

$$\mathsf{LagNegCheck}(v,u)$$

mean the corresponding accepting-history predicate for $B_{\mathrm{Lag}}\left(v\right)$. Their bounded arithmetic representations are denoted

$${\mathsf{LiNegCheck}}^{\mathrm{arith}}(v,u),{\mathsf{LagNegCheck}}^{\mathrm{arith}}(v,u).$$

Proposition 18 

(Li/Lagarias trace exactness).

$$B_{\mathrm{Li}}\left(v\right)\iff \exists u\mathsf{LiNegCheck}(v,u),B_{\mathrm{Lag}}\left(v\right)\iff \exists u\mathsf{LagNegCheck}(v,u),$$

and both accepting-history predicates have bounded arithmetic representations after trace elimination.

Proof. 

The verifiers for $B_{\mathrm{Li}}$ and $B_{\mathrm{Lag}}$ are finite rational verifiers. Their accepting histories are represented by the same complete low-level tableau coding used for ${\mathsf{Bad}}_{\Xi }$, as guaranteed by 21. □

Proposition 19 

(Numeral exactness). For all standard $w,u$,

$$\mathbb{N}\vDash {\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(\overline{w},\overline{u})\iff {\mathsf{BadCheck}}_{\Xi }(w,u).$$

Moreover, every closed numeral instance is decided correctly in the trace language, and after definitional elimination the true closed instances and the negations of the false closed instances are provable in $\mathsf{Q}$. The same assertion holds for the Li-negative, Lagarias-negative, and theta-atlas accepting-history predicates fixed in this paper.

Proof. 

Bounded arithmetic instances reduce to finite computations on numerals. Robinson arithmetic proves the true closed bounded instances and proves the negations of the false ones after the fixed definitional trace notation has been eliminated. The same bounded tableau coding is used for the other fixed verifiers. □

Definition 72 

(Literal arithmetic RH). Define

$$\widehat{\mathsf{RH}}:⟺\forall w\forall u\neg {\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u).$$

Theorem 52 

(Literal ${\Pi }_1$ representative). $\widehat{\mathsf{RH}}$ is a ${\Pi }_1$ sentence in the finite trace extension fixed in 50. After definitional trace elimination, it is represented by a ${\Pi }_1$ sentence in the base arithmetic language. More explicitly, the only unbounded quantifiers are the displayed leading universal number quantifiers

$$\forall w\forall u,$$

and the matrix

$$\neg {\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u)$$

is bounded after the fixed trace notation is eliminated. Moreover,

$$\mathbb{N}\vDash \widehat{\mathsf{RH}}\iff \mathsf{RH}.$$

Proof. 

By 21, the accepting-history relation

$${\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u)$$

is represented, after definitional trace elimination, by a bounded formula. Thus

$$\widehat{\mathsf{RH}}:⟺\forall w\forall u\neg {\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u)$$

has only leading universal number quantifiers followed by a bounded matrix. It is therefore a ${\Pi }_1$ sentence. No pairing of w and u is needed for this classification; if one chooses to pair them, the pairing and projection relations must be read through the same bounded-access coding convention.

For the standard-model equivalence,

$$\mathbb{N}\vDash \widehat{\mathsf{RH}}\iff \forall w\forall u\neg {\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u).$$

By 17, this is equivalent to

$$\forall w\neg {\mathsf{Bad}}_{\Xi }\left(w\right).$$

By 35, this is equivalent to $\mathsf{RH}$. □

Theorem 53 

($\mathsf{Q}$-criterion for the negative channel).

$$\mathsf{Q}⊢\neg \widehat{\mathsf{RH}}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right),$$

where $\widehat{\mathsf{RH}}$ is understood via definitional trace elimination if working in the original language. Consequently

$$\mathsf{Q}⊬\neg \widehat{\mathsf{RH}}\iff \mathsf{RH}.$$

Proof. 

If ${\mathsf{Bad}}_{\Xi }\left(w\right)$, choose u with ${\mathsf{BadCheck}}_{\Xi }(w,u)$. Then by 19,

$$\mathsf{Q}⊢{\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(\overline{w},\overline{u})$$

after definitional trace elimination. Hence

$$\mathsf{Q}⊢\exists x\exists y{\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(x,y),$$

which is classically equivalent to $\neg \widehat{\mathsf{RH}}$.

Conversely, if $\mathsf{Q}⊢\neg \widehat{\mathsf{RH}}$, then by standard soundness of $\mathsf{Q}$ in the standard model,

$$\mathbb{N}\vDash \exists w\exists u{\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u).$$

Hence some standard w satisfies ${\mathsf{Bad}}_{\Xi }\left(w\right)$. The final equivalence follows from 35. □

Theorem 54 

($PA$-criterion under ${\Sigma }_1$-soundness). Assume $\mathsf{PA}$ is ${\Sigma }_1$-sound. Then

$$\mathsf{PA}⊢\neg \widehat{\mathsf{RH}}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right).$$

Consequently

$$\mathsf{PA}⊬\neg \widehat{\mathsf{RH}}\iff \mathsf{RH}.$$

Proof. 

The right-to-left implication follows already from 53. For the converse, $\neg \widehat{\mathsf{RH}}$ is the ${\Sigma }_1$ sentence

$$\exists w\exists u{\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u).$$

If $\mathsf{PA}$ proves it and $\mathsf{PA}$ is ${\Sigma }_1$-sound, then it is true in $\mathbb{N}$, hence some ${\mathsf{Bad}}_{\Xi }\left(w\right)$ holds. □

Theorem 55 

(Proof-status calibration for the literal RH representative). Let $T\supseteq \mathsf{Q}$ be any theory in the language containing $\widehat{\mathsf{RH}}$, understood with the same definitional trace convention as above. Then:

• If $\neg \mathsf{RH}$, then

  $$T⊢\neg \widehat{\mathsf{RH}}.$$
• Hence

  $$T⊬\neg \widehat{\mathsf{RH}}⟹\mathsf{RH}.$$
• Hence

  $${Ind}_T\left(\widehat{\mathsf{RH}}\right)⟹\mathsf{RH}.$$
• If T is consistent and

  $$T⊢\widehat{\mathsf{RH}},$$
  then

  $$\mathsf{RH}.$$
• If T is ${\Sigma }_1$-sound, then

  $$T⊢\neg \widehat{\mathsf{RH}}\iff \neg \mathsf{RH}.$$
  Equivalently,

  $$T⊬\neg \widehat{\mathsf{RH}}\iff \mathsf{RH}.$$

Proof. 

If $\neg \mathsf{RH}$, then by 35 there is w with

$${\mathsf{Bad}}_{\Xi }\left(w\right).$$

By trace representation, choose u with

$${\mathsf{BadCheck}}_{\Xi }(w,u).$$

By numeral exactness,

$$\mathsf{Q}⊢{\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(\overline{w},\overline{u}).$$

Therefore

$$\mathsf{Q}⊢\exists x\exists y{\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(x,y),$$

which is classically equivalent to $\neg \widehat{\mathsf{RH}}$. Hence every $T\supseteq \mathsf{Q}$ proves $\neg \widehat{\mathsf{RH}}$. This proves (i), and (ii) follows by contraposition.

For (iii), independence includes $T⊬\neg \widehat{\mathsf{RH}}$, so (ii) gives RH.

For (iv), suppose T is consistent and $T⊢\widehat{\mathsf{RH}}$. If $\neg \mathsf{RH}$, then by (i) $T⊢\neg \widehat{\mathsf{RH}}$, contradicting consistency. Hence RH.

For (v), the implication $\neg \mathsf{RH}\Rightarrow T⊢\neg \widehat{\mathsf{RH}}$ is (i). Conversely, $\neg \widehat{\mathsf{RH}}$ is a ${\Sigma }_1$ sentence. If T is ${\Sigma }_1$-sound and $T⊢\neg \widehat{\mathsf{RH}}$, then

$$\mathbb{N}\vDash \neg \widehat{\mathsf{RH}}.$$

By 52, this is equivalent to $\neg \mathsf{RH}$. This proves (v). □

13.1. Arithmetizing the Positive Theta-Atlas Channel

The positive theta-atlas channel is an all-stage statement:

$$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

Because $\mathsf{ThetaAtlas}(N,c)$ is decidable, it has a finite accepting-trace relation.

Definition 73 

(Theta-atlas accepting trace). Let

$$\mathsf{ThetaAtlasCheck}(N,c,u)$$

mean that u is a complete accepting low-level computation history of the fixed verifier for $\mathsf{ThetaAtlas}(N,c)$. Let

$${\mathsf{ThetaAtlasCheck}}^{\mathrm{arith}}(N,c,u)$$

be its bounded arithmetic representation.

Proposition 20 

(Theta-atlas trace representation).

$$\mathsf{ThetaAtlas}(N,c)\iff \exists u\mathsf{ThetaAtlasCheck}(N,c,u),$$

and ${\mathsf{ThetaAtlasCheck}}^{\mathrm{arith}}$ is bounded after trace elimination.

Proof. 

The verifier for $\mathsf{ThetaAtlas}(N,c)$ is finite and rational. Its accepting computations are represented by complete low-level computation histories exactly as in 50 and 20. The accepting-history relation is therefore bounded after trace elimination and is exact on standard numerals. □

Definition 74 

(Positive theta arithmetic form). Define the arithmetical all-stage formula

$${\mathsf{RH}}_{\Theta }^{+}:⟺\forall N\left(N<5\vee \exists c\exists u{\mathsf{ThetaAtlasCheck}}^{\mathrm{arith}}(N,c,u)\right).$$

This is the formal arithmetic version of

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

Proposition 21 

(Positive theta arithmetic equivalence).

$$\mathbb{N}\vDash {\mathsf{RH}}_{\Theta }^{+}\iff \mathsf{RH}.$$

Proof. 

By the definition of ${\mathsf{RH}}_{\Theta }^{+}$ and 20,

$$\mathbb{N}\vDash {\mathsf{RH}}_{\Theta }^{+}\iff \forall N\left(N<5\vee \exists c\mathsf{ThetaAtlas}(N,c)\right).$$

This is equivalent to

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

By 18, this is equivalent to RH. □

Remark 56. 

The positive theta arithmetic form is ${\Pi }_2^0$ in the usual arithmetical hierarchy, while $\widehat{\mathsf{RH}}$ is a ${\Pi }_1^0$ representative obtained from the finite negative channel. Both are exact standard-model representatives of RH, but they encode different certificate directions.

Definition 75 

($\omega$-logic over $\mathsf{Q}$). Here ω denotes the least infinite ordinal, identified with the standard natural numbers

$$\omega =\mathbb{N}=\{0,1,2,\dots \}.$$

A ${\mathsf{Q}}^{\omega }$-derivation is a well-founded, possibly infinitary proof tree whose leaves are axioms of $\mathsf{Q}$ or logical axioms, whose finitary inference nodes are the usual first-order inference rules, and which also allows the ω-rule

$${\displaystyle \frac{\{\phi \left(\overline{n}\right):n\in \omega \}}{\forall x\phi \left(x\right)}}.$$

Thus an application of the ω-rule has one premise for each standard natural number n. We write

$${\mathsf{Q}}^{\omega }⊢\sigma$$

if there is such a well-founded ω-proof of σ.

Theorem 57 

($\omega$-resolution).

$${\mathsf{Q}}^{\omega }⊢\widehat{\mathsf{RH}}\iff \mathsf{RH},$$

and

$${\mathsf{Q}}^{\omega }⊢\neg \widehat{\mathsf{RH}}\iff \neg \mathsf{RH}.$$

Also,

$${\mathsf{Q}}^{\omega }⊢{\mathsf{RH}}_{\Theta }^{+}\iff \mathsf{RH}.$$

Proof. 

Let

$$\psi (w,u):=\neg {\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u).$$

Assume RH. Then for every standard pair $w,u$, the bounded sentence

$$\psi (\overline{w},\overline{u})$$

is true in $\mathbb{N}$, hence $\mathsf{Q}$-provable by closed-numeral exactness for bounded verifier formulas. Fixing w, the $\omega$-rule applied to

$$\{\psi (\overline{w},\overline{u}):u\in \omega \}$$

gives

$${\mathsf{Q}}^{\omega }⊢\forall u\psi (\overline{w},u).$$

Applying the $\omega$-rule again over w gives

$${\mathsf{Q}}^{\omega }⊢\widehat{\mathsf{RH}}.$$

If $\neg \mathsf{RH}$, then $\mathsf{Q}⊢\neg \widehat{\mathsf{RH}}$ by 53, hence ${\mathsf{Q}}^{\omega }⊢\neg \widehat{\mathsf{RH}}$.

Conversely, $\omega$-logic over the sound base $\mathsf{Q}$ is sound in the standard model. Soundness is proved by induction on well-founded $\omega$-proof trees; the $\omega$-rule is sound in $\mathbb{N}$ because every element of $\mathbb{N}$ is denoted by a standard numeral. Therefore

$${\mathsf{Q}}^{\omega }⊢\widehat{\mathsf{RH}}\Rightarrow \mathbb{N}\vDash \widehat{\mathsf{RH}}\Rightarrow \mathsf{RH},$$

and

$${\mathsf{Q}}^{\omega }⊢\neg \widehat{\mathsf{RH}}\Rightarrow \neg \mathsf{RH}.$$

For the positive theta form, assume RH. Fix a standard N. If $N<5$, then

$$\mathsf{Q}⊢\overline{N}<5,$$

and therefore

$$\mathsf{Q}⊢\overline{N}<5\vee \exists c\exists u{\mathsf{ThetaAtlasCheck}}^{\mathrm{arith}}(\overline{N},c,u).$$

If $N\ge 5$, then by 18 there is a standard $c_N$ with

$$\mathsf{ThetaAtlas}(N,c_N),$$

and hence a standard accepting trace $u_N$. By numeral exactness,

$$\mathsf{Q}⊢{\mathsf{ThetaAtlasCheck}}^{\mathrm{arith}}(\overline{N},{\overline{c}}_N,{\overline{u}}_N).$$

Thus

$$\mathsf{Q}⊢\exists c\exists u{\mathsf{ThetaAtlasCheck}}^{\mathrm{arith}}(\overline{N},c,u),$$

and hence again

$$\mathsf{Q}⊢\overline{N}<5\vee \exists c\exists u{\mathsf{ThetaAtlasCheck}}^{\mathrm{arith}}(\overline{N},c,u).$$

Applying the $\omega$-rule over N gives

$${\mathsf{Q}}^{\omega }⊢{\mathsf{RH}}_{\Theta }^{+}.$$

Conversely, $\omega$-soundness and 21 give RH. □

Remark 58. 

The use of $\omega$-logic here is purely calibrational; it is not a finitary proof-search procedure and is not used to claim that a recursively axiomatizable theory proves $\widehat{\mathsf{RH}}$ under RH.

14. Selected Packages, Theta Fibres, and Selection Jump

The exact selected package framework records a recurring structure: a proposition has a finite negative certificate channel, an all-stage positive certificate channel, a finite accepting-trace relation for the negative verifier, and an arithmetic representative. The selected RH package now uses the theta-atlas positive channel as its primary positive coordinate. The enriched selected RH package, constructed in Section 12, adjoins additional exact number-theoretic fibres.

14.1. The Selected $\Theta$-Atlas RH Package

Definition 76 

(Theta-atlas positive predicate). Define

$$C_{\Xi }^{\Theta }(n,c):=\mathsf{ThetaAtlas}(n+5,c).$$

Thus

$$\forall n\exists c{C}_{\Xi }^{\Theta }(n,c)$$

is the same as

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

Definition 77 

(The selected $\Theta$-atlas $\Xi$-witness package). Let ${\mathcal{W}}_{\Xi }^{\Theta }$ denote the collection consisting of:

$$\begin{array}{cc}& \mathsf{ThetaAtlas},\mathsf{ThetaPointCert},\mathsf{ThetaMajCert},\hfill \\ & {\mathsf{WindCertRect}}_{\mathrm{AC}},{\mathsf{WindCertRect}}_{\Theta },\hfill \\ & \mathsf{ZeroCert},\mathsf{HalfPlaneCoverCert},\hfill \\ & {\mathsf{Bad}}_{\Xi },{\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}\hfill \end{array}$$

together with the rational theta-kernel, AC, and winding-verifier routines fixed above. Define

$${\mathsf{RH}}^{♯}:=(\mathsf{RH},{\mathcal{W}}_{\Xi }^{\Theta }).$$

When treated as an exact selected package, ${\mathsf{RH}}^{♯}$ is represented by

$${\mathcal{P}}_{\mathsf{RH}}=(\mathsf{RH},B_{\mathsf{RH}},C_{\mathsf{RH}},D_{\mathsf{RH}},\widehat{\mathsf{RH}}),$$

where

$$B_{\mathsf{RH}}\left(w\right):={\mathsf{Bad}}_{\Xi }\left(w\right),$$

$$C_{\mathsf{RH}}(n,c):=C_{\Xi }^{\Theta }(n,c)=\mathsf{ThetaAtlas}(n+5,c),$$

$$D_{\mathsf{RH}}(w,u):={\mathsf{BadCheck}}_{\Xi }(w,u),$$

and

$$\widehat{\mathsf{RH}}:=\widehat{\mathsf{RH}}.$$

Convention 59 

(Resolver notation for ${\mathsf{RH}}^{♯}$). The symbol

$${\mathsf{RH}}^{♯}=(\mathsf{RH},{\mathcal{W}}_{\Xi }^{\Theta })$$

denotes the selected RH object together with its coordinate-bearing witness apparatus. Whenever ${\mathsf{RH}}^{♯}$ is used as an argument of a resolver whose domain is the class $\mathfrak{Pkg}$ of exact selected packages, it is understood to mean the exact selected package

$${\mathcal{P}}_{\mathsf{RH}}=(\mathsf{RH},{\mathsf{Bad}}_{\Xi },C_{\Xi }^{\Theta },{\mathsf{BadCheck}}_{\Xi },\widehat{\mathsf{RH}}).$$

Thus expressions such as

$$\mathcal{R}\left({\mathsf{RH}}^{♯}\right)$$

are abbreviations for

$$\mathcal{R}\left({\mathcal{P}}_{\mathsf{RH}}\right).$$

When the enriched package is intended, we write ${\mathcal{P}}_{\mathsf{RH}}^{*}$.

Theorem 60 

(Exact selected theta-atlas RH package). The package

$${\mathcal{P}}_{\mathsf{RH}}=(\mathsf{RH},{\mathsf{Bad}}_{\Xi },C_{\Xi }^{\Theta },{\mathsf{BadCheck}}_{\Xi },\widehat{\mathsf{RH}})$$

is an exact selected package. Explicitly,

$$\mathsf{RH}\iff \forall w\neg {\mathsf{Bad}}_{\Xi }\left(w\right),$$

$$\mathsf{RH}\iff \forall n\exists c{C}_{\Xi }^{\Theta }(n,c),$$

$${\mathsf{Bad}}_{\Xi }\left(w\right)\iff \exists u{\mathsf{BadCheck}}_{\Xi }(w,u),$$

and

$$\mathsf{RH}\iff \mathbb{N}\vDash \widehat{\mathsf{RH}}.$$

Proof. 

