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Radio interferometry often faces challenges in correcting distortions of the wavefront caused by the instrument and the propagation medium. In such cases, closure invariants—quantities that remain unchanged despite severe corruption of individual measurements—play a crucial role. These invariants are true observables of the underlying physical system. Among the most well-known are closure phases and closure amplitudes, foundational tools in radio astronomy since the 1950s. Their importance was underscored recently in the groundbreaking images of the supermassive black holes in M87 and the Milky Way, where closure invariants were key to overcoming calibration limitations. Despite their proven utility, the deeper reasons behind their invariance and their true power remain underexplored. I will present recent progress towards a unified theoretical framework for interferometric invariants. On the practical front, a direct link between these invariants and the morphological features of the source has long been elusive. I will discuss new machine learning approaches that can reconstruct high-fidelity images using only closure invariants—bypassing the need for calibration entirely. This approach not only offers deeper insights into the source structure but also opens a new, independent pathway for exploring black hole-scale morphology.