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Abstract
This article develops an operator–geometric reconstruction of the Riemann Hypothesis within the χ‑coordinate architecture of the ZEBTS‑SUPER framework. The central idea is that the analytic domain of the Riemann zeta function can be transferred to a compact geometric manifold—the χ‑sphere—via the χ‑lifting transformation 𝜒 = exp ( 𝑠 − 1 / 2 ) . Under this mapping, the classical critical line becomes the χ‑equator, and the nontrivial zeros of the zeta function become geometric points equipped with a coherent operator, spectral, dynamical, and resolvent structure. On the χ‑sphere, we introduce the χ‑operator 𝑇 𝜒 = − Δ 𝜒 + 𝑉 𝜒 , where the potential is supported near the χ‑equator. This operator is self‑adjoint, elliptic, and has a discrete spectrum. Its eigenfunctions, resolvent residues, and semigroup flow exhibit a stabilization phenomenon: χ‑spectral and χ‑dynamical objects concentrate on the χ‑equator. This behavior is enforced by Agmon localization, the Eχ‑restriction on perturbations, and the geometric identity linking the critical line to the χ‑equator. As a consequence, χ‑spectral zeros—localized eigenfunctions corresponding to χ‑images of zeta zeros—coincide with χ‑dynamic attractors, χ‑resolvent residues, and χ‑falsifiable χ‑structures. The key result is the χ‑∞ Boundary Stability Law, which states that the Riemann Hypothesis is equivalent to the χ‑equator being the unique χ‑∞ stable boundary manifold. This interpretation admits experimental and numerical verification: χ‑flows, χ‑spectral maps, χ‑resolvent densities, and χ‑discretizations yield binary predictions capable of corroborating or refuting RH. The article concludes with the χ‑∞ Boundary Stability Law as an equivalent stability‑theoretic formulation of the Riemann Hypothesis.