The first equivalence is 35. The second is 18, with the shift $N=n+5$. The third is 17. The fourth is 52. The convention above identifies this exact package with ${\mathsf{RH}}^{♯}$ whenever ${\mathsf{RH}}^{♯}$ is used in resolver-level statements. □

Remark 61 (Enriched criterion package) 

The package ${\mathcal{P}}_{\mathsf{RH}}$ is the minimal selected RH package used for the basic resolver formalism. The enriched package ${\mathcal{P}}_{\mathsf{RH}}^{*}$, constructed in Section 12, adjoins the replay-equivalent $\Xi$-winding, Li-negative, Lagarias-negative, theta-atlas, zero-count, and half-plane-cover channels. These additions do not form escape routes from the nullity theorem; they are exact sign fibres of the enriched package.

14.2. Trace-Disjunct Discipline

Definition 78 

(Broad bridge trace). Let T be a usual deductive theory with proof predicate ${\mathrm{Proof}}_T(p,x)$. For a sentence P, define

$${\mathsf{Trace}}_T^{+}\left(P\right):⟺\exists b\exists p\exists q\left({\mathrm{Proof}}_T(p,b)\wedge {\mathrm{Proof}}_T(q,b\to P)\right),$$

where b ranges over sentence codes and $b\to P$ denotes the primitive recursive operation forming the code of the implication from the sentence coded by b to P. Define ${\mathsf{Trace}}_T^{-}\left(P\right)$ similarly with P replaced by $\neg P$.

Theorem 62 

(Trace-disjunct vacuity). For every T and P,

$${\mathsf{Trace}}_T^{+}\left(P\right)\iff T⊢P,$$

and

$${\mathsf{Trace}}_T^{-}\left(P\right)\iff T⊢\neg P.$$

Proof. 

If ${\mathsf{Trace}}_T^{+}\left(P\right)$, then for some sentence B,

$$T⊢B\mathrm{and}T⊢B\to P.$$

By modus ponens, $T⊢P$. Conversely, if $T⊢P$, take $B=P$, take p to be a proof of P, and take q to be a proof of the tautology $P\to P$. The negative case is identical. □

14.3. Selection Jump

The following index-set phenomenon is repeatedly used. Once a stagewise certificate relation has even one successful total selector, recognizing all successful selectors is already ${\Pi }_2^0$-complete.

Definition 79 

(Stagewise-local success class). Let $\mathsf{Cert}(n,c)$ be decidable. Define

$$S_{\mathsf{Cert}}=\{e:\forall n\exists t({\mathsf{T}}_{\mathrm{K}}(e,n,t)\wedge \mathsf{Cert}(n,{\mathsf{U}}_{\mathrm{K}}\left(t\right)))\}.$$

Theorem 63 

(Selection Jump). If $S_{\mathsf{Cert}}\ne ⌀$, then $S_{\mathsf{Cert}}$ is ${\Pi }_2^0$-complete.

Proof. 

First,

$$e\in S_{\mathsf{Cert}}$$

means

$$\forall n\exists t({\mathsf{T}}_{\mathrm{K}}(e,n,t)\wedge \mathsf{Cert}(n,{\mathsf{U}}_{\mathrm{K}}\left(t\right))).$$

If $\mathsf{Cert}$ is primitive recursive, this is immediately ${\Pi }_2^0$. If $\mathsf{Cert}$ is only decidable, fix a machine deciding it and let $A_{\mathsf{Cert}}(n,c,v)$ be the primitive-recursive accepting-history predicate for that machine. Then

$$\mathsf{Cert}(n,c)\iff \exists v{A}_{\mathsf{Cert}}(n,c,v),$$

so

$$e\in S_{\mathsf{Cert}}\iff \forall n\exists t\exists v\left({\mathsf{T}}_{\mathrm{K}}(e,n,t)\wedge A_{\mathsf{Cert}}(n,{\mathsf{U}}_{\mathrm{K}}\left(t\right),v)\right).$$

Pairing $t,v$ shows $S_{\mathsf{Cert}}\in {\Pi }_2^0$.

Assume $S_{\mathsf{Cert}}\ne ⌀$, and choose $e_0\in S_{\mathsf{Cert}}$. Let

$$A=\{a:\forall n\exists mR(a,n,m\left)\right\}$$

be an arbitrary ${\Pi }_2^0$ set with decidable R. Map a effectively to an index $f\left(a\right)$ for the program which, on input n, first searches for an m satisfying $R(a,n,m)$. Once such an m is found, it simulates $e_0\left(n\right)$ and outputs whatever $e_0\left(n\right)$ outputs.

If $a\in A$, then for every n the first search terminates, and since $e_0\in S_{\mathsf{Cert}}$, the subsequent computation eventually outputs a valid $\mathsf{Cert}(n,·)$-certificate. Hence $f\left(a\right)\in S_{\mathsf{Cert}}$. If $a\notin A$, then for some n no such m exists, and the program $f\left(a\right)$ never reaches the simulation of $e_0\left(n\right)$. Thus $f\left(a\right)\notin S_{\mathsf{Cert}}$. Therefore

$$a\in A\iff f\left(a\right)\in S_{\mathsf{Cert}}.$$

This gives ${\Pi }_2^0$-hardness, and hence ${\Pi }_2^0$-completeness. □

Remark 64 (Nonuniformity of the seed) 

The hardness reduction uses a fixed successful selector $e_0\in S_{\mathsf{Cert}}$. Thus the theorem is an ordinary completeness theorem for a nonempty fibre, not a uniform construction of a seed. In the RH application, the existence of such a seed is itself equivalent to occupation of the positive sign fibre.

Definition 80 

(Selected theta-atlas RH sign fibres). Define the positive theta-atlas realizer fibre

$${\mathcal{F}}_{\Xi ,\Theta }^{+}=\left\{e:\forall n\exists t\left({\mathsf{T}}_{\mathrm{K}}(e,n,t)\wedge \mathsf{ThetaAtlas}(n+5,{\mathsf{U}}_{\mathrm{K}}\left(t\right))\right)\right\}.$$

Define the negative fibre

$${\mathcal{F}}_{\Xi }^{-}=\{w:{\mathsf{Bad}}_{\Xi }\left(w\right)\}.$$

Theorem 65 

(Exact selected theta-atlas RH fibres).

$${\mathcal{F}}_{\Xi ,\Theta }^{+}\ne ⌀\iff \mathsf{RH},{\mathcal{F}}_{\Xi }^{-}\ne ⌀\iff \neg \mathsf{RH}.$$

If $\mathsf{RH}$, then ${\mathcal{F}}_{\Xi ,\Theta }^{+}$ is ${\Pi }_2^0$-complete. If $\neg \mathsf{RH}$, then ${\mathcal{F}}_{\Xi }^{-}$ contains a finite selected witness.

Proof. 

If $e\in {\mathcal{F}}_{\Xi ,\Theta }^{+}$, then

$$\forall n\exists c\mathsf{ThetaAtlas}(n+5,c),$$

so

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c),$$

and hence RH by 18. Conversely, if RH holds, then the all-stage theta-atlas statement holds. Since $\mathsf{ThetaAtlas}$ is decidable, least search produces a total certificate generator, so ${\mathcal{F}}_{\Xi ,\Theta }^{+}\ne ⌀$. Selection Jump gives ${\Pi }_2^0$-completeness.

The negative equivalence is 35. □

Remark 66 (Positive fibre complexity) 

The ${\Pi }_2^0$-completeness of a positive realizer fibre is an index-set complexity statement about recognizing successful certificate-generating programs. It is not, by itself, an unprovability or independence theorem for RH. In the enriched package, it records a complexity consequence of positive fibre occupation.

Theorem 67 

(Selection Jump does not imply unprovability). There is a decidable $C(n,c)$ such that

$$\mathsf{Q}⊢\forall n\exists cC(n,c),$$

but

$$S_C=\{e:\forall n\exists t({\mathsf{T}}_{\mathrm{K}}(e,n,t)\wedge C(n,{\mathsf{U}}_{\mathrm{K}}\left(t\right)))\}$$

is ${\Pi }_2^0$-complete.

Proof. 

Let

$$C(n,c)\iff c=0.$$

Then $\mathsf{Q}$ proves the universal statement. The constant-zero program is a successful seed, so Selection Jump applies. □

Remark 68. 

Selection Jump is an index-set complexity theorem about recognizing successful stagewise realizer programs. It is not, by itself, a theorem about unprovability or independence of the target proposition. The theta-atlas fibre ${\mathcal{F}}_{\Xi ,\Theta }^{+}$ is ${\Pi }_2^0$-complete if RH holds, but that fact alone does not imply any proof-theoretic status for RH.

15. Global Selected-Resolution Nonclosure

This section collects uniform nonclosure barriers. These barriers are not used to prove RH, disprove RH, or establish independence of RH. They calibrate what kinds of terminal classification principles cannot exist uniformly over broad selected-package families.

The theta-atlas and criterion-absorption layers clarify the interpretation of this section. The nonclosure theorems do not say that the theta, winding, Li, Lagarias, zero-count, half-plane-cover, or other exact number-theoretic criteria are outside the analysis. They are precisely the RH-specific coordinate-bearing criteria analyzed by the paper. Once exact, they are absorbed into the enriched selected package. The nonclosure results say that the unoccupied enriched criterion architecture itself does not produce a terminal selected-logical label in the pure replacement-invariant regime.

15.1. Finite-Prefix Nonclosure

Definition 81 

(Finite prefix). For decidable $R(n,c)$, define

$${\mathsf{Pref}}_R\left(M\right):=\forall n<M\exists cR(n,c).$$

Theorem 69 

(Finite-prefix nonclosure). For every M, there is a decidable $R_M(n,c)$ such that

$${\mathsf{Pref}}_{R_M}\left(M\right)$$

holds but

$$\forall n\exists c{R}_M(n,c)$$

fails.

Proof. 

Let

$$R_M(n,c)\iff n<M\wedge c=0.$$

Then the prefix below M holds, but stage M has no witness. □

Remark 70. 

A finite collection of theta-atlas certificates proves zero-freeness only on the corresponding finite collection of windows. By itself, no finite prefix can close the all-stage RH channel. This is a logical statement about the form

$$\forall N\exists c,$$

not a numerical statement about any particular window. A complete all-stage stream is not an escape from nullity; it is occupation of the positive sign fibre.

15.2. Computable Triage and Independence

Definition 82 

(Coded ${\Pi }_1$-selected packages). Let ${\left({\delta }_e\left(v\right)\right)}_{e\in \mathbb{N}}$ enumerate bounded arithmetic formulas with one free variable. Set

$${\pi }_e:=\forall v{\delta }_e\left(v\right).$$

Define

$$P_e:⟺\mathbb{N}\vDash {\pi }_e,$$

$$B_e\left(w\right):⟺\neg {\delta }_e\left(w\right),$$

and

$$C_e(n,c):⟺c=0\wedge \forall w<n{\delta }_e\left(w\right).$$

Then

$$P_e\iff \forall w\neg B_e\left(w\right)\iff \forall n\exists c{C}_e(n,c).$$

Moreover $B_e\left(w\right)$ has the trivial trace representation

$$B_e\left(w\right)\iff \exists u(u=0\wedge \neg {\delta }_e\left(w\right)).$$

Thus this is a primitive recursive family of exact selected packages in the sense of 1, with ${\widehat{P}}_e={\pi }_e$.

Definition 83 

(Index independence over T). Let $T\supseteq \mathsf{Q}$ be recursively axiomatizable. Define

$${Ind}_T\left(e\right):⟺T⊬{\pi }_e\mathrm{and}T⊬\neg {\pi }_e.$$

For sentences σ, ${Ind}_T\left(\sigma \right)$ has the meaning fixed in Section 3.

Theorem 71 

(No computable proof-disproof-independence triage). Let $T\supseteq \mathsf{Q}$ be recursively axiomatizable. There is no total computable function

$$M:\mathbb{N}\to \{+,-,0\}\times \mathbb{N}$$

such that, writing $M\left(e\right)=({\ell }_e,a_e)$:

• ${\ell }_e=+$ implies $\mathbb{N}\vDash {\pi }_e$;
• ${\ell }_e=-$ implies $a_e$ is a genuine counterexample:

  $$\mathbb{N}\vDash \neg {\delta }_e\left(a_e\right);$$
• ${\ell }_e=0$ implies ${Ind}_T\left(e\right)$.

Proof. 

If ${\pi }_e$ is false, choose a with $\neg {\delta }_e\left(a\right)$. Since ${\delta }_e$ is bounded,

$$\mathsf{Q}⊢\neg {\delta }_e\left(\overline{a}\right),$$

so

$$T⊢\neg {\pi }_e.$$

Thus ${Ind}_T\left(e\right)$ is false. Soundness of M also forbids +. Hence $M\left(e\right)$ must output −.

If ${\pi }_e$ is true, $M\left(e\right)$ cannot output −. Therefore

$$\mathbb{N}\vDash {\pi }_e\iff {\ell }_e\ne -.$$

This would decide true ${\Pi }_1$ arithmetic, impossible. Indeed

$$e\notin {\mathsf{K}}_0\iff \forall t\neg {\mathsf{T}}_{\mathrm{K}}(e,0,t)$$

is a true-${\Pi }_1$ condition. □

Theorem 72 

(Independence is not decidable uniformly). If $T\supseteq \mathsf{Q}$ is sound and recursively axiomatizable, then

$$\{e:{Ind}_T\left(e\right)\}$$

is not decidable.

Proof. 

If ${Ind}_T$ were decidable, decide truth of ${\pi }_e$ as follows. If ${Ind}_T\left(e\right)$, then ${\pi }_e$ must be true, since false ${\Pi }_1$ sentences are refutable already in $\mathsf{Q}\subseteq T$. If not independent, dovetail proof searches for ${\pi }_e$ and $\neg {\pi }_e$; one appears, and soundness gives the truth value. This decides true ${\Pi }_1$ arithmetic. □

15.3. Reflection, Tarski, and Diagonal Barriers

Theorem 73 

(Reflection collapse). Let $T\supseteq I{\Sigma }_1$ be a recursively axiomatizable theory equipped with the standard provability predicate satisfying the usual Hilbert–Bernays–Löb derivability conditions. Suppose an arithmetic predicate $\mathsf{Term}\left(x\right)$ is adequate along a primitive recursive ${\Pi }_1$-universal embedding f, meaning that for each coded ${\Pi }_1$ sentence ${\phi }_e$,

$$T⊢{\mathrm{Prov}}_T\left({\phi }_e\right)\to \mathsf{Term}\left(\overline{f\left(e\right)}\right),$$

and

$$T⊢\mathsf{Term}\left(\overline{f\left(e\right)}\right)\to {\phi }_e.$$

Then T proves ${\Pi }_1$-reflection for itself on the embedded family. If T is consistent and sufficiently strong for Gödel’s second theorem, no such $\mathsf{Term}$ exists for a ${\Pi }_1$-universal family containing $0=1$ as a degenerate ${\Pi }_1$ sentence.

Proof. 

Compose the displayed implications. This gives

$$T⊢{\mathrm{Prov}}_T\left({\phi }_e\right)\to {\phi }_e$$

for all ${\Pi }_1$ sentences in the embedded family. Taking ${\phi }_e$ to be the degenerate ${\Pi }_1$ sentence $0=1$ gives

$$T⊢{\mathrm{Prov}}_T(0=1)\to 0=1.$$

Since $T⊢0\ne 1$, this yields

$$T⊢\neg {\mathrm{Prov}}_T(0=1),$$

i.e.

$$T⊢\mathrm{Con}\left(T\right),$$

contradicting Gödel’s second incompleteness theorem for consistent recursively axiomatizable $T\supseteq I{\Sigma }_1$ with the usual derivability conditions. □

Definition 84 

(Exact terminality on an image). Let $\mathcal{F}$ be a primitive recursive set of arithmetic sentence codes, and let

$$f:\mathcal{F}\to \mathbb{N}$$

be a primitive recursive injection. An arithmetic predicate $\mathsf{Term}\left(x\right)$ is exact on $f\left[\mathcal{F}\right]$ for a semantic success predicate $\mathsf{Suc}$ if for every $y\in \mathcal{F}$,

$$\mathbb{N}\vDash \mathsf{Term}\left(\overline{f\left(y\right)}\right)\iff \mathsf{Suc}\left(f\left(y\right)\right).$$

Definition 85 

(Truth-faithful and truth-universal images). Let $\mathcal{F}$ be a primitive recursive set of arithmetic sentence codes, and let

$$f:\mathcal{F}\to \mathbb{N}$$

be a primitive recursive injection into a class of nodes, packages, or terminal objects. A semantic success predicate $\mathsf{Suc}$ is truth-faithful on the image of f if

$$\mathsf{Suc}\left(f\right(y\left)\right)\iff \mathbb{N}\vDash y(y\in \mathcal{F}).$$

If $\mathcal{F}$ is the set of all arithmetic sentence codes, the image $f\left[\mathcal{F}\right]$ is called truth-universal.

Theorem 74 

(Tarski threshold). Let $\mathcal{F}$ be a primitive recursive set of arithmetic sentence codes, and let

$$f:\mathcal{F}\to \mathbb{N}$$

be a primitive recursive injection. Suppose

$$\mathsf{Suc}\left(f\right(y\left)\right)\iff \mathsf{Truth}\left(y\right)(y\in \mathcal{F}).$$

Then an arithmetic predicate $\mathsf{Term}\left(x\right)$ exact on $f\left[\mathcal{F}\right]$ exists iff

$$\{y\in \mathcal{F}:\mathsf{Truth}(y\left)\right\}$$

is arithmetical. In particular, no arithmetic exact terminality predicate exists on a truth-universal primitive recursive image of all arithmetic sentence codes.

Proof. 

If $\mathsf{Term}$ is exact, then truth on $\mathcal{F}$ is arithmetically defined by

$$y\in \mathcal{F}\wedge \exists z(z=f(y)\wedge \mathsf{Term}(z\left)\right).$$

Here $z=f\left(y\right)$ abbreviates the arithmetical graph of the primitive recursive function f.

Conversely, suppose the truth set on $\mathcal{F}$ is arithmetical. Let $\theta \left(y\right)$ be an arithmetic formula defining

$$\{y\in \mathcal{F}:\mathsf{Truth}(y\left)\right\}.$$

Define an arithmetic predicate on the primitive recursive image of f by

$$\mathsf{Term}\left(x\right)\iff \exists y\left(y\in \mathcal{F}\wedge f\left(y\right)=x\wedge \theta \left(y\right)\right).$$

Because f is injective on $\mathcal{F}$, this predicate is exact on $f\left[\mathcal{F}\right]$. The full-sentence case is Tarski’s undefinability theorem: truth for all arithmetic sentences is not arithmetical. □

Theorem 75 

(Diagonal barrier). If $(\mathcal{N},\mathsf{Suc})$ is diagonally universal, i.e. there is a primitive recursive d such that

$$\mathsf{Suc}\left(d\left(\phi \right)\right)\iff \phi \left(\overline{d\left(\phi \right)}\right)$$

for every one-variable arithmetic formula φ, then no arithmetic $\mathsf{Term}\left(x\right)$ is exact for $\mathsf{Suc}$ on all of $\mathcal{N}$.

