Logical stratification of RH-type propositions and exact-count witness calculi for the Riemann Ξ-function: selected classical RH, repaired entire admissibility, natural Π0 2-complete companion families, literal Π1 arithmetization, local half-plane sweep certificates, and a formal diagnosis of laddered non-closure

This paper separates several logically distinct RH-related objects that are often conflated: a raw formula in a language containing \(\zeta(\cdot)\); a canonical completion-family truth problem attached to a logically prior zeta-clause package; a natural analytic family of exact-count atlases for the actual Riemann \(\Xi\)-function; the selected classical RH proposition attached to the unique analytically coherent evaluator; a repaired entire-backend admissibility predicate for project-level witnesses; a legacy support-strict predicate relative to the clause-prior Dirichlet package; and a literal \(\Pi_1\) arithmetic representative of the selected witness calculus. At the clause-prior level we work with the Dirichlet evaluator \(\EDir\), whose only zeta clause is the Dirichlet-series clause on \(\Rea(s)>1\). Under strong Kleene semantics with strict atomic predicates, the raw strip formula \[ \RHzero:\quad \forall s\Bigl((0<\Rea(s)<1\wedge \zeta(s)=0)\Rightarrow \Rea(s)=\tfrac12\Bigr) \] is gappy in \(\EDir\), whereas the definedness-guarded version is true. Across arbitrary strip-total completions every subset of the critical strip is realizable as a strip zero-set of \(\zeta\), so the raw formula is maximally completion-sensitive. At the canonical completion-family level we define a divergence-coded family \((\mathcal E_e)_{e\in\N}\) and prove \[ \IRHzero=\IRHdef=\Tot. \] Hence the true-fiber is \(\Pi^0_2\)-complete, the false-fiber is \(\Sigma^0_2\)-complete, and the corresponding total judgment functions equal \(\chi_{\Tot}\). At the analytic level, Appendix~\ref{app:ac-winding} proves a rectangle-uniform exact-count certificate theorem for the actual Riemann \(\Xi\)-function using explicit enclosure routines for \(\zeta,\zeta',\zeta''\), \(\Gamma,\psi,\psi'\), and the completed \(\Xi\)-backend. This yields decidable exact-count predicates on admissible rational rectangles, a canonical least exact-count atlas \(a_\Xi\), and a \(\Pi^0_2\)-complete atlas index set \[ \AtlasXi\equiv_m\Tot. \] Classical RH is the selected proposition \[ \RH:\Longleftrightarrow \EAC\models \RHzero \] and is equivalent to \[ Z(\Xi;\U)=\varnothing, \qquad \forall n\,\exists c\,\ZeroCert(n,c), \qquad \forall w\,\neg \BadXi(w), \] where \(\BadXi\) is a decidable exact-count bad-witness predicate on backend-neutral witness codes. A central correction of the present version is the separation of two admissibility notions on the same witness codes. First, Appendix~\ref{app:theta-backend} constructs a global entire \(\Theta\)-backend for \(\Xi\), proves a replay theorem \[ \BadXi(w)\Longrightarrow \AdmXiTheta(w), \] and therefore shows that the repaired entire-admissible bad-witness predicate \[ \BadXiTheta(w)\;:\Longleftrightarrow\;\BadXi(w)\wedge \AdmXiTheta(w) \] is extensionally equivalent to \(\BadXi(w)\). Hence \[ \RH \iff \forall w\,\neg \BadXiTheta(w). \] Second, the legacy support-strict predicate \(\AdmXiDir(w)\) requires certificate verification to stay inside the logically prior Dirichlet half-plane. We prove \[ \BadXi(w)\Longrightarrow \neg \AdmXiDir(w), \] so the legacy filtered class \[ \BadXiDir(w)\;:\Longleftrightarrow\;\BadXi(w)\wedge \AdmXiDir(w) \] is empty: \[ \forall w\,\neg \BadXiDir(w). \] This is an unconditional theorem about the clause-prior support-mismatch filter, not an