Geometrically Localized Wormholes with Compact Redshift Support: A Minimal Exotic Matter Framework

This research presents a novel wormhole geometry constructed from toroidal coordinates and engineered through controlled redshift function shaping. We evolve the redshift profile through a series of refinements—global smooth, Gaussian-localized, exponentially decaying, and finally compactly supported bump functions—to spatially confine exotic matter near the throat. Key achievements include: Derivation of the full Einstein tensor from the metric Detailed NEC analysis using null vectors Numerical simulation and visualization of exotic energy localization Volume-integrated estimate of total exotic matter: finite and sharply bounded High-resolution geometric rendering colored by energy density The accompanying three-part paper set includes: Geometric Foundations Mathematical Breakdown Geometrically Localized Wormholes This work is intended to contribute a physically meaningful, mathematically rigorous step toward traversable wormhole models with minimal quantum violations, offering potential applications in quantum gravity, field theory, and cosmological model design.