Proof. 

Apply diagonal universality to $\phi \left(x\right)\equiv \neg \mathsf{Term}\left(x\right)$. Exactness would give, for

$$e=d(\neg \mathsf{Term}),$$

the contradiction

$$\mathsf{Term}\left(e\right)\iff \mathsf{Suc}\left(e\right)\iff \neg \mathsf{Term}\left(e\right).$$

□

15.4. Finite-Interface Nonclosure in Broad and Theta-Cosine Classes

The original broad finite-interface nonclosure statement is retained. We also add a theta-cosine perturbation version, which is closer to the analytic identity of $\Xi$: perturbations arise from real compactly supported theta-kernel changes.

Definition 86 

($\xi$-symmetric ambient class). Let

$$\mathcal{X}=\{F:\mathbb{C}\to \mathbb{C}\mathrm{entire}:F(1-s)=F\left(s\right),F\left(\overline{s}\right)=\overline{F\left(s\right)}\}.$$

Lemma 22 

(Infinite-dimensional broad perturbation space). Let

$$B\left(s\right):=exp\left({(s-1/2)}^2\right),$$

and

$$\mathcal{V}:=\left\{\right(P\left(s\right)+P(1-s)\left)B\right(s):P\in \mathbb{R}[s\left]\right\}.$$

Then $\mathcal{V}\subseteq \mathcal{X}$ is infinite-dimensional.

Proof. 

The symmetries are immediate. The functions

$${(s-1/2)}^{2k}B\left(s\right)$$

are linearly independent over $\mathbb{R}$, and each belongs to $\mathcal{V}$ by taking

$$P\left(s\right)={\displaystyle \frac{1}{2}}{(s-1/2)}^{2k}.$$

□

Definition 87 

(Finite real-linear interface). A finite real-linear interface is a finite list of real-linear maps

$${\Lambda }_1,\dots ,{\Lambda }_N:{Span}_{\mathbb{R}}\left\{\xi \right\}+\mathcal{V}\to \mathbb{C},$$

where $\mathbb{C}$ is regarded as a real vector space.

Theorem 76 

(Broad finite-interface nonclosure). For every finite real-linear interface ${\Lambda }_1,\dots ,{\Lambda }_N$, there exists $F\in \mathcal{X}$ such that

$${\Lambda }_j\left(F\right)={\Lambda }_j\left(\xi \right)(j=1,\dots ,N),$$

but F has a real off-critical zero in

$$(0,1)\setminus \{1/2\}.$$

Proof. 

The finite interface imposes finitely many real-linear constraints on the infinite-dimensional space $\mathcal{V}$. Choose nonzero $Q\in \mathcal{V}$ in the common kernel of all imposed constraints. Since Q is not identically zero, its zeros on the real interval $(0,1)$ are isolated unless it vanishes identically, which it does not. Hence choose

$$\rho \in (0,1)\setminus \{1/2\}$$

with $Q\left(\rho \right)\ne 0$. By 2,

$$\xi \left(\rho \right)>0.$$

Also $Q\left(\rho \right)\in \mathbb{R}$, because $Q\in \mathcal{X}$. Set

$$c=-\xi \left(\rho \right)/Q\left(\rho \right)\in \mathbb{R},$$

and

$$F=\xi +cQ.$$

Then $F\left(\rho \right)=0$ and all interface values match those of $\xi$. □

Definition 88 

(Theta-cosine perturbation space). Let

$${\mathcal{H}}_{\Theta }=\left\{H_h\left(z\right)={\int }_0^{\infty }h\left(u\right)cos\left(zu\right)\mathrm{d}u:h\in C_c^{\infty }((0,\infty ),\mathbb{R})\right\}.$$

Each $H_h$ is entire, even, and satisfies

$$H_h\left(\overline{z}\right)=\overline{H_h\left(z\right)}.$$

Define the theta-cosine perturbation class

$${\mathcal{X}}_{\Theta }=\{\Xi +H:H\in {\mathcal{H}}_{\Theta }\}.$$

For $F\in {\mathcal{X}}_{\Theta }$, define the corresponding completed s-plane function by

$${\Lambda }_F\left(s\right):=F(-\mathrm{i}(s-1/2)).$$

Then

$${\Lambda }_F(1-s)={\Lambda }_F\left(s\right),{\Lambda }_F\left(\overline{s}\right)=\overline{{\Lambda }_F\left(s\right)}.$$

Lemma 23 

(Infinite-dimensional theta-cosine perturbation space). The real vector space ${\mathcal{H}}_{\Theta }$ is infinite-dimensional.

Proof. 

Choose pairwise disjoint nonempty compact intervals

$$I_1,I_2,\dots \subset (0,\infty )$$

and nonzero functions

$$h_k\in C_c^{\infty }(I_k,\mathbb{R}).$$

If

$${\displaystyle \sum _{k=1}^m}{a}_k{H}_{h_k}=0$$

as an entire function, then

$${\int }_0^{\infty }\left({\displaystyle \sum _{k=1}^m}{a}_k{h}_k\left(u\right)\right)cos\left(zu\right)\mathrm{d}u=0$$

for all z. The cosine transform of a compactly supported smooth function is entire, and injectivity of the Fourier transform on compactly supported distributions implies

$${\displaystyle \sum _{k=1}^m}{a}_k{h}_k\left(u\right)=0.$$

Since the supports are disjoint and the $h_k$’s are nonzero, all $a_k=0$. Thus the functions $H_{h_k}$ are linearly independent. □

Definition 89 

(Theta-cosine finite real-linear interface). A theta-cosine finite real-linear interface is a finite list of real-linear maps

$${\Lambda }_1,\dots ,{\Lambda }_N:{Span}_{\mathbb{R}}\{\Xi \}+{\mathcal{H}}_{\Theta }\to \mathbb{C},$$

where $\mathbb{C}$ is regarded as a real vector space.

Theorem 77 

(Theta-cosine finite-interface nonclosure). For every theta-cosine finite real-linear interface

$${\Lambda }_1,\dots ,{\Lambda }_N,$$

there exists

$$F\in {\mathcal{X}}_{\Theta }$$

such that

$${\Lambda }_j\left(F\right)={\Lambda }_j(\Xi )(j=1,\dots ,N),$$

but the corresponding completed s-plane function

$${\Lambda }_F\left(s\right)=F(-\mathrm{i}(s-1/2))$$

has a real off-critical zero

$$s_0\in (0,1)\setminus \{1/2\}.$$

Equivalently, F has a zero at

$$z_0=\mathrm{i}(1/2-s_0),$$

with

$$0<|Im{z}_0|<1/2.$$

This zero lies on the imaginary axis

$$Re{z}_0=0,$$

and therefore lies on the boundary of the Ξ-region

$$\mathcal{U}=\{x+\mathrm{i}y:x>0,0<y<1/2\},$$

not in $\mathcal{U}$ itself. The theorem is a finite-interface obstruction to RH-like zero-freeness of the associated completed s-plane function, not an interior-$\mathcal{U}$ zero construction.

Proof. 

The interface imposes finitely many real-linear constraints on the infinite-dimensional space ${\mathcal{H}}_{\Theta }$. Choose

$$0\ne H\in {\mathcal{H}}_{\Theta }$$

in the common kernel of all ${\Lambda }_j{|}_{{\mathcal{H}}_{\Theta }}$.

Since H is an entire nonzero function, it cannot vanish on the entire imaginary interval

$$\{\mathrm{i}y:0<y<1/2\}.$$

Choose

$$y_0\in (0,1/2)$$

such that

$$H\left(\mathrm{i}{y}_0\right)\ne 0.$$

Because h is real and

$$cos\left(\mathrm{i}{y}_{0}u\right)=cosh\left(y_{0}u\right)\in \mathbb{R},$$

one has

$$H\left(\mathrm{i}{y}_0\right)\in \mathbb{R}.$$

Also

$$\Xi \left(\mathrm{i}{y}_0\right)=\xi (1/2-y_0)>0$$

by 2. Set

$$c=-{\displaystyle \frac{\Xi \left(\mathrm{i}{y}_0\right)}{H\left(\mathrm{i}{y}_0\right)}}\in \mathbb{R},$$

and put

$$F=\Xi +cH.$$

Then

$$F\left(\mathrm{i}{y}_0\right)=0.$$

Since H is in the common kernel,

$${\Lambda }_j\left(F\right)={\Lambda }_j(\Xi )+c{\Lambda }_j\left(H\right)={\Lambda }_j(\Xi )$$

for all j.

The corresponding s-plane point is

$$s_0=1/2-y_0\in (0,1/2),$$

so it is a real off-critical zero of ${\Lambda }_F$. □

Corollary 9 

(Finite theta-kernel data do not force RH-like zero-freeness). No finite collection of real-linear theta-kernel tests, finitely many values of Ξ, finitely many derivatives of Ξ, or finitely many compactly supported real-linear pairings against the theta kernel can force RH-like zero-freeness of the associated completed s-plane functions inside the theta-cosine perturbation class

$${\mathcal{X}}_{\Theta }.$$

More precisely, for every such finite collection there is $F\in {\mathcal{X}}_{\Theta }$ matching the finite data of Ξ whose associated completed function

$${\Lambda }_F\left(s\right)=F(-\mathrm{i}(s-1/2))$$

has a real off-critical zero in the critical strip. The corresponding z-plane zero lies on the boundary line $Rez=0$, not in the interior of $\mathcal{U}$.

Proof. 

Each such finite collection is a theta-cosine finite real-linear interface. Apply 77. □

Remark 78 (Scope of finite-interface nonclosure) 

The theta-cosine nonclosure theorem preserves the completed functional equation and real symmetry, and its perturbations arise from real compactly supported cosine-kernel changes. It does not claim preservation of Euler products, Dirichlet coefficients, Selberg-class axioms, or prime-number arithmetic. Its role is to show that finite analytic interface data, even when theta-cosine in form, do not determine RH-like zero-freeness of the associated completed s-plane function. The specific obstruction constructed above is a real off-critical strip zero; in the z-plane it lies on the boundary line $Rez=0$, not inside $\mathcal{U}$.

Remark 79 (Nonclosure and absorbed criteria) 

The finite-interface nonclosure theorem concerns finite pieces of analytic data. It does not say that an exact all-stage theta-atlas criterion is outside the analysis. The all-stage theta-atlas criterion is absorbed as the positive fibre

$$\forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c),$$

which is exactly RH. Finite-interface nonclosure says that finite fragments do not occupy that fibre.

16. No-Transfer Benchmark Theorem

The no-transfer theorem is a benchmark calibration showing that exact selected-package architecture is compatible with truth, falsity, provability, refutability, and, under the usual hypotheses, independence. It is not the final nullity theorem; the final theorem quantifies over arbitrary sound fully invariant resolvers and then derives invariance for pure selected-parametric resolvers. When proof-status labels are involved, the final nullity theorem is stated for consistent background theories T.

Definition 90 

(Exact two-sided selected architecture). A proposition P, represented when necessary by an arithmetic sentence, has an exact two-sided selected architecture if there are decidable $R_P(n,c)$ and $B_P\left(w\right)$ such that

$$P\iff \forall n\exists c{R}_P(n,c),$$

and

$$\neg P\iff \exists w{B}_P\left(w\right).$$

It carries its positive realizer fibre

$${\mathcal{F}}_P^{+}=\{e:\forall n\exists t({\mathsf{T}}_{\mathrm{K}}(e,n,t)\wedge R_P(n,{\mathsf{U}}_{\mathrm{K}}\left(t\right)))\},$$

and its negative fibre

$${\mathcal{F}}_P^{-}=\{w:B_P\left(w\right)\}.$$

Theorem 80 

(Provable benchmark). There is a proposition $P_{\top }$ with exact two-sided selected architecture, with ${\Pi }_2^0$-complete positive realizer fibre, such that

$$\mathsf{Q}⊢P_{\top }.$$

Proof. 

Let

$$P_{\top }:=\forall n\exists c(c=0).$$

Let

$$R_{\top }(n,c)\iff c=0,B_{\top }\left(w\right)\iff 0=1.$$

Then $\mathsf{Q}⊢P_{\top }$, and the architecture is exact. The positive realizer fibre is nonempty, so it is ${\Pi }_2^0$-complete by 63. □

Theorem 81 

(Refutable benchmark). There is a proposition $P_{\perp }$ with exact two-sided selected architecture such that

$$\mathsf{Q}⊢\neg P_{\perp }.$$

Proof. 

Let

$$P_{\perp }:=0=1.$$

Let

$$R_{\perp }(n,c)\iff 0=1,B_{\perp }\left(w\right)\iff w=0.$$

Then

$$P_{\perp }\iff \forall n\exists c{R}_{\perp }(n,c),$$

both sides being false, and

$$\neg P_{\perp }\iff \exists w{B}_{\perp }\left(w\right),$$

both sides being true. Also

$$\mathsf{Q}⊢\neg P_{\perp }.$$

□

Theorem 82 

(Independent benchmark). Let T be a consistent recursively axiomatizable extension of sufficient arithmetic, equipped with a witnessed proof predicate whose checking relation is primitive recursive, proves exactly the same theorem set as the ordinary recursive axiomatization of T, and is represented, under the coding conventions of this paper, by a bounded arithmetic formula. There is a T-independent ${\Pi }_1$ sentence $G_T$ with exact two-sided selected architecture.

Proof. 

Use Rosser’s construction for T, with proof codes expanded to contain computation histories witnessing that alleged axiom instances enter the recursive axiom enumeration. The witnessed proof predicate is assumed to prove exactly the same theorem set as the original recursively axiomatized theory T, while making proof checking primitive recursive in the witnessed sense stipulated. Let the resulting Rosser sentence be written

$$G_T\equiv \forall w{\delta }_T\left(w\right),$$

where ${\delta }_T$ is bounded. By Rosser’s theorem, $G_T$ is independent of T, assuming T is consistent.

Define

$$P:=G_T,\widehat{P}:=G_T.$$

Let

$$B_P\left(w\right)\iff \neg {\delta }_T\left(w\right),$$

$$C_P(n,c)\iff c=0\wedge \forall w<n{\delta }_T\left(w\right),$$

and

$$D_P(w,u)\iff (u=0\wedge \neg {\delta }_T\left(w\right)).$$

Then

$$B_P\left(w\right)\iff \exists u{D}_P(w,u),$$

and

$$P\iff \forall w{\delta }_T\left(w\right)\iff \forall w\neg B_P\left(w\right).$$

Also,

$$\forall n\exists c{C}_P(n,c)$$

holds iff every finite initial segment satisfies ${\delta }_T$, which is equivalent in the standard model to

$$\forall w{\delta }_T\left(w\right).$$

Thus

$$P\iff \forall w\neg B_P\left(w\right)\iff \forall n\exists c{C}_P(n,c)\iff \mathbb{N}\vDash \widehat{P}.$$

This is an exact selected architecture for the T-independent sentence $G_T$. □

Definition 91 

(Architecture-forced conclusion). A fixed-instance conclusion $\Phi \left(P\right)$ is architecture-forced if it holds for every proposition P with exact two-sided selected architecture and the associated universal theorems above.

Theorem 83 

(No-transfer theorem). Let $T\supseteq \mathsf{Q}$, and restrict attention to propositions P represented by arithmetic sentences when proof-theoretic predicates are mentioned. None of the following fixed-instance conclusions is architecture-forced:

$$P,\neg P,T⊬P,T⊬\neg P,{Ind}_T\left(P\right).$$

Consequently, the selected-resolution architecture, Selection Jump, finite-prefix failure, truth-enriched barriers, reflection collapse under its stated hypotheses, diagonal barriers, computable triage collapse, theta-cosine finite-interface nonclosure, criterion absorption, and admissibility-filter laundering cannot by themselves imply proof, disproof, or theory-relative independence of an isolated sentence. For consistent sufficiently strong recursively axiomatizable T, the independent benchmark shows that the same architecture is also compatible with T-independence.

Proof. 

The conclusion P is refuted by $P_{\perp }$. The conclusion $\neg P$ is refuted by $P_{\top }$. The conclusion $T⊬P$ is refuted by $P_{\top }$ whenever $T\supseteq \mathsf{Q}$. The conclusion $T⊬\neg P$ is refuted by $P_{\perp }$. The conclusion ${Ind}_T\left(P\right)$ is refuted by either $P_{\top }$ or $P_{\perp }$, since both are decided by $T\supseteq \mathsf{Q}$. The independent benchmark supplies compatibility with independence under the usual consistency and strength assumptions. □

17. Universal A-Priori Selected-Logical Nullity

The preceding no-transfer theorem is a benchmark calibration. The final theorem is stronger. It does not say merely that a chosen list of methods fails, nor that a chosen theorem stack is incomplete. It says that every sound resolver whose output is invariant under arbitrary replacement of exact selected packages is identically empty. Then the selected-parametricity theorem derives this invariance for pure selected-logical resolvers. The enriched version says the same for enriched exact criterion packages: adjoining exact number-theoretic channels does not remove nullity at the pure invariant level.

When the fixed-instance label set includes proof-status labels

$${\mathbf{Prov}}_T,{\mathbf{Ref}}_T,{\mathbf{Unprov}}_T,{\mathbf{Unref}}_T,{\mathbf{Ind}}_T,$$

the universal selected-resolution nullity theorem is stated for consistent background theories $T\supseteq \mathsf{Q}$. This is an unconditional metatheorem about all consistent such theories. The consistency antecedent is essential: if T is inconsistent, then $T⊢\widehat{P}$ and $T⊢\neg \widehat{P}$ for every arithmetic representative $\widehat{P}$, so ${\mathbf{Prov}}_T$ and ${\mathbf{Ref}}_T$ become common sound labels for every package under the label semantics used here.

The nullity is recorded in four matched layers:

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,$$

$${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀,$$

and, for every consistent $T\supseteq \mathsf{Q}$,

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀,$$

$$\bigcap _{{\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}}{Act}_T\left({\mathcal{P}}^{*}\right)=⌀.$$

17.1. Clause-Prior Bivalent Nullity

Definition 92 

(Clause-prior bivalent resolvent). Let σ be a sentence interpreted in a partial evaluator $\mathcal{E}$ with strong Kleene truth values. Define the bivalent resolvent of σ in $\mathcal{E}$ by

$${BivRes}_{\mathcal{E}}\left(\sigma \right):=\{\mathsf{T}:〚\sigma {〛}_{\mathcal{E}}=\mathsf{T}\}\cup \{\mathsf{F}:〚\sigma {〛}_{\mathcal{E}}=\mathsf{F}\}.$$

Thus ${BivRes}_{\mathcal{E}}\left(\sigma \right)$ records whether the evaluator itself supplies a bivalent sign for σ.

