On Super-Hyperbolic Wormhole Geometries and their Deformations

In this study, we developed a unified geometric and relativistic framework for a new family of traversable wormholes constructed from a super-hyperbolic deformation of the classical catenoid, whose geometric embedding combines an explicit local curvature parameter $\kappa$, which secures a regular throat, with the Mittag-Leffler-based super-hyperboloid deformation function ${\mathfrak{S}C}_{\alpha }(v)$, where the deformation parameter $\alpha >1$ governs the radial onset and far-field growth of the embedding beyond the throat, continuously approaching the classical catenoidal geometry in the boundary limit $\alpha \to 1^{+}$. Allowing a non-constant redshift $\mathrm{\Phi }(r)$, we systematically evaluate closed analytical expressions for the shape function, curvature tensors, matter sources, and junction conditions, culminating in an exact thin-shell equation of motion and stability criterion. This work establishes a mathematically consistent and physically viable framework for the extension of wormhole theory, providing a new avenue for exploring exotic geometries within general relativity and identifying the conditions under which traversable configurations remain locally stable.