We separate four levels associated with the Riemann Hypothesis and strengthen the selected \(\Xi\)-certificate layer by an explicit theta-kernel atlas theorem. The resulting framework is a theory of RH-specific analytic and number-theoretic reasoning as exact selected sign-fibre reasoning. The theta kernel, theta-atlas certificates, winding certificates, Li-negative certificates, Lagarias-negative certificates, zero-count certificates, and half-plane-cover certificates are coordinate-bearing RH-specific channels. We make precise their absorption into one enriched selected RH criterion package. The number theory does not escape selected-logical nullity; it supplies the exact sign fibres to which nullity applies.First, at the raw formula level, the Dirichlet evaluator interprets \(\zeta(s)\) only by the Dirichlet series on \(\Rea(s)>1\). Under strict atomic strong Kleene semantics, the usual strip formula \[ \RHzero: \quad \forall s\bigl((0<\Rea(s)<1\wedge\zeta(s)=0)\Rightarrow\Rea(s)=1/2\bigr) \] has value \(\Gap\). Moreover, arbitrary strip-total completions realize arbitrary zero sets in the critical strip. Therefore no bivalent RH sign is invariant across such completions. Analytic continuation is treated as the selector forming the classical proposition \(\RH\).Second, after analytic selection, we work with \[ \Xi(z)=\xi(1/2+\ii z), \qquad \mathcal U=\{x+\ii y:x>0,\ 0<y<1/2\}, \] so that \[ \RH\iff Z(\Xi;\mathcal U)=\varnothing. \] The main analytic/certification contribution in the present framework is the conversion of the classical theta-kernel representation \[ \Xi(z)=4\int_0^\infty \Phi_+(u)\cos(zu)\,\dd u \] into a finite rational atlas-certificate system with explicit majorants and quantitative cell-count bounds. Here \[ \Phi_+(u) = \sum_{n=1}^{\infty} \left( 2\pi^2 n^4 e^{9u/2} - 3\pi n^2 e^{5u/2} \right)e^{-\pi n^2e^{2u}}, \] and we derive explicit majorants \[ \sup_{|z|\le R}|\Xi^{(j)}(z)| \le \mathfrak M_j(R), \] where \[ \mathfrak M_j(R) = 4\left( 2\pi^2G_4I_j(R+9/2) + 3\pi G_2I_j(R+5/2) \right), \] \[ G_m=\sum_{n=1}^{\infty}n^m e^{-\pi(n^2-1)}, \qquad I_j(\alpha)=\int_0^\infty u^j e^{\alpha u}e^{-\pi e^{2u}}\,\dd u. \] For rational radii \(R\), and more generally for any supplied rational upper radius \(R^+\ge R_0\), all ingredients admit finite rational upper certificates. If the analytic radius is \(R_0\), then the certified estimate is \[ \sup_{|z|\le R_0}|\Xi^{(j)}(z)| \le \mathfrak M_j(R^+). \]For \[ K_N= \left[\frac1N,N\right] \times \left[\frac1N,\frac12-\frac1N\right]\subset\mathcal U \qquad(N\ge5), \] we define a finite decidable rational certificate predicate \[ \ThetaAtlas(N,c) \] certifying zero-freeness of \(\Xi\) on \(K_N\). It is sound and existentially complete on zero-free windows; from supplied rational lower-bound data it is effectively constructible. It is quantitative: if \[ \delta_N:=\min_{z\in K_N}|\Xi(z)|>0 \] and \[ M_N\ge \sup_{|z|\le N+2}|\Xi'(z)| \] is certified, then an accepted atlas exists with explicitly bounded cell count \[ \left( 1+ \left\lceil \frac{2\sqrt2\,M_N^+}{\Delta_N} \left(N-\frac1N\right) \right\rceil \right) \left( 1+ \left\lceil \frac{2\sqrt2\,M_N^+}{\Delta_N} \left(\frac12-\frac2N\right) \right\rceil \right), \] where \[ M_N^+=\max(1,M_N), \qquad \Delta_N=\min(1,\delta_N). \] In finite rational certificate records, \(\sqrt2\) is replaced by any certified rational upper bound \(q_2>0\) satisfying \(q_2^2>2\). Thus \[ \RH\iff \forall N\ge5\,\exists c\,\ThetaAtlas(N,c). \] This is the coordinate-bearing positive \(\Xi\)-channel used later by the selected package.Third, exact winding certificates provide the finite negative channel \[ \neg\RH\iff\exists w\,\BadXi(w), \] and the theta-atlas channel provides the all-stage positive channel \[ \RH\iff\forall N\ge5\,\exists c\,\ThetaAtlas(N,c). \] The quantitative boundary-mesh bound for winding certificates includes an explicit certified tube-radius term, because a finite boundary mesh must be fine enough not only for variation control of \(\Xi\), but also for the enclosing domain disks to remain inside the certified compact guard. We also compare the \(\Xi\)-negative channel with classical RH criteria: Li-negative certificates and Lagarias-negative certificates are exact finite negative channels for \(\neg\RH\), and the corresponding witness fibres are pairwise computably replay-equivalent.Fourth, these number-theoretic criteria are absorbed into one enriched selected RH criterion package. The finite negative fibres include: \[ \exists w\,\BXi(w) \iff \exists v\,\BLi(v) \iff \exists v\,\BLag(v) \iff \neg\RH. \] The all-stage positive fibres include: \[ \forall n\,\exists c\,\CTheta(n,c) \iff \forall n\,\exists