Theorem 84 

(Clause-prior bivalent nullity). For the Dirichlet evaluator ${\mathcal{E}}_{\mathrm{Dir}}$,

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀.$$

Proof. 

By 2,

$$〚{\mathsf{RH}}_0{〛}_{{\mathcal{E}}_{\mathrm{Dir}}}=\mathsf{G}.$$

Hence neither $\mathsf{T}$ nor $\mathsf{F}$ belongs to ${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)$. □

Corollary 10 

(Zero is completion-posterior). For every $a\in \mathcal{S}$, the atomic formula

$$\zeta \left(a\right)=0$$

is not bivalent in ${\mathcal{E}}_{\mathrm{Dir}}$. Therefore a bivalent strip zero-set for ζ is not a clause-prior object. It exists only after a completion of the Dirichlet clause package.

Proof. 

If $a\in \mathcal{S}$, then $\zeta \left(a\right)$ is undefined in ${\mathcal{E}}_{\mathrm{Dir}}$. Since equality is strict, the atom $\zeta \left(a\right)=0$ has value $\mathsf{G}$. □

17.2. Completion-Invariant Nullity

Definition 93 

(Completion-invariant resolvent). Let $\mathcal{C}$ be a class of completions of a partial evaluator, and let σ be a sentence. Define

$${CompInvRes}_{\mathcal{C}}\left(\sigma \right):=\{\mathsf{T}:\forall \mathcal{E}\in \mathcal{C}〚\sigma {〛}_{\mathcal{E}}=\mathsf{T}\}\cup \{\mathsf{F}:\forall \mathcal{E}\in \mathcal{C}〚\sigma {〛}_{\mathcal{E}}=\mathsf{F}\}.$$

Thus ${CompInvRes}_{\mathcal{C}}\left(\sigma \right)$ records the bivalent signs that are invariant over the completion class.

Theorem 85 

(Completion-invariant nullity). Let

$${\mathcal{C}}_{\mathcal{S}}:=\{{\mathcal{E}}_{\kappa }:\kappa :\mathcal{S}\to \mathbb{C}\}$$

be the class of strip-total completions. Then

$${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀.$$

If the base evaluator is included, then

$${Spec}_{\left\{{\mathcal{E}}_{\mathrm{Dir}}\right\}\cup {\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=\{\mathsf{G},\mathsf{T},\mathsf{F}\}.$$

Proof. 

By the arbitrary strip zero-set realization theorem 6, there are completions in ${\mathcal{C}}_{\mathcal{S}}$ for which ${\mathsf{RH}}_0$ is true and completions in which it is false. Therefore neither $\mathsf{T}$ nor $\mathsf{F}$ is invariant over ${\mathcal{C}}_{\mathcal{S}}$.

Including ${\mathcal{E}}_{\mathrm{Dir}}$ adds the gap value by 2. □

17.3. Analytic Selection as a Formation Operation

Definition 94 

(Analytic selector). Let ${Sel}_{\mathrm{AC}}$ denote the operation that restricts the completion class to the unique meromorphic continuation agreeing with the Dirichlet series on $Re\left(s\right)>1$. Thus

$${Sel}_{\mathrm{AC}}\left({\mathsf{RH}}_0\right)$$

is the selected classical proposition

$$\mathsf{RH}:⟺{\mathcal{E}}_{\mathrm{AC}}\vDash {\mathsf{RH}}_0.$$

Theorem 86 

(Analytic selection forms the classical proposition). The analytic selector ${Sel}_{\mathrm{AC}}$ is a formation operation:

$${Sel}_{\mathrm{AC}}\left({\mathsf{RH}}_0\right)=\mathsf{RH}.$$

It does not convert a clause-prior or completion-invariant sign into a theorem, because both prior resolvents are null:

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,{CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀.$$

Proof. 

By the identity theorem, there is exactly one meromorphic continuation to $\mathbb{C}\setminus \left\{1\right\}$ agreeing with the Dirichlet series on the half-plane $Re\left(s\right)>1$. This defines ${\mathcal{E}}_{\mathrm{AC}}$, and hence defines

$$\mathsf{RH}:⟺{\mathcal{E}}_{\mathrm{AC}}\vDash {\mathsf{RH}}_0.$$

The two displayed nullities are . □

17.4. Selected-Logical Theorem Stacks and Maximal Invariant Closure

Definition 95 

(Selected-logical theorem stack). A theorem stack Γ about exact selected packages is calledselected-logicalif every theorem in Γ is invariant under replacement of one exact selected package by another exact selected package.

Equivalently, Γ is selected-logical if every exact selected package satisfies Γ. Thus selected-logical theorems are theorems of the selected package form itself, not theorems depending on the accidental identity of a particular selected package.

Definition 96 

(Full selected-logical invariant closure). Let

$${\Gamma }_{\mathrm{sel}}^{max}$$

denote the full selected-logical invariant closure of exact selected packages: the class of all sentences, schemes, and theorem forms in the language of exact selected packages that are satisfied by every exact selected package.

Equivalently, ${\Gamma }_{\mathrm{sel}}^{max}$ is the complete a-priori theory of exact selected package form under arbitrary replacement of one exact selected package by another. It is not merely the collection of selected-logical theorems proved up to a stage; by definition it contains every selected-logical invariant consequence.

Definition 97 

(Fixed-instance label schemas). Let $T\supseteq \mathsf{Q}$ be a recursively axiomatizable theory. Let

$${\mathcal{L}}_T=\{\mathbf{True},\mathbf{False},{\mathbf{Prov}}_T,{\mathbf{Ref}}_T,{\mathbf{Unprov}}_T,{\mathbf{Unref}}_T,{\mathbf{Ind}}_T\}.$$

For an exact selected package

$$\mathcal{P}=(P,B,C,D,\widehat{P}),$$

these labels mean respectively:

$$P,\neg P,T⊢\widehat{P},T⊢\neg \widehat{P},T⊬\widehat{P},T⊬\neg \widehat{P},{Ind}_T\left(\widehat{P}\right).$$

When the universal common-label nullity theorem is invoked for this label set, T is assumed consistent.

Definition 98 

(Uniform selected-logical fixed-instance resolvent). Let Γ be a selected-logical theorem stack. Define

$${LogRes}_T^{\mathrm{unif}}\left(\Gamma \right):=\{\Lambda \in {\mathcal{L}}_T:\Gamma {\vDash }_{\mathrm{sel}}^{\mathrm{unif}}\Lambda \},$$

where

$$\Gamma {\vDash }_{\mathrm{sel}}^{\mathrm{unif}}\Lambda$$

means that every exact selected package satisfying Γ satisfies the fixed-instance label schema Λ.

For the selected RH package

$${\mathsf{RH}}^{♯}=(\mathsf{RH},{\mathcal{W}}_{\Xi }^{\Theta }),$$

write

$${LogRes}_T^{\mathrm{unif}}({\Gamma }_{\mathrm{sel}}^{max};{\mathsf{RH}}^{♯})$$

for the uniform fixed-instance resolvent of the maximal invariant selected-logical closure as applied to the selected RH package. This notation records the package at which the invariant analysis is being discussed. It does not mean the collection of all metamathematical facts about the particular sentence $\widehat{\mathsf{RH}}$, and it does not mean the consequence relation obtained by adding the full instance-specific identity of ${\mathsf{RH}}^{♯}$ as an extra premise. It means that ${\mathsf{RH}}^{♯}$ is being viewed as an instance of the general exact selected-package form, while the forcing relation remains uniform over all exact selected packages. By 60 and the resolver notation convention, this use of ${\mathsf{RH}}^{♯}$ means the exact package ${\mathcal{P}}_{\mathsf{RH}}$.

17.5. Universal Selected-Resolution Nullity

Definition 99 

(Resolver-level package class). In the resolver and proof-status sections, let

$$\mathfrak{Pkg}$$

denote the class of arithmetically represented exact selected packages. Thus an element of $\mathfrak{Pkg}$ is a tuple

$$\mathcal{P}=(P,B_P,C_P,D_P,\widehat{P})$$

such that:

• P is a proposition;
• $\widehat{P}$ is an arithmetic sentence;
• $B_P\left(w\right)$ is a decidable finite negative-witness predicate;
• $C_P(n,c)$ is a decidable all-stage positive-certificate predicate;
• $D_P(w,u)$ is a decidable accepting-trace predicate for the negative verifier;

and, in the intended standard semantics,

$$P\iff \forall w\neg B_P\left(w\right),$$

$$P\iff \forall n\exists c{C}_P(n,c),$$

$$B_P\left(w\right)\iff \exists u{D}_P(w,u),$$

and

$$P\iff \mathbb{N}\vDash \widehat{P}.$$

This is the package class used whenever the proof-status labels

$${\mathbf{Prov}}_T,{\mathbf{Ref}}_T,{\mathbf{Unprov}}_T,{\mathbf{Unref}}_T,{\mathbf{Ind}}_T$$

are mentioned.

Definition 100 

(Resolver-level enriched package class). Let

$${\mathfrak{Pkg}}^{*}$$

denote the class of arithmetically represented enriched exact selected packages in the sense of 67. If a fixed enriched arity $(I,J)$ is under discussion, ${\mathfrak{Pkg}}^{*}$ may be read as the class of packages with that fixed arity; all benchmark packages used below are then taken with that same arity. For

$${\mathcal{P}}^{*}=(P,\widehat{P},{(B_i,D_i)}_{i<I},{\left(C_j\right)}_{j<J}),$$

the actual fixed-instance label set ${Act}_T\left({\mathcal{P}}^{*}\right)\subseteq {\mathcal{L}}_T$ is defined exactly as for ordinary packages:

$$\mathbf{True}\in {Act}_T\left({\mathcal{P}}^{*}\right)\iff P,$$

$$\mathbf{False}\in {Act}_T\left({\mathcal{P}}^{*}\right)\iff \neg P,$$

$${\mathbf{Prov}}_T\in {Act}_T\left({\mathcal{P}}^{*}\right)\iff T⊢\widehat{P},$$

$${\mathbf{Ref}}_T\in {Act}_T\left({\mathcal{P}}^{*}\right)\iff T⊢\neg \widehat{P},$$

and similarly for

$${\mathbf{Unprov}}_T,{\mathbf{Unref}}_T,{\mathbf{Ind}}_T.$$

Convention 87 

(Enriched arity in nullity arguments). In enriched selected-resolution nullity statements, ${\mathfrak{Pkg}}^{*}$ is read either as a fixed finite-arity class or as a disjoint union of such classes. In the fixed-arity reading, the arity $(I,J)$ is fixed before the resolver is considered, and all benchmark packages used in the nullity proof have exactly that same arity. This prevents arity from becoming an observable package coordinate.

Remark 88. 

Earlier selected-bridge statements may be read for abstract exact packages. The resolver-level nullity theorem, however, uses proof-theoretic labels and therefore requires an arithmetic representative $\widehat{P}$. Accordingly, $\mathfrak{Pkg}$ denotes the arithmetically represented exact selected packages in the resolver sections, while ${\mathfrak{Pkg}}^{*}$ denotes the arithmetically represented enriched exact selected packages.

Definition 101 

(Actual fixed-instance label set). Let $T\supseteq \mathsf{Q}$ be a recursively axiomatizable theory. For an exact selected package

$$\mathcal{P}=(P,B_P,C_P,D_P,\widehat{P}),$$

define

$${Act}_T\left(\mathcal{P}\right)\subseteq {\mathcal{L}}_T$$

by:

$$\mathbf{True}\in {Act}_T\left(\mathcal{P}\right)\iff P,$$

$$\mathbf{False}\in {Act}_T\left(\mathcal{P}\right)\iff \neg P,$$

$${\mathbf{Prov}}_T\in {Act}_T\left(\mathcal{P}\right)\iff T⊢\widehat{P},$$

$${\mathbf{Ref}}_T\in {Act}_T\left(\mathcal{P}\right)\iff T⊢\neg \widehat{P},$$

$${\mathbf{Unprov}}_T\in {Act}_T\left(\mathcal{P}\right)\iff T⊬\widehat{P},$$

$${\mathbf{Unref}}_T\in {Act}_T\left(\mathcal{P}\right)\iff T⊬\neg \widehat{P},$$

and

$${\mathbf{Ind}}_T\in {Act}_T\left(\mathcal{P}\right)\iff T⊬\widehat{P}\mathrm{and}T⊬\neg \widehat{P}.$$

For enriched packages, ${Act}_T\left({\mathcal{P}}^{*}\right)$ is defined by the same label clauses using the proposition and arithmetic representative of ${\mathcal{P}}^{*}$.

Definition 102 

(Selected-label resolver). A selected-label resolver over exact selected packages is any assignment

$$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right).$$

No computability, definability, finiteness, recursive axiomatizability, proof-search, or syntactic-effectivity restriction is imposed on $\mathcal{R}$.

Definition 103 

(Sound selected-label resolver). A resolver $\mathcal{R}$ is T-sound if, for every exact selected package $\mathcal{P}$,

$$\mathcal{R}\left(\mathcal{P}\right)\subseteq {Act}_T\left(\mathcal{P}\right).$$

Thus every label output by $\mathcal{R}$ is actually correct for the package to which it is assigned.

Definition 104 

(Full selected-package replacement invariance). A resolver $\mathcal{R}$ is fully selected-package invariant if, for all exact selected packages

$$\mathcal{P},\mathcal{Q}\in \mathfrak{Pkg},$$

one has

$$\mathcal{R}\left(\mathcal{P}\right)=\mathcal{R}\left(\mathcal{Q}\right).$$

This is the resolver-level form of selected-logical invariance: the output depends only on what is common to all exact selected packages, not on the accidental identity of a particular package.

Remark 89 (Arbitrary resolvers versus pure selected-parametric resolvers) 

Full selected-package invariance is a semantic replacement condition. For arbitrary resolvers it is a hypothesis, not a conclusion. Many sound resolvers are not fully invariant. For example, the resolver

$${\mathcal{R}}_0\left(\mathcal{P}\right)=\left\{\begin{array}{cc}\left\{\mathbf{False}\right\},\hfill & B_P\left(0\right),\hfill \\ ⌀,\hfill & \neg B_P\left(0\right),\hfill \end{array}\right.$$

is sound for exact selected packages, because $B_P\left(0\right)$ implies $\exists w{B}_P\left(w\right)$, hence $\neg P$. But it is not invariant under arbitrary package replacement.

The point of the selected-parametric meta theory is that pure selected-logical resolvers are not arbitrary. Their package variable is opaque and their language contains no operation capable of observing $B_P\left(w\right)$, $C_P(n,c)$, $D_P(w,u)$, package codes, truth values, proof statuses, or coordinate-bearing data. Therefore, for pure selected-parametric resolvers, full replacement invariance is derived by the selected parametricity theorem below.

Warning 90 (Why consistency of T is required for proof-status nullity) 

If T is inconsistent, then for every arithmetic sentence $\sigma$,

$$T⊢\sigma \mathrm{and}T⊢\neg \sigma .$$

Hence for every package $\mathcal{P}$, the labels

$${\mathbf{Prov}}_T,{\mathbf{Ref}}_T$$

belong to ${Act}_T\left(\mathcal{P}\right)$. The constant resolver

$$\mathcal{R}\left(\mathcal{P}\right)=\{{\mathbf{Prov}}_T,{\mathbf{Ref}}_T\}$$

is then T-sound and fully selected-package invariant but nonempty. Thus the universal selected-resolution nullity theorem for the label set ${\mathcal{L}}_T$ must be read as an unconditional metatheorem about all consistent $T\supseteq \mathsf{Q}$, not as a statement about inconsistent theories.

Definition 105 

(Enriched selected-label resolver). An enriched selected-label resolver is an assignment

$${\mathcal{R}}^{*}:{\mathfrak{Pkg}}^{*}\to \mathcal{P}\left({\mathcal{L}}_T\right).$$

It is T-sound if

$${\mathcal{R}}^{*}\left({\mathcal{P}}^{*}\right)\subseteq {Act}_T\left({\mathcal{P}}^{*}\right)({\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}).$$

It is fully enriched-package invariant if

$${\mathcal{R}}^{*}\left({\mathcal{P}}^{*}\right)={\mathcal{R}}^{*}\left({\mathcal{Q}}^{*}\right)({\mathcal{P}}^{*},{\mathcal{Q}}^{*}\in {\mathfrak{Pkg}}^{*}).$$

17.5.1. Selected Parametricity and Fibre-Observation Effects

Full replacement invariance is not a property of arbitrary resolvers. It can, however, be made a theorem for a deliberately restricted resolver language whose package arguments are opaque. This is the selected analogue of representation independence or relational parametricity in programming-language semantics.

Definition 106 

(Stipulated pure selected-parametric resolver language). The pure selected-parametric resolver language is the smallest resolver language generated by the following clauses:

• it has an opaque package variable $\mathcal{P}$;
• it may use selected bridge schemes that are valid for every exact selected package, such as the displayed package-form equivalences in 2;
• it may use ordinary logical rules and replacement-stable set operations on resolver outputs when those operations do not introduce package-specific observations;
• every primitive rule, axiom, and constructor is stable under arbitrary replacement of one exact selected package by another;
• it has no eliminator for concrete fibre facts

  $$B_P\left(w\right),C_P(n,c),D_P(w,u);$$
• it has no package-code equality, package-code reflection, semantic truth oracle, proof-status oracle, or coordinate-bearing analytic or arithmetic primitive attached to a particular package.

A resolver is pure selected-parametric if it is generated by this language. The enriched pure resolver language is defined similarly, with the enriched package variable opaque and with no eliminator for any of its concrete negative or positive fibres.

Definition 107 

(Pure selected-parametric resolver). A resolver is pure selected-parametric if it is generated in the resolver language of 106. Thus the package variable is opaque and every formation rule is stable under arbitrary replacement of exact selected packages. The pure language contains selected bridge schemes and theorem forms valid uniformly for every exact selected package, but contains no operation that can inspect the identity, code, concrete fibre occupation, arithmetic representative, semantic truth value, proof status, or coordinate-bearing data of a particular package.

Theorem 91 

(Selected parametricity). Every pure selected-parametric resolver is fully selected-package invariant. Likewise, every pure enriched selected-parametric resolver is fully enriched-package invariant.

Proof. 

This is the abstraction theorem for the stipulated pure resolver language of 106. The proof is by induction on the generation of resolver terms or resolver constructions in that language. Each primitive axiom, bridge scheme, and inference rule is replacement-stable by definition of the language. The package variable is opaque, and there is no primitive capable of observing the package identity, code, concrete fibre occupation, arithmetic representative, semantic truth value, proof status, or coordinate-bearing data of a particular package. Therefore replacing one exact selected package by any other leaves the denotation of every generated resolver unchanged. The enriched case is identical: the enriched package variable is opaque, and the pure enriched language has no eliminator for its concrete fibres or coordinates. □

Corollary 11 

(Pure selected-parametric nullity). Let $T\supseteq \mathsf{Q}$ be consistent. Every T-sound pure selected-parametric resolver is identically empty. Every T-sound pure enriched selected-parametric resolver is identically empty.

