We observe that Young’s double-slit experiment is incomplete because it uses only one screen to capture its result and relies on linear approximations of interference. This paper presents the exact geometric formulation of monochromatic wave interference, which is intuitively related to water waves, revealing a family of hyperbolic interference patterns that have been overlooked for centuries. These nonlinear patterns suggest that a single screen is insufficient to capture the complete result of the experiment. We propose that multiple or moving screens are necessary for a comprehensive understanding of Young’s double-slit experiment and the behavior of passing photons. This is Part I of a three-paper series. This note develops the core idea and motivation related to material-dependent effects in single-slit diffraction and frames the claim at existence level (interaction occurs inside the slit, consistent with Einstein’s view). Related: DIY optics preprint (DOI 10.5281/zenodo.17059754) and report (DOI 10.5281/zenodo.17118246 )