c\,\CZero(n,c) \iff \forall n\,\exists c\,\CHP(n,c) \iff \RH. \] 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 \(\neg\RH\). A complete positive certificate stream is literal occupation of the positive fibre and is therefore exactly \(\RH\). Before such occupation is supplied, the mere availability, exactness, and replay equivalence of the criteria remain selected-logically null.Fifth, arithmetizing the finite negative verifier gives a literal \(\Pi_1\) sentence \[ \RHarith\equiv \forall w\,\forall u\,\neg\BadXiCheckArith(w,u) \] satisfying \[ \mathbb N\models\RHarith\iff\RH. \] The arithmetic formula is obtained from complete low-level Turing/register-machine accepting computation histories; no high-level halting assertion is hidden in the matrix. The final version fixes a concrete bounded tableau-access convention, for instance a Gödel-\(\beta\)-style bounded access relation, and every subroutine not among the fixed bounded access primitives is displayed by its own finite subtrace. Thus the matrix is bounded after definitional trace elimination.For every \(T\supseteq\mathsf Q\), \[ \neg\RH\Rightarrow T\vdash\neg\RHarith, \] so \[ T\nvdash\neg\RHarith\Rightarrow\RH. \] If \(T\) is \(\Sigma_1\)-sound, then \[ T\vdash\neg\RHarith\iff\neg\RH. \]The final layer is universal selected-logical nullity, together with its enriched criterion version. At the level of arbitrary resolvers, the semantic nullity theorem has the following unconditional metatheoretic form: for every consistent theory \(T\supseteq\mathsf Q\), \[ \SoundT(\Resolver)\wedge\InvSel(\Resolver) \Rightarrow \Resolver\equiv\varnothing. \] Here \(\InvSel\) is full replacement invariance: the resolver is unchanged under arbitrary replacement of one exact selected package by another. This condition is not asserted for arbitrary coordinate-bearing RH-specific resolvers. Rather, it is the semantic target of the pure selected-logical regime. The consistency hypothesis on \(T\) is essential for proof-status labels: if \(T\) were inconsistent, then \(\mathbf{Prov}_T\) and \(\mathbf{Ref}_T\) would be common sound labels for every package under the label semantics used in this paper.The meta theory then discharges the invariance premise for pure selected-parametric resolvers. In the stipulated pure resolver language the package variable is opaque: one may use only selected bridge forms that are valid for all exact selected packages, and one may not inspect concrete fibres, package codes, truth or proof-status oracles, or coordinate-bearing analytic data. The selected parametricity theorem proves, by induction on the generation of pure resolver terms, \[ \PureSel(\Resolver)\Rightarrow\InvSel(\Resolver). \] Consequently, for every consistent \(T\supseteq\mathsf Q\), \[ \SoundT(\Resolver)\wedge\PureSel(\Resolver) \Rightarrow \Resolver\equiv\varnothing. \] Thus the invariance premise is a semantic hypothesis for arbitrary resolvers, but it is a theorem for the intended pure selected-logical regime.This gives a nullity-matching structure: \[ \BivRes_{\EDir}(\RHzero)=\varnothing, \] \[ \CompInvRes_{\mathscr C_{\Strip}}(\RHzero)=\varnothing, \] and, for every consistent \(T\supseteq\mathsf Q\), \[ \bigcap_{\mathcal P\in\Pkg}\Act_T(\mathcal P)=\varnothing, \] \[ \bigcap_{\mathcal P^\ast\in\PkgStar}\Act_T(\mathcal P^\ast)=\varnothing. \] The first is raw Dirichlet-clause bivalence nullity; the second is completion-invariant bivalence nullity; the third is ordinary exact-package selected-label nullity; and the fourth is enriched exact-package selected-label nullity. The canonical resolver induced by the full invariant selected-logical closure \(\Gamselmax\) is invariant by construction, so for every consistent \(T\supseteq\mathsf Q\), \[ \LogRes_T^{\Unif}(\Gamselmax;\RHsharp)=\varnothing. \] For the enriched selected RH criterion package \(\PRHstar\), \[ \SoundT(\Resolver^\ast)\wedge\PureSel^\ast(\Resolver^\ast) \Rightarrow \Resolver^\ast(\PRHstar)=\varnothing \] for every consistent \(T\supseteq\mathsf Q\).Consequently, the theta-atlas theorem and the universal nullity theorem are complementary. The theta, winding, Li, Lagarias, zero-count, and half-plane-cover criteria supply coordinate-bearing RH-specific fibres. The universal nullity theorem says that the unoccupied enriched fibre architecture, when viewed in the pure replacement-invariant selected-logical regime, carries no invariant truth/proof-status label. Exact number-theoretic criteria do not bypass nullity; they become its fibres. Thus the paper is a theory of RH-specific analytic reasoning with an effect discipline: non-null RH-specific conclusions occur by fibre occupation or by leaving the pure invariant regime.