Proof. 

By 91, a pure selected-parametric resolver is fully selected-package invariant, and a pure enriched selected-parametric resolver is fully enriched-package invariant. Apply the universal selected-resolution nullity theorem and its enriched version below. □

Definition 108 

(Fibre and coordinate observation effects). A resolver construction has a negative-fibre observation effect if it may query concrete facts of the form $B_P\left(w\right)$ or accepting traces for such facts. It has a positive-fibre observation effect if it may use a complete all-stage stream of facts $C_P(n,c_n)$. It has a coordinate-bearing observation effect if it may use package-specific analytic or arithmetic data, such as the actual theta kernel of Ξ, actual winding certificates, Li coefficients, Lagarias inequalities, Euler-product data, or explicit-formula data.

Remark 92 (Effect discipline) 

Pure selected-parametric resolvers are invariant and hence null if sound for consistent T. A non-null resolver must either fail soundness, fail full replacement invariance, use an observation effect, or rely on an inconsistent background T in which proof-status labels collapse. Negative-fibre observation can yield non-null output only by finite occupation of a negative sign fibre. Positive-fibre observation can yield non-null output only by all-stage occupation of a positive sign fibre. Coordinate-bearing observation is package-specific and therefore lies outside the fully invariant selected-logical resolver regime.

Lemma 24 

(Common-label intersection criterion). Let $\mathfrak{C}$ be a class of objects, let L be a set of labels, and let

$$Act\left(X\right)\subseteq L$$

be the actual label set of $X\in \mathfrak{C}$. A sound fully invariant resolver can output only labels in

$$\bigcap _{X\in \mathfrak{C}}Act\left(X\right).$$

Consequently, if this intersection is empty, every sound fully invariant resolver is identically empty.

Proof. 

A fully invariant resolver has a constant output $S\subseteq L$. Soundness gives

$$S\subseteq Act\left(X\right)(X\in \mathfrak{C}),$$

hence

$$S\subseteq \bigcap _{X\in \mathfrak{C}}Act\left(X\right).$$

If the intersection is empty, then $S=⌀$. □

Lemma 25 

(Abstract invariant-nullity lemma). Let $\mathfrak{C}$ be a class of objects, let L be a set of labels, and let

$$Act\left(X\right)\subseteq L$$

be the actual label set of $X\in \mathfrak{C}$. Suppose that for every

$$\lambda \in L$$

there exists $X_{\lambda }\in \mathfrak{C}$ such that

$$\lambda \notin Act\left(X_{\lambda }\right).$$

If

$$R:\mathfrak{C}\to \mathcal{P}\left(L\right)$$

is sound,

$$R\left(X\right)\subseteq Act\left(X\right)(X\in \mathfrak{C}),$$

and invariant,

$$R\left(X\right)=R\left(Y\right)(X,Y\in \mathfrak{C}),$$

then

$$R\left(X\right)=⌀(X\in \mathfrak{C}).$$

Proof. 

By invariance there is a fixed set $S\subseteq L$ such that

$$R\left(X\right)=S(X\in \mathfrak{C}).$$

If $\lambda \in S$, then soundness gives

$$\lambda \in Act\left(X\right)(X\in \mathfrak{C}).$$

This contradicts the existence of $X_{\lambda }$ with

$$\lambda \notin Act\left(X_{\lambda }\right).$$

Thus $S=⌀$. □

Lemma 26 

(Exact-package label separation). Let $T\supseteq \mathsf{Q}$ be consistent. For every label

$$\Lambda \in {\mathcal{L}}_T,$$

there exists an exact selected package ${\mathcal{P}}_{\Lambda }$ such that

$$\Lambda \notin {Act}_T\left({\mathcal{P}}_{\Lambda }\right).$$

Proof. 

Two benchmark packages suffice.

First, let

$$P_{\top }:=\forall n\exists c(c=0).$$

Define

$$B_{\top }\left(w\right)\iff 0=1,C_{\top }(n,c)\iff c=0,$$

$$D_{\top }(w,u)\iff 0=1,{\widehat{P}}_{\top }:=\forall n\exists c(c=0).$$

Then $P_{\top }$ is an exact selected package and

$$\mathsf{Q}⊢{\widehat{P}}_{\top }.$$

Since $T\supseteq \mathsf{Q}$,

$$T⊢{\widehat{P}}_{\top }.$$

By consistency,

$$T⊬\neg {\widehat{P}}_{\top }.$$

Thus $P_{\top }$ refutes the labels

$$\mathbf{False},{\mathbf{Ref}}_T,{\mathbf{Unprov}}_T,{\mathbf{Ind}}_T.$$

Second, let

$$P_{\perp }:=0=1.$$

Define

$$B_{\perp }\left(w\right)\iff w=0,C_{\perp }(n,c)\iff 0=1,$$

$$D_{\perp }(w,u)\iff (w=0\wedge u=0),{\widehat{P}}_{\perp }:=0=1.$$

Then $P_{\perp }$ is an exact selected package and

$$\mathsf{Q}⊢\neg {\widehat{P}}_{\perp }.$$

Hence

$$T⊢\neg {\widehat{P}}_{\perp }.$$

By consistency,

$$T⊬{\widehat{P}}_{\perp }.$$

Thus $P_{\perp }$ refutes the labels

$$\mathbf{True},{\mathbf{Prov}}_T,{\mathbf{Unref}}_T,{\mathbf{Ind}}_T.$$

Every label in ${\mathcal{L}}_T$ is refuted by at least one of the two benchmark packages. □

Theorem 93 

(Ordinary common-label nullity). Let $T\supseteq \mathsf{Q}$ be consistent. Then

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀.$$

Proof. 

By 26, every label $\Lambda \in {\mathcal{L}}_T$ fails for at least one exact selected package. Hence no label lies in the intersection of all actual label sets. □

Theorem 94 

(Universal selected-resolution nullity). Let $T\supseteq \mathsf{Q}$ be consistent. Suppose

$$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right)$$

is T-sound and fully selected-package invariant. The invariance assumption is a hypothesis of the theorem for arbitrary resolvers, not a conclusion about all resolvers. Then

$$\mathcal{R}\left(\mathcal{P}\right)=⌀$$

for every exact selected package $\mathcal{P}\in \mathfrak{Pkg}$.

In particular,

$$\mathcal{R}\left({\mathsf{RH}}^{♯}\right)=\mathcal{R}\left({\mathcal{P}}_{\mathsf{RH}}\right)=⌀$$

for the selected RH package, where the equality is the convention identifying ${\mathsf{RH}}^{♯}$ with its exact selected package when used as a resolver argument.

Proof. 

By 93,

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀.$$

Now use the additional invariance hypothesis. Full selected-package invariance makes the resolver output a fixed set

$$S\subseteq {\mathcal{L}}_T$$

on all of $\mathfrak{Pkg}$. If $\Lambda \in S$, then T-soundness would require

$$\Lambda \in {Act}_T\left(\mathcal{P}\right)(\mathcal{P}\in \mathfrak{Pkg}).$$

Thus

$$S\subseteq \bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀.$$

So $S=⌀$. □

Theorem 95 

(Pure selected-resolution nullity). Let $T\supseteq \mathsf{Q}$ be consistent. If

$$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right)$$

is T-sound and pure selected-parametric, then

$$\mathcal{R}\left(\mathcal{P}\right)=⌀(\mathcal{P}\in \mathfrak{Pkg}).$$

In particular,

$$\mathcal{R}\left({\mathsf{RH}}^{♯}\right)=⌀.$$

Proof. 

By selected parametricity, $\mathcal{R}$ is fully selected-package invariant. Apply 94. □

Theorem 96 

(Universal enriched selected-resolution nullity). Let $T\supseteq \mathsf{Q}$ be consistent. Suppose

$${\mathcal{R}}^{*}:{\mathfrak{Pkg}}^{*}\to \mathcal{P}\left({\mathcal{L}}_T\right)$$

is T-sound and fully enriched-package invariant. Again, the invariance assumption is a hypothesis of the theorem for arbitrary enriched resolvers, not a property proved for all enriched resolvers. Then

$${\mathcal{R}}^{*}\left({\mathcal{P}}^{*}\right)=⌀({\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}).$$

In particular,

$${\mathcal{R}}^{*}\left({\mathcal{P}}_{\mathsf{RH}}^{*}\right)=⌀.$$

Proof. 

Full enriched-package invariance makes the resolver output a fixed set

$$S\subseteq {\mathcal{L}}_T$$

on every enriched exact selected package. Soundness requires every label in S to be correct for every enriched exact selected package.

Use enriched versions of the true and false benchmark packages. If a fixed enriched arity $(I,J)$ is under discussion, the following constructions use exactly that same arity.

For the true benchmark, take

$$P_{\top }:=\forall n\exists c(c=0),$$

and let

$${\widehat{P}}_{\top }:=\forall n\exists c(c=0).$$

For every negative channel $i<I$, set

$$B_i\left(w\right)\equiv 0=1,D_i(w,u)\equiv 0=1.$$

For every positive channel $j<J$, set

$$C_j(n,c)\equiv c=0.$$

This enriched package is exact and, by consistency of T, refutes the labels

$$\mathbf{False},{\mathbf{Ref}}_T,{\mathbf{Unprov}}_T,{\mathbf{Ind}}_T.$$

For the false benchmark, take

$$P_{\perp }:=0=1,$$

and let

$${\widehat{P}}_{\perp }:=0=1.$$

For every negative channel $i<I$, set

$$B_i\left(w\right)\equiv w=0,D_i(w,u)\equiv (w=0\wedge u=0).$$

For every positive channel $j<J$, set

$$C_j(n,c)\equiv 0=1.$$

This enriched package is exact and, by consistency of T, refutes the labels

$$\mathbf{True},{\mathbf{Prov}}_T,{\mathbf{Unref}}_T,{\mathbf{Ind}}_T.$$

Thus no label in ${\mathcal{L}}_T$ is correct for all enriched exact selected packages. Hence $S=⌀$. □

Theorem 97 

(Enriched common-label nullity). Let $T\supseteq \mathsf{Q}$ be consistent. Then

$$\bigcap _{{\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}}{Act}_T\left({\mathcal{P}}^{*}\right)=⌀.$$

Proof. 

The true and false enriched benchmark packages in the proof of 96 separate every label in ${\mathcal{L}}_T$. Hence no label is common to all enriched exact selected packages. □

Theorem 98 

(Pure enriched selected-resolution nullity). Let $T\supseteq \mathsf{Q}$ be consistent. If

$${\mathcal{R}}^{*}:{\mathfrak{Pkg}}^{*}\to \mathcal{P}\left({\mathcal{L}}_T\right)$$

is T-sound and pure enriched selected-parametric, then

$${\mathcal{R}}^{*}\left({\mathcal{P}}^{*}\right)=⌀({\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}).$$

In particular,

$${\mathcal{R}}^{*}\left({\mathcal{P}}_{\mathsf{RH}}^{*}\right)=⌀.$$

Proof. 

By selected parametricity, a pure enriched selected-parametric resolver is fully enriched-package invariant. Apply 96. □

Theorem 99 

(Nullity matching). Let $T\supseteq \mathsf{Q}$ be consistent. The following four invariant resolvents are empty:

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,$$

$${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀,$$

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀,$$

and

$$\bigcap _{{\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}}{Act}_T\left({\mathcal{P}}^{*}\right)=⌀.$$

Moreover, every sound pure selected-parametric resolver factors through the third intersection, and every sound pure enriched selected-parametric resolver factors through the fourth intersection. Hence all such resolvers are identically null.

Proof. 

The first statement is 84. The second is 85. The third is 93. The fourth is 97.

By selected parametricity, a pure selected-parametric resolver is fully replacement invariant. Therefore its output is constant and, if it is sound, must lie in the common-label intersection

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right).$$

This intersection is empty. The enriched case is identical. □

Corollary 12 

(Number-theoretic criteria are subject to enriched nullity). Let $T\supseteq \mathsf{Q}$ be consistent. The enriched selected RH criterion package ${\mathcal{P}}_{\mathsf{RH}}^{*}$, containing the Ξ-winding, Li-negative, Lagarias-negative, theta-atlas, zero-count, and half-plane-cover fibres, satisfies

$${\mathcal{R}}^{*}\left({\mathcal{P}}_{\mathsf{RH}}^{*}\right)=⌀$$

for every T-sound pure enriched selected-parametric resolver

$${\mathcal{R}}^{*}:{\mathfrak{Pkg}}^{*}\to \mathcal{P}\left({\mathcal{L}}_T\right).$$

More generally, the same conclusion holds for every T-sound fully enriched-package invariant resolver. Thus the exact number-theoretic criteria do not bypass selected-logical nullity; they are part of the enriched package to which nullity applies.

Proof. 

The pure version is 98; the arbitrary invariant version is 96. The fact that ${\mathcal{P}}_{\mathsf{RH}}^{*}$ is enriched exact is 44. □

Corollary 13 

(No invariant bivalent truth resolver). Let a partial truth resolver be a partial assignment

$$\tau :\mathfrak{Pkg}\rightharpoonup \{\mathbf{True},\mathbf{False}\}.$$

Assume that τ is sound and fully selected-package invariant in the sense that whenever it is defined, it gives the same correct truth label under arbitrary replacement of exact selected packages. Then τ is nowhere defined.

In particular,

$$\tau \left({\mathsf{RH}}^{♯}\right)\uparrow .$$

The same statement holds for enriched packages with $\mathfrak{Pkg}$ replaced by ${\mathfrak{Pkg}}^{*}$. It also holds for pure selected-parametric partial truth resolvers, because their invariance is derived by selected parametricity.

Proof. 

If $\tau$ were defined at one package, full selected-package invariance would make it defined with the same value at every package. If the common value were $\mathbf{True}$, soundness would fail at $P_{\perp }$. If the common value were $\mathbf{False}$, soundness would fail at $P_{\top }$. Hence $\tau$ is nowhere defined. The enriched case uses the enriched benchmark packages from 96. The pure case follows from selected parametricity. □

Corollary 14 

(Non-nullity is fibre occupation or invariance failure). Let $T\supseteq \mathsf{Q}$ be consistent. Suppose

$${\mathcal{R}}^{*}:{\mathfrak{Pkg}}^{*}\to \mathcal{P}\left({\mathcal{L}}_T\right)$$

is T-sound and

$${\mathcal{R}}^{*}\left({\mathcal{P}}^{*}\right)\ne ⌀$$

for some enriched exact selected package ${\mathcal{P}}^{*}$. Then ${\mathcal{R}}^{*}$ is not fully enriched-package invariant. In particular, if a resolver is T-sound and pure enriched selected-parametric, such a non-null output is impossible. Any sound non-null construction at an enriched package must leave the pure replacement-invariant regime, for example by an observation effect, package-specific dependence, or an explicit oracle/sign assumption.

For the enriched selected RH package ${\mathcal{P}}_{\mathsf{RH}}^{*}$, a non-null sign outcome provided by one of its exact criterion channels is RH-specific coordinate-bearing reasoning. It is not a pure invariant selected label; it is sign-fibre occupation:

$$\exists w{B}_i\left(w\right)\iff \neg \mathsf{RH}$$

for a negative fibre, or

$$\forall n\exists c{C}_j(n,c)\iff \mathsf{RH}$$

for a positive fibre.

Proof. 

The first assertion is the contrapositive of 96. A pure enriched selected-parametric resolver is fully enriched-package invariant by 91. Thus any sound non-null pure resolver is impossible, and any sound non-null non-pure resolver must leave the pure regime by some observation effect or package-specific dependence. The second assertion is exactly the absorption theorem 44. □

Theorem 100 

(Invariant resolution dichotomy). Let $T\supseteq \mathsf{Q}$ be consistent, and let

$$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right)$$

be any selected-label resolver. Then at least one of the following holds:

• $\mathcal{R}$ is not T-sound;
• $\mathcal{R}$ is not fully selected-package invariant;
• $\mathcal{R}$ is identically null:

  $$\forall \mathcal{P}\in \mathfrak{Pkg}\mathcal{R}\left(\mathcal{P}\right)=⌀.$$

The analogous dichotomy holds for enriched selected-label resolvers on ${\mathfrak{Pkg}}^{*}$.

In particular, no resolver can be simultaneously sound, fully selected-package invariant, and non-null at ${\mathsf{RH}}^{♯}$, and no enriched resolver can be simultaneously sound, fully enriched-package invariant, and non-null at ${\mathcal{P}}_{\mathsf{RH}}^{*}$. If the resolver is pure selected-parametric, the second alternative is unavailable by selected parametricity.

Proof. 

If $\mathcal{R}$ is both T-sound and fully selected-package invariant, then 94 gives the identically null alternative. Thus any non-null resolver must fail soundness or fail full selected-package invariance. The enriched case is identical, using 96. For pure selected-parametric resolvers, full invariance follows from 91. □

Corollary 15 

(Maximal selected-logical uniform fixed-instance nullity). Let $T\supseteq \mathsf{Q}$ be a consistent recursively axiomatizable theory. Then

$${LogRes}_T^{\mathrm{unif}}\left({\Gamma }_{\mathrm{sel}}^{max}\right)=⌀.$$

Consequently, for the selected RH package,

$${LogRes}_T^{\mathrm{unif}}({\Gamma }_{\mathrm{sel}}^{max};{\mathsf{RH}}^{♯})=⌀.$$

Proof. 

The uniform consequence relation determined by ${\Gamma }_{\mathrm{sel}}^{max}$ induces a resolver

$${\mathcal{R}}_{max}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right)$$

by setting

$${\mathcal{R}}_{max}\left(\mathcal{P}\right)=\{\Lambda \in {\mathcal{L}}_T:{\Gamma }_{\mathrm{sel}}^{max}{\vDash }_{\mathrm{sel}}^{\mathrm{unif}}\Lambda \}.$$

By the definition of uniform selected-logical forcing, this resolver is fully selected-package invariant. It is also sound, since a uniformly forced label must hold for every exact selected package. Therefore

$${\mathcal{R}}_{max}\left(\mathcal{P}\right)=⌀$$

for every exact selected package $\mathcal{P}$, by 94. This is exactly the displayed nullity. □

Corollary 16 

(Universal selected-logical nullity for RH). Let $T\supseteq \mathsf{Q}$ be a consistent recursively axiomatizable theory. For the selected RH package

$${\mathsf{RH}}^{♯}=(\mathsf{RH},{\mathcal{W}}_{\Xi }^{\Theta }),$$

the following universal nullity holds: for every selected-label resolver

$$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right),$$

if $\mathcal{R}$ is T-sound and fully selected-package invariant, then

$$\mathcal{R}\left({\mathsf{RH}}^{♯}\right)=⌀.$$

If $\mathcal{R}$ is pure selected-parametric, the full-invariance premise is derived from selected parametricity, so

$${\mathsf{Sound}}_T\left(\mathcal{R}\right)\wedge \mathsf{PureSel}\left(\mathcal{R}\right)\Rightarrow \mathcal{R}\left({\mathsf{RH}}^{♯}\right)=⌀.$$

For the enriched selected RH criterion package ${\mathcal{P}}_{\mathsf{RH}}^{*}$, every T-sound fully enriched-package invariant resolver satisfies

$${\mathcal{R}}^{*}\left({\mathcal{P}}_{\mathsf{RH}}^{*}\right)=⌀.$$

If ${\mathcal{R}}^{*}$ is pure enriched selected-parametric, then again full invariance is derived. In particular,

$${LogRes}_T^{\mathrm{unif}}({\Gamma }_{\mathrm{sel}}^{max};{\mathsf{RH}}^{♯})=⌀.$$

Proof. 

The selected RH package is an exact selected package by 60. Therefore 94 applies directly. The pure case follows from 95. The enriched statement is 96 applied to ${\mathcal{P}}_{\mathsf{RH}}^{*}$, which is enriched exact by 44; the pure enriched case is 98. The ${\Gamma }_{\mathrm{sel}}^{max}$-statement follows from 15. □

17.6. The Combined Nullity Theorem

Theorem 101 

(Universal a-priori selected-logical nullity theorem). The following two nullities hold unconditionally:

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,$$

$${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀.$$

Moreover, for every consistent recursively axiomatizable theory $T\supseteq \mathsf{Q}$, the ordinary and enriched common-label nullities hold:

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀,\bigcap _{{\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}}{Act}_T\left({\mathcal{P}}^{*}\right)=⌀.$$

Consequently:

• for every arbitrary resolver

  $$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right),$$
  if $\mathcal{R}$ is T-sound and fully selected-package invariant, then

  $$\mathcal{R}\left(\mathcal{P}\right)=⌀(\mathcal{P}\in \mathfrak{Pkg});$$
• for every pure selected-parametric resolver

  $$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right),$$
  if $\mathcal{R}$ is T-sound, then

  $$\mathcal{R}\left(\mathcal{P}\right)=⌀(\mathcal{P}\in \mathfrak{Pkg});$$
• for every arbitrary enriched resolver

  $${\mathcal{R}}^{*}:{\mathfrak{Pkg}}^{*}\to \mathcal{P}\left({\mathcal{L}}_T\right),$$
  if ${\mathcal{R}}^{*}$ is T-sound and fully enriched-package invariant, then

  $${\mathcal{R}}^{*}\left({\mathcal{P}}^{*}\right)=⌀({\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*});$$
• for every pure enriched selected-parametric resolver

  $${\mathcal{R}}^{*}:{\mathfrak{Pkg}}^{*}\to \mathcal{P}\left({\mathcal{L}}_T\right),$$
  if ${\mathcal{R}}^{*}$ is T-sound, then

  $${\mathcal{R}}^{*}\left({\mathcal{P}}^{*}\right)=⌀({\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}).$$

In particular,

$$\mathcal{R}\left({\mathsf{RH}}^{♯}\right)=⌀$$

for every sound fully selected-package invariant resolver and for every sound pure selected-parametric resolver, and

$${\mathcal{R}}^{*}\left({\mathcal{P}}_{\mathsf{RH}}^{*}\right)=⌀$$

for every sound fully enriched-package invariant resolver and for every sound pure enriched selected-parametric resolver. Taking $\mathcal{R}$ to be the canonical resolver induced by the full invariant selected-logical closure gives

$${LogRes}_T^{\mathrm{unif}}({\Gamma }_{\mathrm{sel}}^{max};{\mathsf{RH}}^{♯})=⌀.$$

Proof. 

The first statement is 84. The second is 85. The ordinary and enriched common-label nullities are . The arbitrary-resolver nullity conclusions are . The pure conclusions follow by selected parametricity, 91, and are recorded in . The ${\Gamma }_{\mathrm{sel}}^{max}$-resolvent statement is 15. □

Remark 102 (Exact criteria are fibres, not bypasses) 

The nullity theorem is a theorem about pure invariant resolvers and, more generally, arbitrary resolvers satisfying full replacement invariance. It is not a dismissal of RH-specific analytic or number-theoretic reasoning. It does not say that a finite negative witness, if found, would fail to disprove RH; nor does it say that a complete positive theta-atlas stream, if produced, would fail to prove RH. Those are RH-specific sign-fibre occupations:

$$\exists w{B}_i\left(w\right)\iff \neg \mathsf{RH},\forall n\exists c{C}_j(n,c)\iff \mathsf{RH}.$$

Rather, the nullity theorem says that the unoccupied exact criterion architecture does not itself produce a fixed-instance label at the pure selected-logical level. Exact number-theoretic criteria do not bypass nullity; they become the fibres to which nullity applies.

18. Laddering and Regular Terminal-Route Normal Forms

Definition 109 

(Stagewise auxiliary target). An auxiliary target B for proving P is stagewise if it comes with a decidable predicate $R_B(n,c)$ such that

$$B\iff \forall n\exists c{R}_B(n,c).$$

Theorem 103 

(Stagewise target displacement). If an auxiliary target B for RH is stagewise, then proving B is proving a new all-stage selected closure:

$$\forall n\exists c{R}_B(n,c).$$

The corresponding realizer class

$$\{e:\forall n\exists t({\mathsf{T}}_{\mathrm{K}}(e,n,t)\wedge R_B(n,{\mathsf{U}}_{\mathrm{K}}\left(t\right)))\}$$

is either empty or ${\Pi }_2^0$-complete.

Proof. 

The first statement is the definition of stagewise. The second is 63. □

Definition 110 

(Regular terminal route). A regular terminal route for a selected proposition is a formally specified route whose terminal force is presented by one of the following explicitly listed mechanisms:

• a finite decidable selected artifact;
• a finite formal proof of a selected-level bridge theorem;
• an exact stagewise witness architecture

  $$\forall n\exists cR(n,c)$$
  with decidable R;
• a computable total classifier or generator;
• a same-theory certification principle over a recursively axiomatizable theory;
• an arithmetic truth-definitional terminality predicate;
• a diagonal-universal exact terminality predicate;
• a finite-prefix or finite-interface terminal premise;
• an admissibility-filter inference;
• an explicitly stated oracle or sign axiom;
• an absorbed exact criterion route, namely instance-specific number-theoretic information presented as an exact finite negative channel or an exact all-stage positive channel.

Definition 111 

(Regular-route outcomes). A regular terminal routefinitely terminatesif it supplies a finite selected artifact whose validity is decidable and whose validity implies a sign channel. Itsingularly compressesif it supplies a finite non-laundered selected theorem implying a selected sign channel. Itladdersif its correctness reduces to an exact stagewise condition. Itlaundersif its terminal force is merely a selected closure channel or a premise equivalent to one. Itcollapsesif it is generic enough to induce one of the global barriers proved above. It isabsorbed as an exact criterion fibreif it uses instance-specific number-theoretic information presented as an exact finite negative channel or exact all-stage positive channel. Such a route does not escape nullity; a non-null outcome occurs only by occupation of that fibre. An explicitly stated oracle or sign axiom is classified asoracular; it resolves only by assumption and is not a non-oracular selected-resolution mechanism.

Theorem 104 

(Regular terminal-route normal form). Every regular terminal route, in the precise stipulated sense of Section 18, finitely terminates, singularly compresses, ladders, launders, collapses, is absorbed as an exact criterion fibre, or is oracular. In the ladder case, the associated effective success class is either empty or ${\Pi }_2^0$-complete.

Proof. 

This is an exhaustion by the defining trace type of a regular route. Finite selected artifacts finitely terminate. Finite non-laundered selected bridge theorems singularly compress. Stagewise witness architectures ladder, and their effective success classes are empty or ${\Pi }_2^0$-complete by 63. Terminal premises equivalent to selected closure launder. Generic computable, reflective, truth-definitional, diagonal, finite-prefix, finite-interface, triage, or legacy filtered routes collapse by the corresponding theorems in . Instance-specific number-theoretic criteria presented as exact finite negative channels or all-stage positive channels are absorbed as exact criterion fibres by 46. Explicit sign axioms are oracular. These cases exhaust the formal definition of regularity. □

Remark 105 (Formal status of the normal form) 

thm:regular-route-normal-form is a taxonomic normal form relative to the definition of regular route. Its substantive mathematical content is not the exhaustion-by-definition itself, but the individual nonclosure and certificate theorems cited in its proof: theta-atlas completeness, winding-certificate soundness and completeness, criterion absorption, Selection Jump, computable triage failure, reflection collapse under its stated hypotheses, Tarski undefinability, diagonal contradiction, finite-prefix failure, finite-interface nonclosure, and legacy-filter laundering.

19. Main Synthesis

Theorem 106 

(Selected RH fibre theorem). For the selected RH package,

$$\neg \mathsf{RH}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right),$$

and

$$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

Equivalently,

$$\mathsf{RH}\iff \forall n\exists c{C}_{\Xi }^{\Theta }(n,c).$$

Thus the selected negative side is finite-certificate semidecidable, while the selected positive side is all-stage and is supplied by explicit theta-atlas certificates.

Proof. 

The negative equivalence is 35. The positive equivalence is 18, with $C_{\Xi }^{\Theta }(n,c)=\mathsf{ThetaAtlas}(n+5,c)$. □

Remark 107 (Finite negative side versus all-stage positive side) 

The finite negative channel and the all-stage positive theta-atlas channel have different certificate geometries. A negative certificate is a single exact positive winding count in one admissible rectangle. A positive certificate stream requires one zero-free atlas for each window $K_N$. The quantitative atlas bounds depend on ${min}_{K_N}|\Xi |$ and a derivative majorant, while the quantitative winding bounds depend on ${min}_{\partial Q}|\Xi |$, a derivative majorant on a compact guard, and a certified tube radius for the boundary mesh.

Theorem 108 

(Enriched selected RH criterion fibre theorem). For the enriched selected RH criterion package ${\mathcal{P}}_{\mathsf{RH}}^{*}$,

$$\exists w{B}_{\Xi }\left(w\right)\iff \exists v{B}_{\mathrm{Li}}\left(v\right)\iff \exists v{B}_{\mathrm{Lag}}\left(v\right)\iff \neg \mathsf{RH},$$

and

$$\forall n\exists c{C}_{\Theta }(n,c)\iff \forall n\exists c{C}_0(n,c)\iff \forall n\exists c{C}_{\mathrm{hp}}(n,c)\iff \mathsf{RH}.$$

The positive realizer fibres associated to $C_{\Theta },C_0,C_{\mathrm{hp}}$ are empty if $\neg \mathsf{RH}$, and are ${\Pi }_2^0$-complete if $\mathsf{RH}$. The finite negative fibres associated to $B_{\Xi },B_{\mathrm{Li}},B_{\mathrm{Lag}}$ are pairwise computably replay-equivalent.

Proof. 

This is . □

Theorem 109 

(Selected RH Selection Jump). Let

$${\mathcal{F}}_{\Xi ,\Theta }^{+}=\left\{e:\forall n\exists t\left({\mathsf{T}}_{\mathrm{K}}(e,n,t)\wedge \mathsf{ThetaAtlas}(n+5,{\mathsf{U}}_{\mathrm{K}}\left(t\right))\right)\right\}.$$

Then

$${\mathcal{F}}_{\Xi ,\Theta }^{+}\ne ⌀\iff \mathsf{RH}.$$

If $\mathsf{RH}$, then ${\mathcal{F}}_{\Xi ,\Theta }^{+}$ is ${\Pi }_2^0$-complete.

Proof. 

This is 65. □

Theorem 110 

(Global selected-resolution nonclosure). There is no universal terminal chart for selected witness packages containing the coded ${\Pi }_1$-selected class by any of the following mechanisms, under the hypotheses stated in the corresponding component theorems:

• total computable proof-disproof-independence triage;
• decidable independence classification over a sound r.e. theory T;
• same-theory self-certifying terminality on a ${\Pi }_1$-universal image, under the reflection-collapse hypotheses of 73;
• truth-definitional terminality on a truth-universal image;
• diagonal-universal terminality;
• fixed finite-prefix closure;
• fixed finite real-linear interface data in the broad ξ-symmetric ambient class;
• fixed finite real-linear theta-cosine interface data in ${\mathcal{X}}_{\Theta }$, for RH-like zero-freeness of the associated completed s-plane functions;
• legacy Dirichlet support-strict filtering.

Proof. 

These are respectively . □

Theorem 111 

(Main theorem). For every consistent recursively axiomatizable $T\supseteq \mathsf{Q}$, the selected RH object

$${\mathsf{RH}}^{♯}=(\mathsf{RH},{\mathcal{W}}_{\Xi }^{\Theta })$$

and its enriched criterion package ${\mathcal{P}}_{\mathsf{RH}}^{*}$ satisfy:

(I)

Clause-prior and completion-prior nullity.

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,{CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀.$$

(II)

Analytic selection.Analytic continuation is a formation operation:

$${Sel}_{\mathrm{AC}}\left({\mathsf{RH}}_0\right)=\mathsf{RH}.$$

(III)

$\Xi$-plane formulation.With

$$\Xi \left(z\right)=\xi (1/2+\mathrm{i}z),\mathcal{U}=\{x+\mathrm{i}y:x>0,0<y<1/2\},$$

one has

$$\mathsf{RH}\iff Z(\Xi ;\mathcal{U})=⌀.$$

(IV)

Explicit theta majorants.For every real $R\ge 0$ and $j\ge 0$,

$$\underset{\left|z\right|\le R}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le {\mathfrak{M}}_j\left(R\right),$$

where

$${\mathfrak{M}}_j\left(R\right)=4\left(2{\pi }^2{G}_4{I}_j(R+9/2)+3\pi G_2{I}_j(R+5/2)\right),$$

$$G_m={\displaystyle \sum _{n=1}^{\infty }}{n}^m{e}^{-\pi (n^2-1)},I_j\left(\alpha \right)={\int }_0^{\infty }{u}^j{e}^{\alpha u}{e}^{-\pi e^{2u}}\mathrm{d}u.$$

For rational radii R, and for any supplied rational upper radius $R^{+}\ge R_0$, all ingredients admit finite rational upper certificates.

(V)

Quantitative theta-atlas positive channel.For

$$K_N=[1/N,N]\times [1/N,1/2-1/N]\subset \mathcal{U}(N\ge 5),$$

there is a decidable rational certificate predicate

$$\mathsf{ThetaAtlas}(N,c)$$

such that

$$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

If

$${\delta }_N=\underset{K_N}{min}|\Xi |>0$$

and

$$M_N\ge \underset{\left|z\right|\le N+2}{sup}\left|{\Xi }^{\prime }\right|$$

is certified, then an accepted atlas exists with the explicit cell-count bound of 2. In finite rational records, the displayed $\sqrt{2}$ is replaced by any certified rational upper bound $q_2>0$ satisfying $q_2^2>2$. The bound is existential in terms of the actual ${\delta }_N$; effective construction is obtained from supplied rational lower bounds by 19.

(VI)

Exact finite negative fibre.

$$\neg \mathsf{RH}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right).$$

(VII)

Winding certificate calculus.The predicates ${\mathsf{WindCertRect}}_{\mathrm{AC}}$ and ${\mathsf{WindCertRect}}_{\Theta }$ are finite, rational, decidable exact-count predicates. Accepted certificates prove exact zero counts by a tube homotopy to a rational polygon and the argument principle. If a rectangle boundary is zero-free, a certificate exists. Quantitative mesh and precision bounds are given by 31; these bounds include an explicit certified tube-radius term ensuring that the boundary subarc disks remain inside the certified compact guard. As with the theta-atlas theorem, the displayed bound is an existential completeness bound when written in terms of the actual separation

$$d_Q=\underset{\partial Q}{min}|\Xi |.$$

Effective construction of the winding certificate requires supplied rational boundary lower-bound data, as in 32.

(VIII)

Classical-criterion witness transfer.The Ξ-winding negative channel, Li-negative channel, and Lagarias-negative channel are exact finite negative channels for $\neg \mathsf{RH}$, and their finite witness fibres are pairwise computably replay-equivalent.

(IX)

Number-theoretic criterion absorption.The exact Ξ-winding, Li-negative, and Lagarias-negative predicates are replay-equivalent finite negative fibres for $\neg \mathsf{RH}$:

$$\exists w{B}_{\Xi }\left(w\right)\iff \exists v{B}_{\mathrm{Li}}\left(v\right)\iff \exists v{B}_{\mathrm{Lag}}\left(v\right)\iff \neg \mathsf{RH}.$$

The theta-atlas, zero-count, and half-plane-cover predicates are all-stage positive fibres for $\mathsf{RH}$:

$$\forall n\exists c{C}_{\Theta }(n,c)\iff \forall n\exists c{C}_0(n,c)\iff \forall n\exists c{C}_{\mathrm{hp}}(n,c)\iff \mathsf{RH}.$$

Thus these number-theoretic criteria are absorbed into the enriched selected RH criterion package ${\mathcal{P}}_{\mathsf{RH}}^{*}$. They do not stand outside nullity as escape routes; their non-null force occurs exactly by sign-fibre occupation.

(X)

Literal arithmetic representative.

$$\widehat{\mathsf{RH}}:⟺\forall w\forall u\neg {\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u)$$

is ${\Pi }_1$ and satisfies

$$\mathbb{N}\vDash \widehat{\mathsf{RH}}\iff \mathsf{RH}.$$

The bounded matrix is an accepting-history checker for complete low-level Turing/register-machine computation tableaux, with concrete bounded β-style access coding and no hidden high-level halting assertion.

(XI)

Proof-status calibration.For every $T\supseteq \mathsf{Q}$:

$$\neg \mathsf{RH}\Rightarrow T⊢\neg \widehat{\mathsf{RH}},$$

hence

$$T⊬\neg \widehat{\mathsf{RH}}\Rightarrow \mathsf{RH},{Ind}_T\left(\widehat{\mathsf{RH}}\right)\Rightarrow \mathsf{RH}.$$

If T is consistent and $T⊢\widehat{\mathsf{RH}}$, then $\mathsf{RH}$. If T is ${\Sigma }_1$-sound, then

$$T⊢\neg \widehat{\mathsf{RH}}\iff \neg \mathsf{RH}.$$

(XII)

Positive Selection Jump.If $\mathsf{RH}$, then the positive theta-atlas realizer fibre ${\mathcal{F}}_{\Xi ,\Theta }^{+}$ is ${\Pi }_2^0$-complete. More generally, the positive realizer fibres associated to the enriched positive channels $C_{\Theta },C_0,C_{\mathrm{hp}}$ are empty if $\neg \mathsf{RH}$ and ${\Pi }_2^0$-complete if $\mathsf{RH}$.

(XIII)

Admissibility separation.The repaired entire theta filter is extensionally equivalent to the selected bad-witness predicate, whereas the legacy Dirichlet support-strict filter is empty on selected bad witnesses. Its missing coverage premise is equivalent to RH.

(XIV)

Global nonclosure.The global nonclosure barriers of 110 hold under their respective stated hypotheses. In particular, no computable triage, truth-definitional terminality predicate, diagonal-universal classifier, finite-prefix mechanism, finite-interface mechanism, or legacy support-strict filter gives a universal terminal chart for selected resolution; same-theory self-certifying terminality collapses to reflection under the hypotheses of 73. The theta-cosine finite-interface theorem shows that finite real-linear theta-kernel data do not determine RH-like zero-freeness of the associated completed s-plane functions even inside a functional-equation and real-symmetry preserving perturbation class.

(XV)

Nullity matching.For every consistent $T\supseteq \mathsf{Q}$,

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,$$

$${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀,$$

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀,$$

and

$$\bigcap _{{\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}}{Act}_T\left({\mathcal{P}}^{*}\right)=⌀.$$

Pure selected-parametric resolvers factor through these common-label intersections and hence are null if sound.

(XVI)

Universal selected-logical nullity.For every resolver

$$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right),$$

if $\mathcal{R}$ is T-sound and fully selected-package invariant, then

$$\mathcal{R}\left(\mathcal{P}\right)=⌀(\mathcal{P}\in \mathfrak{Pkg}).$$

For pure selected-parametric resolvers, full replacement invariance is derived:

$$\mathsf{PureSel}\left(\mathcal{R}\right)\Rightarrow {\mathsf{Inv}}_{\mathrm{sel}}\left(\mathcal{R}\right),$$

so

$${\mathsf{Sound}}_T\left(\mathcal{R}\right)\wedge \mathsf{PureSel}\left(\mathcal{R}\right)\Rightarrow \mathcal{R}\left(\mathcal{P}\right)=⌀(\mathcal{P}\in \mathfrak{Pkg}).$$

In particular,

$$\mathcal{R}\left({\mathsf{RH}}^{♯}\right)=⌀.$$

Consequently, because the canonical resolver induced by ${\Gamma }_{\mathrm{sel}}^{max}$ is uniform by construction,

$${LogRes}_T^{\mathrm{unif}}({\Gamma }_{\mathrm{sel}}^{max};{\mathsf{RH}}^{♯})=⌀.$$

(XVII)

Enriched selected-logical nullity and fibre occupation.For every

$${\mathcal{R}}^{*}:{\mathfrak{Pkg}}^{*}\to \mathcal{P}\left({\mathcal{L}}_T\right),$$

if ${\mathcal{R}}^{*}$ is T-sound and fully enriched-package invariant, then

$${\mathcal{R}}^{*}\left({\mathcal{P}}^{*}\right)=⌀({\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}).$$

For pure enriched selected-parametric resolvers, full enriched-package invariance is derived by selected parametricity. In particular, every sound pure enriched selected-parametric resolver satisfies

$${\mathcal{R}}^{*}\left({\mathcal{P}}_{\mathsf{RH}}^{*}\right)=⌀.$$

The theta-atlas, winding, Li, Lagarias, zero-count, and half-plane-cover criteria are therefore the RH-specific coordinate-bearing criteria of the enriched package, not external exceptions to nullity. A non-null sign outcome from them is precisely sign-fibre occupation:

$$\exists w{B}_i\left(w\right)\iff \neg \mathsf{RH}$$

for a negative fibre, or

$$\forall n\exists c{C}_j(n,c)\iff \mathsf{RH}$$

for a positive fibre.

Proof. 

Item (I) is thm:clause-prior-bivalent-nullity and thm:completion-invariant-nullity.

Item (II) is thm:selection-forms-proposition.

Item (III) is lem:rh-xi.

Item (IV) is thm:theta-majorants and prop:maj-cert.

Item (V) follows from prop:atlas-decidable, thm:atlas-soundness, thm:quant-atlas-rect, cor:quant-atlas-KN, and thm:theta-atlas-rh, together with the effective conversion theorem thm:lower-bound-to-atlas.

Item (VI) is thm:badxi.

Item (VII) follows from prop:wind-decidable, thm:exact-count, thm:theta-backend, thm:quantitative-winding-certificates, and thm:lower-bound-to-winding.

Item (VIII) follows from thm:negative-channel-equivalence and cor:replay-equivalence.

Item (IX) is thm:criterion-absorption.

Item (X) is thm:rharith.

Item (XI) is thm:proof-status-calibration.

Item (XII) follows from thm:exact-theta-fibres and thm:fibre-complexity-absorption.

Item (XIII) follows from thm:theta-filter, prop:support-mismatch, cor:legacy-filter-empty, and thm:legacy-laundering.

Item (XIV) is the collection of global barriers summarized in thm:global-nonclosure, with the hypotheses of the component theorems as stated in thm:triage, thm:ind-undecidable, thm:reflection, thm:tarski, thm:diagonal, thm:finite-prefix, thm:finite-interface, thm:theta-cosine-finite-interface, and thm:legacy-laundering.

Item (XV) is thm:nullity-matching.

Item (XVI) follows from thm:universal-selected-resolution-nullity, thm:selected-parametricity, thm:pure-selected-resolution-nullity, thm:selected-logical-nullity, and cor:selected-logical-nullity-rh.

Item (XVII) follows from thm:universal-enriched-selected-resolution-nullity, thm:pure-enriched-selected-resolution-nullity, cor:number-theoretic-criteria-subject-to-nullity, and thm:criterion-absorption. □

Remark 112 (Scope of the main theorem) 

The main theorem is not the assertion

$${Ind}_T\left(\widehat{\mathsf{RH}}\right)$$

for any particular T, and it is not a proof or disproof of RH. It asserts:

• the raw Dirichlet-clause formula has no bivalent sign;
• arbitrary strip completions have no invariant sign;
• analytic continuation forms the classical proposition;
• the theta kernel gives explicit quantitative positive certificates;
• winding certificates give finite negative witnesses;
• classical criteria give replay-equivalent negative witness channels;
• exact number-theoretic criteria are absorbed as enriched selected sign fibres;
• the literal ${\Pi }_1$ representative has sharp negative proof-status calibration;
• raw, completion, ordinary selected, and enriched selected nullities match;
• every sound resolver invariant under arbitrary replacement of exact selected packages is identically empty for every consistent $T\supseteq \mathsf{Q}$;
• every sound pure selected-parametric resolver is identically empty for every consistent $T\supseteq \mathsf{Q}$, because replacement invariance is derived by selected parametricity;
• the same nullity holds for enriched exact selected packages.

The theorem therefore separates RH-specific sign-fibre occupation from pure invariant selected-logical label output. The number theory supplies exact coordinate-bearing fibres; the universal nullity theorem applies to the unoccupied enriched fibre architecture in the pure replacement-invariant regime.

20. Conclusions

The selected RH problem has exact finite/universal witness fibres:

$$\neg \mathsf{RH}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right),$$

and

$$\mathsf{RH}\iff \forall N\ge 5\exists c\mathsf{ThetaAtlas}(N,c).$$

The first is finite and semidecidable. The second is all-stage and is realized by an explicit quantitative theta-kernel atlas calculus.

The positive theta-atlas channel uses

$$\Xi \left(z\right)=4{\int }_0^{\infty }{\Phi }_{+}\left(u\right)cos\left(zu\right)\mathrm{d}u$$

and explicit majorants

$$\underset{\left|z\right|\le R}{sup}\left|{\Xi }^{\left(j\right)}\left(z\right)\right|\le {\mathfrak{M}}_j\left(R\right).$$

For rational radii, and for supplied rational upper radii, these majorants admit finite rational upper certificates. For the compact windows

$$K_N=[1/N,N]\times [1/N,1/2-1/N],$$

if

$${\delta }_N=\underset{K_N}{min}|\Xi |>0$$

and

$$M_N\ge \underset{\left|z\right|\le N+2}{sup}\left|{\Xi }^{\prime }\right|$$

is certified, then a theta-atlas certificate exists with explicit cell-count and precision bounds. Thus the positive side is not merely a formal compactness assertion; it is a finite rational certificate calculus with quantitative parameters. Effective construction of those certificates requires supplied rational lower bounds for the relevant minima; the lower-bound-to-atlas theorem isolates this data explicitly.

The negative side is finite. If RH is false, an off-critical zero of $\Xi$ in $\mathcal{U}$ lies inside some rational rectangle with zero-free boundary, and a winding certificate proves a positive zero count. The quantitative winding mesh bound includes a certified tube-radius parameter, reflecting the geometric requirement that the finite boundary-subarc disks remain inside the chosen compact guard. Written in terms of the actual boundary separation, the quantitative bound is existential; from a supplied rational lower bound for that separation, the lower-bound-to-winding theorem gives an effective certificate construction. This finite negative channel is RH-specific analytic reasoning in the sense of this paper: it uses the actual $\Xi$-function and occupies the negative sign fibre. It gives

$$\neg \mathsf{RH}\iff \exists w{\mathsf{Bad}}_{\Xi }\left(w\right).$$

The finite negative channel is replay-equivalent to finite negative channels from Li’s criterion and Lagarias’s criterion.

The central absorption statement is:

$$\exists w{B}_{\Xi }\left(w\right)\iff \exists v{B}_{\mathrm{Li}}\left(v\right)\iff \exists v{B}_{\mathrm{Lag}}\left(v\right)\iff \neg \mathsf{RH},$$

and

$$\forall n\exists c{C}_{\Theta }(n,c)\iff \forall n\exists c{C}_0(n,c)\iff \forall n\exists c{C}_{\mathrm{hp}}(n,c)\iff \mathsf{RH}.$$

The theta-atlas, winding, Li, Lagarias, zero-count, and half-plane-cover criteria are therefore not external escape routes from selected-logical nullity. Once made exact, they are absorbed into the enriched selected RH package as sign fibres.

A finite $\Xi$-winding witness, a Li-negative witness, or a Lagarias-negative witness is literal occupation of the negative fibre and is therefore exactly $\neg \mathsf{RH}$. A complete theta-atlas stream, zero-count stream, or half-plane-cover stream is literal occupation of the positive fibre and is therefore exactly $\mathsf{RH}$. Before such occupation is supplied, the mere availability of the criteria, their exact equivalence to RH or $\neg \mathsf{RH}$, and their replay equivalence do not yield a fixed-instance truth, falsity, proof, refutation, nonproof, nonrefutation, or independence label at the pure invariant selected-logical level.

The selected arithmetic representative

$$\widehat{\mathsf{RH}}=\forall w\forall u\neg {\mathsf{BadCheck}}_{\Xi }^{\mathrm{arith}}(w,u)$$

is literally ${\Pi }_1$ and is true in $\mathbb{N}$ exactly when RH holds. The bounded verifier matrix is a local consistency checker for complete low-level Turing/register-machine accepting computation histories, using concrete bounded $\beta$-style access coding. The negative proof-status calibration is sharp. If RH is false, then the finite negative witness channel produces a standard accepting trace, and hence every $T\supseteq \mathsf{Q}$ proves

$$\neg \widehat{\mathsf{RH}}.$$

Consequently,

$$T⊬\neg \widehat{\mathsf{RH}}\Rightarrow \mathsf{RH},{Ind}_T\left(\widehat{\mathsf{RH}}\right)\Rightarrow \mathsf{RH}.$$

If T is consistent and $T⊢\widehat{\mathsf{RH}}$, then RH. If T is ${\Sigma }_1$-sound, then

$$T⊢\neg \widehat{\mathsf{RH}}\iff \neg \mathsf{RH}.$$

If the positive theta-atlas side is occupied, then its program-realizer fibre is ${\Pi }_2^0$-complete by Selection Jump:

$$\mathsf{RH}\Rightarrow {\mathcal{F}}_{\Xi ,\Theta }^{+}\mathrm{is}{\Pi }_2^0-\mathrm{complete}.$$

More generally, the enriched positive fibres associated to

$$C_{\Theta },C_0,C_{\mathrm{hp}}$$

are empty if $\neg \mathsf{RH}$ and ${\Pi }_2^0$-complete if $\mathsf{RH}$. This is a global index-set complexity phenomenon: recognizing successful positive certificate generators has maximal ${\Pi }_2^0$ complexity once the positive sign fibre is occupied.

At the global level, selected resolution has no universal terminal chart. There is no total computable proof-disproof-independence triage; independence over a sound r.e. theory is not uniformly decidable on coded ${\Pi }_1$-selected packages; same-theory terminality collapses to reflection under its stated proof-theoretic hypotheses; truth-definitional terminality meets Tarski; diagonal-universal terminality is inconsistent; finite prefixes do not close universal selected channels; finite-interface data do not determine RH-like zero-freeness in broad symmetric classes, and in the theta-cosine case they do not determine RH-like zero-freeness of the associated completed s-plane function; and legacy Dirichlet support filtering launders RH.

The endpoint is the nullity-matching and universal selected-resolution theorem. The matched nullities are:

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,$$

$${CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀,$$

and, for every consistent $T\supseteq \mathsf{Q}$,

$$\bigcap _{\mathcal{P}\in \mathfrak{Pkg}}{Act}_T\left(\mathcal{P}\right)=⌀,$$

$$\bigcap _{{\mathcal{P}}^{*}\in {\mathfrak{Pkg}}^{*}}{Act}_T\left({\mathcal{P}}^{*}\right)=⌀.$$

The first is raw Dirichlet bivalence nullity. The second is completion-invariant bivalence nullity. The third is ordinary selected-package common-label nullity. The fourth is enriched selected-package common-label nullity. These nullities do not erase RH-specific analytic reasoning; they identify the pure invariant residue left after concrete RH-specific fibre occupation is not supplied.

Let

$${\mathcal{L}}_T=\{\mathbf{True},\mathbf{False},{\mathbf{Prov}}_T,{\mathbf{Ref}}_T,{\mathbf{Unprov}}_T,{\mathbf{Unref}}_T,{\mathbf{Ind}}_T\}.$$

For arbitrary resolvers

$$\mathcal{R}:\mathfrak{Pkg}\to \mathcal{P}\left({\mathcal{L}}_T\right),$$

the semantic theorem is the following unconditional metatheorem about all consistent background theories:

$$\begin{array}{|c|}\hline T\supseteq \mathsf{Q}\mathrm{consistent}⟹\left({\mathsf{Sound}}_T\left(\mathcal{R}\right)\wedge {\mathsf{Inv}}_{\mathrm{sel}}\left(\mathcal{R}\right)\Rightarrow \forall \mathcal{P}\in \mathfrak{Pkg}\mathcal{R}\left(\mathcal{P}\right)=⌀\right).\\ \hline\end{array}$$

The consistency antecedent is essential for this proof-status label set. For an inconsistent T, ${\mathbf{Prov}}_T$ and ${\mathbf{Ref}}_T$ are common labels for every package.

For pure selected-parametric resolvers, the invariance premise is derived:

$$\begin{array}{|c|}\hline \mathsf{PureSel}\left(\mathcal{R}\right)\Rightarrow {\mathsf{Inv}}_{\mathrm{sel}}\left(\mathcal{R}\right).\\ \hline\end{array}$$

Therefore, for every consistent $T\supseteq \mathsf{Q}$,

$$\begin{array}{|c|}\hline {\mathsf{Sound}}_T\left(\mathcal{R}\right)\wedge \mathsf{PureSel}\left(\mathcal{R}\right)\Rightarrow \forall \mathcal{P}\in \mathfrak{Pkg}\mathcal{R}\left(\mathcal{P}\right)=⌀.\\ \hline\end{array}$$

This statement imposes no computability or definability restriction on arbitrary resolvers in the conditional theorem, and it derives invariance for the pure resolver language by opacity of the package variable. A non-null sound outcome must leave the pure invariant regime by observing fibres, using coordinate-bearing data, adding an oracle/sign axiom, relying on an inconsistent proof-status background, or otherwise failing full replacement invariance.

For the selected RH package,

$${\mathsf{RH}}^{♯}=(\mathsf{RH},{\mathcal{W}}_{\Xi }^{\Theta }),$$

identified at resolver level with

$${\mathcal{P}}_{\mathsf{RH}}=(\mathsf{RH},{\mathsf{Bad}}_{\Xi },C_{\Xi }^{\Theta },{\mathsf{BadCheck}}_{\Xi },\widehat{\mathsf{RH}}),$$

this gives, for every consistent $T\supseteq \mathsf{Q}$,

$$\begin{array}{|c|}\hline {\mathsf{Sound}}_T\left(\mathcal{R}\right)\wedge \mathsf{PureSel}\left(\mathcal{R}\right)\Rightarrow \mathcal{R}\left({\mathsf{RH}}^{♯}\right)=⌀.\\ \hline\end{array}$$

Taking $\mathcal{R}$ to be the canonical resolver induced by the complete invariant theory ${\Gamma }_{\mathrm{sel}}^{max}$ yields

$$\begin{array}{|c|}\hline {LogRes}_T^{\mathrm{unif}}({\Gamma }_{\mathrm{sel}}^{max};{\mathsf{RH}}^{♯})=⌀.\\ \hline\end{array}$$

Here invariance holds by construction of the uniform ${\Gamma }_{\mathrm{sel}}^{max}$-resolver: the forcing relation quantifies over exact selected packages uniformly and does not inspect the coordinate-bearing identity of ${\mathsf{RH}}^{♯}$. Thus at the complete a-priori selected-logical invariant level, the RH truth/proof-status coordinate is absent.

The enriched version is equally important. Let

$${\mathfrak{Pkg}}^{*}$$

be the class of arithmetically represented enriched exact selected packages. For the enriched selected RH criterion package

$${\mathcal{P}}_{\mathsf{RH}}^{*},$$

which contains the $\Xi$-winding, Li-negative, Lagarias-negative, theta-atlas, zero-count, and half-plane-cover fibres, every T-sound pure enriched selected-parametric resolver satisfies, for every consistent $T\supseteq \mathsf{Q}$,

$$\begin{array}{|c|}\hline {\mathcal{R}}^{*}\left({\mathcal{P}}_{\mathsf{RH}}^{*}\right)=⌀.\\ \hline\end{array}$$

Therefore the exact number-theoretic criteria do not bypass selected-logical nullity; they are part of the enriched package to which nullity applies.

$$\begin{array}{|c|}\hline \begin{array}{c}\mathrm{This}\mathrm{is}\mathrm{a}\mathrm{theory}\mathrm{of}\mathrm{RH}-\mathrm{specific}\mathrm{analytic}\mathrm{and}\mathrm{number}-\mathrm{theoretic}\mathrm{reasoning}\\ \mathrm{as}\mathrm{exact}\mathrm{selected}\mathrm{sign}-\mathrm{fibre}\mathrm{reasoning}.\end{array}\\ \hline\end{array}$$

$$\begin{array}{|c|}\hline \begin{array}{c}\mathrm{The}\mathrm{number}\mathrm{theory}\mathrm{does}\mathrm{not}\mathrm{escape}\mathrm{nullity};\\ \mathrm{it}\mathrm{supplies}\mathrm{the}\mathrm{exact}\mathrm{fibres}\mathrm{to}\mathrm{which}\mathrm{nullity}\mathrm{applies}.\end{array}\\ \hline\end{array}$$

$$\begin{array}{|c|}\hline \begin{array}{c}\mathrm{Nullity}\mathrm{governs}\mathrm{the}\mathrm{unoccupied}\mathrm{pure}\mathrm{invariant}\mathrm{architecture},\\ \mathrm{not}\mathrm{occupied}\mathrm{RH}-\mathrm{specific}\mathrm{fibres}.\end{array}\\ \hline\end{array}$$

Together with the earlier two nullities,

$${BivRes}_{{\mathcal{E}}_{\mathrm{Dir}}}\left({\mathsf{RH}}_0\right)=⌀,{CompInvRes}_{{\mathcal{C}}_{\mathcal{S}}}\left({\mathsf{RH}}_0\right)=⌀,$$

this gives the four-level nullity structure:

$$\begin{array}{|c|}\hline \begin{array}{cc}& \mathrm{raw}\mathrm{Dirichlet}\mathrm{bivalence}\mathrm{nullity}\hfill \\ & +\mathrm{completion}-\mathrm{invariant}\mathrm{bivalence}\mathrm{nullity}\hfill \\ & +\mathrm{ordinary}\mathrm{selected}-\mathrm{package}\mathrm{common}-\mathrm{label}\mathrm{nullity}\mathrm{for}\mathrm{consistent}T\hfill \\ & +\mathrm{enriched}\mathrm{selected}-\mathrm{package}\mathrm{common}-\mathrm{label}\mathrm{nullity}\mathrm{for}\mathrm{consistent}T.\hfill \end{array}\\ \hline\end{array}$$

Appendix A.1. Elementary Rational Interval Primitives

Lemma A1 

(Elementary constants and functions). There are terminating rational interval algorithms which, given a precision parameter p, produce rational intervals of width at most $2^{-p}$ containing each of

$$\pi ,log2,{\gamma }_{\mathrm{E}}.$$

There are also terminating interval algorithms which, given rational input intervals separated from their singularities, enclose

$$expx,logx,arctanx,sinx,cosx$$

to any prescribed rational width. The corresponding complex disk algorithms for $expz$, $Logz$, $sinz$, and $cosz$ are obtained by Taylor series or exponential formulas with rational disk arithmetic.

Proof. 

For arctan, use

$$arctanx={\displaystyle \sum _{j=0}^{m-1}}{\displaystyle \frac{{(-1)}^j{x}^{2j+1}}{2j+1}}+R_m\left(x\right),\left|R_m\left(x\right)\right|\le {\displaystyle \frac{{\left|x\right|}^{2m+1}}{2m+1}}$$

for $\left|x\right|\le 1$, together with Machin’s identity

$${\displaystyle \frac{\pi }{4}}=4arctan{\displaystyle \frac{1}{5}}-arctan{\displaystyle \frac{1}{239}}.$$

For logarithms, reduce by powers of 2 and use

$$logx=2{\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{y^{2j+1}}{2j+1}},y={\displaystyle \frac{x-1}{x+1}},$$

with a geometric tail bound. For exp, sin, and cos, use Taylor series with factorial remainder bounds on bounded rational intervals. For ${\gamma }_{\mathrm{E}}$, use the Euler–Maclaurin expansion

$$H_N=logN+{\gamma }_{\mathrm{E}}+{\displaystyle \frac{1}{2N}}-{\displaystyle \sum _{r=1}^{m-1}}{\displaystyle \frac{B_{2r}}{2r{N}^{2r}}}+R_{m,N},$$

with

$$|R_{m,N}|\le {\displaystyle \frac{|B_{2m}|}{2m{N}^{2m}}}.$$

All Bernoulli numbers used are rational and computed recursively. Complex disk versions follow from the same power series and disk arithmetic. □

Appendix A.2. Euler–Maclaurin Enclosures for Zeta Derivatives

Let

$$D_s=\overline{B}(s_0,r)$$

be a rational disk separated from $s=1$, and put

$${\sigma }_{-}=Re\left(s_0\right)-r.$$

Choose $m\ge 1$ with

$${\sigma }_{-}+2m>1.$$

Throughout this appendix,

$${\left(s\right)}_k=s(s+1)\dots (s+k-1)(k\ge 1),$$

with

$${\left(s\right)}_0=1,$$

denotes the rising Pochhammer symbol.

Euler–Maclaurin gives

$$\zeta \left(s\right)={\displaystyle \sum _{k=1}^{N-1}}{k}^{-s}+{\displaystyle \frac{N^{1-s}}{s-1}}+{\displaystyle \frac{1}{2}}{N}^{-s}+{\displaystyle \sum _{\ell =1}^m}{\displaystyle \frac{B_{2\ell }}{(2\ell )!}}{\left(s\right)}_{2\ell -1}{N}^{-s-(2\ell -1)}+R_{m,N}\left(s\right),$$

where

$$R_{m,N}\left(s\right)=-{\displaystyle \frac{1}{\left(2m\right)!}}{\int }_N^{\infty }{B}_{2m}\left(\left\{x\right\}\right){\left(s\right)}_{2m}{x}^{-s-2m}\mathrm{d}x.$$

For $0\le q\le 3$, differentiation under the integral gives

$$R_{m,N}^{\left(q\right)}\left(s\right)=-{\displaystyle \frac{1}{\left(2m\right)!}}{\int }_N^{\infty }{B}_{2m}\left(\left\{x\right\}\right){\displaystyle \sum _{a=0}^q}\left({\displaystyle \genfrac{}{}{0pt}{}{q}{a}}\right)P_m^{\left(a\right)}\left(s\right){(-logx)}^{q-a}{x}^{-s-2m}\mathrm{d}x,$$

where

$$P_m\left(s\right)={\left(s\right)}_{2m}.$$

Let ${\mathfrak{B}}_{2m}$ be a rational upper bound for

$$\underset{0\le u\le 1}{sup}\left|B_{2m}\left(u\right)\right|.$$

Let $C_a$ be a rational upper bound for

$$\underset{s\in D_s}{sup}|P_m^{\left(a\right)}\left(s\right)|.$$

Then

$$|R_{m,N}^{\left(q\right)}\left(s\right)|\le {\displaystyle \frac{{\mathfrak{B}}_{2m}}{\left(2m\right)!}}{\displaystyle \sum _{a=0}^q}\left({\displaystyle \genfrac{}{}{0pt}{}{q}{a}}\right)C_a{J}_{q-a}^{+}({\sigma }_{-}+2m,N),$$

where

$$J_q(a,N)={\int }_N^{\infty }{x}^{-a}{(logx)}^q\mathrm{d}x$$

and $J_q^{+}(a,N)$ is any rational upper enclosure. Repeated integration by parts gives the exact formula

$$J_q(a,N)=N^{1-a}{\displaystyle \sum _{\ell =0}^q}{\displaystyle \frac{q!}{(q-\ell )!}}{\displaystyle \frac{{(logN)}^{q-\ell }}{{(a-1)}^{\ell +1}}}(a>1).$$

Theorem A1 

(Zeta-jet enclosure). Given a rational disk $D_s$ separated from 1, and a precision parameter, there is a terminating rational algorithm producing disks enclosing

$${\zeta }^{\left(q\right)}\left(D_s\right),0\le q\le 3.$$

For point disks $D_s=\left\{s_0\right\}$, the output radii can be made smaller than any prescribed positive rational number.

Proof. 

The Euler–Maclaurin expression equals the analytic continuation of $\zeta \left(s\right)$ on $D_s$, since both sides agree in the half-plane of convergence and both are meromorphic away from $s=1$. The displayed remainder bounds are uniform on $D_s$ and tend effectively to 0 as $N\to \infty$. The finite terms are computed by rational disk arithmetic using

$$k^{-s}=exp(-slogk).$$

This gives terminating enclosures through order 3, with arbitrary precision at points. □

Appendix A.3. Gamma and Polygamma Enclosures

For $Re\left(z\right)>0$, use

$${\displaystyle \frac{1}{\Gamma \left(z\right)}}=z{e}^{{\gamma }_{\mathrm{E}}z}{\displaystyle \prod _{k=1}^{\infty }}(1+z/k)e^{-z/k},$$

and

$$\psi \left(z\right)=-{\gamma }_{\mathrm{E}}-{\displaystyle \frac{1}{z}}+{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \frac{1}{k}}-{\displaystyle \frac{1}{k+z}}\right),$$

$${\psi }^{\prime }\left(z\right)={\displaystyle \frac{1}{z^2}}+{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{{(k+z)}^2}},$$

$${\psi }^{\prime \prime }\left(z\right)=-{\displaystyle \frac{2}{z^3}}-2{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{{(k+z)}^3}}.$$

On a rational disk

$$D_z=\overline{B}(c,r)⋐\{Rez>0\},$$

put

$$a=Re\left(c\right)-r>0,M=|c|+r.$$

Choose $N>2M$. The tails are bounded by:

$$\left|{\displaystyle \sum _{k=N}^{\infty }}\left(Log(1+z/k)-z/k\right)\right|\le {\displaystyle \frac{2M^2}{N-1}},$$

$$\left|{\displaystyle \sum _{k=N}^{\infty }}{\displaystyle \frac{z}{k(k+z)}}\right|\le M{\int }_{N-1}^{\infty }{\displaystyle \frac{\mathrm{d}x}{x(x+a)}},$$

$$\left|{\displaystyle \sum _{k=N}^{\infty }}{\displaystyle \frac{1}{{(k+z)}^2}}\right|\le {\displaystyle \frac{1}{N-1+a}},$$

and

$$\left|2{\displaystyle \sum _{k=N}^{\infty }}{\displaystyle \frac{1}{{(k+z)}^3}}\right|\le {\displaystyle \frac{1}{{(N-1+a)}^2}}.$$

Lemma A2 

(Finite-subdivision inversion for nonvanishing analytic images). Let $D=\overline{B}(c,r)$ be a rational disk and let f be a computable holomorphic function on a neighborhood of D. Suppose

$$0\notin f\left(D\right).$$

Assume that point enclosures for $f\left(z_0\right)$ and compact derivative bounds for $f^{\prime }$ on D are computably available by finite rational certificates. Then there is a terminating finite rational subdivision procedure producing rational subdisks $D_1,\dots ,D_m$ covering D and rational disks $E_1,\dots ,E_m$ such that

$$f\left(D_j\right)\subseteq E_j,0\notin E_j(1\le j\le m).$$

Consequently $1/f\left(D\right)$ is enclosed by inverting the finitely many disks $E_j$ and enclosing their finite union.

This lemma is used only as a finite certificate-existence and backend termination principle. It explicitly avoids the false inference that a compact nonzero image $f\left(D\right)$ must be contained in one disk avoiding 0.

Proof. 

Since f is continuous and D is compact,

$$\delta :=\underset{z\in D}{min}|f\left(z\right)|>0.$$

Let $M_D$ be a certified upper bound for $|f^{\prime }|$ on D. Choose rational subdisks $D_j=\overline{B}(c_j,r_j)$ covering D with

$$M_D{r}_j<\delta /8.$$

For each center $c_j$, compute a point enclosure

$$f\left(c_j\right)\in \overline{B}(a_j,{ϵ}_j)$$

with

$${ϵ}_j<\delta /8.$$

Then

$$f\left(D_j\right)\subseteq \overline{B}(a_j,{ϵ}_j+M_D{r}_j)$$

and

$${ϵ}_j+M_D{r}_j<\delta /4.$$

Also

$$|a_j|\ge |f\left(c_j\right)|-{ϵ}_j\ge \delta -\delta /8=7\delta /8.$$

Thus

$${ϵ}_j+M_D{r}_j<|a_j|,$$

so the disk

$$E_j:=\overline{B}(a_j,{ϵ}_j+M_D{r}_j)$$

avoids 0. The construction terminates because $\delta >0$, even though $\delta$ need not be known in advance: one may dovetail finite subdivisions and point-precision refinements until the rational inequalities

$$rad\left(E_j\right)<|a_j|$$

are verified for all subdisks. Compactness and the preceding argument guarantee that some finite stage succeeds.

For a disk $E_j=\overline{B}(a_j,{\rho }_j)$ with ${\rho }_j<|a_j|$, inversion is soundly enclosed by

$${\displaystyle \frac{1}{E_j}}\subseteq \overline{B}\left({\displaystyle \frac{\overline{a_j}}{|a_j{|}^2-{\rho }_j^2}},{\displaystyle \frac{{\rho }_j}{|a_j{|}^2-{\rho }_j^2}}\right),$$

or by any equivalent rational disk inversion formula checked by outward rational arithmetic. A rational disk enclosing the finite union of the inverted disks is obtained by a finite rational containment computation. □

Theorem A2 

(Gamma-polygamma enclosure). Given a rational disk $D_z$ contained in the open right half-plane and a precision parameter, there is a terminating rational algorithm producing disks enclosing

$$\Gamma \left(D_z\right),\psi \left(D_z\right),{\psi }^{\prime }\left(D_z\right),{\psi }^{\prime \prime }\left(D_z\right).$$

For point disks, the output radii can be made smaller than any prescribed positive rational number.

Proof. 

The displayed finite products, finite sums, and tail estimates converge locally uniformly on the right half-plane. Principal logarithms are used on right-half-plane disks, avoiding branch ambiguity for the factors

$$1+z/k,$$

because $D_z⋐\{Rez>0\}$ and $k\ge 1$. The tails tend effectively to 0, and all finite operations are rational disk computations.

The polygamma enclosures are obtained directly from the displayed series and tail bounds. For $\Gamma \left(D_z\right)$, there are two equivalent validated routes.

First, one may compute $1/\Gamma \left(D_z\right)$ by the reciprocal Weierstrass product. The function $1/\Gamma \left(z\right)$ is nonzero on the right half-plane. For positive-radius disks, however, one must not assume that a single disk enclosure of $1/\Gamma \left(D_z\right)$ avoids 0. Instead, apply A2 to

$$f\left(z\right)=1/\Gamma \left(z\right)$$

on $D_z$. The product formula and derivative bounds obtained from

$${(1/\Gamma )}^{\prime }\left(z\right)=-\left(\psi \left(z\right)\right)/\Gamma \left(z\right)$$

or from differentiating the logarithm of the product give the needed point enclosures and local derivative bounds. The lemma yields finitely many subdisks whose reciprocal-product image enclosures avoid 0. Inverting those finitely many enclosures and enclosing their finite union gives a sound enclosure for $\Gamma \left(D_z\right)$.

Second, one may instead compute a branch of $Log\Gamma$ on the simply connected disk $D_z$. Since $\Gamma$ has no zeros or poles in the right half-plane and $D_z⋐\{Rez>0\}$, such a branch exists. The derivative of this branch is $\psi$, already computably enclosed above, and a point value may be computed from the reciprocal product or from a standard Stirling expansion. Integrating/enclosing the branch on $D_z$ and exponentiating gives an enclosure for $\Gamma \left(D_z\right)$. This route also avoids any invalid one-disk inversion inference.

Either route is terminating by compactness and local uniform convergence. For point disks, the product and series tails can be made arbitrarily small, giving arbitrarily sharp enclosures. □

Appendix A.4. Composition to ξ and Ξ

Define

$$A\left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2),\xi \left(s\right)=A\left(s\right)\zeta \left(s\right),\Xi \left(z\right)=\xi (1/2+\mathrm{i}z).$$

Let

$$L\left(s\right)={\displaystyle \frac{A^{\prime }\left(s\right)}{A\left(s\right)}}={\displaystyle \frac{1}{s}}+{\displaystyle \frac{1}{s-1}}-{\displaystyle \frac{1}{2}}log\pi +{\displaystyle \frac{1}{2}}\psi (s/2).$$

Then

$$L^{\prime }\left(s\right)=-{\displaystyle \frac{1}{s^2}}-{\displaystyle \frac{1}{{(s-1)}^2}}+{\displaystyle \frac{1}{4}}{\psi }^{\prime }(s/2),$$

and

$$L^{\prime \prime }\left(s\right)={\displaystyle \frac{2}{s^3}}+{\displaystyle \frac{2}{{(s-1)}^3}}+{\displaystyle \frac{1}{8}}{\psi }^{\prime \prime }(s/2).$$

Consequently,

$$A^{\prime }\left(s\right)=A\left(s\right)L\left(s\right),$$

$$A^{\prime \prime }\left(s\right)=A\left(s\right)(L{\left(s\right)}^2+L^{\prime }\left(s\right)),$$

$$A^{\prime \prime \prime }\left(s\right)=A\left(s\right)(L{\left(s\right)}^3+3L\left(s\right)L^{\prime }\left(s\right)+L^{\prime \prime }\left(s\right)).$$

Combining with the product rule gives ${\xi }^{\left(j\right)}$ through order 3, and

$${\Xi }^{\left(j\right)}\left(z\right)={\mathrm{i}}^j{\xi }^{\left(j\right)}(1/2+\mathrm{i}z).$$

Theorem A3 

(AC $\Xi$-jet enclosure). Let $K⋐\mathcal{U}$ be a rational compact guard. For $0\le j\le 3$, there is a terminating rational algorithm which, given a rational disk

$$D=\overline{B}(z_0,r)\subset K,$$

produces a rational disk enclosing

$${\Xi }^{\left(j\right)}\left(D\right).$$

For point disks, the output radius can be made smaller than any prescribed positive rational number.

Proof. 

Map D to

$$D_s=1/2+\mathrm{i}D.$$

For $D\subset K⋐\mathcal{U}$, this s-disk is separated from 1, and $D_s/2$ lies in the open right half-plane. Apply A1 and A2, then combine the resulting enclosures by the displayed formulas. All operations are finite rational disk operations. □

Theorem A4 

(Compact sup bounds and shrinking-domain enclosures). Let $K⋐\mathcal{U}$ be a rational compact guard. For every $0\le j\le 3$, there is a terminating rational algorithm producing

$$M_{j,K}>0$$

with

$$\underset{z\in K}{sup}|{\Xi }^{\left(j\right)}\left(z\right)|\le M_{j,K}.$$

For $j=0,1,2$, given a rational disk

$$D=\overline{B}(z_0,r)\subseteq K$$

and precision p, there is a terminating routine producing

$$E_{j,p}\left(D\right)$$

with

$${\Xi }^{\left(j\right)}\left(D\right)\subseteq E_{j,p}\left(D\right)$$

and

$$rad\left(E_{j,p}\left(D\right)\right)\le 2^{-p}+M_{j+1,K}r.$$

Proof. 

Cover K by finitely many rational disks contained in $\mathcal{U}$ and apply A3; finite maxima give sup bounds. For shrinking-domain enclosures, compute a point enclosure

$${\Xi }^{\left(j\right)}\left(z_0\right)\in \overline{B}(a,2^{-p})$$

and use

$$|{\Xi }^{\left(j\right)}\left(z\right)-{\Xi }^{\left(j\right)}\left(z_0\right)|\le M_{j+1,K}|z-z_0|\le M_{j+1,K}r.$$

□