Projective Wavefront Interferometry, Visibility-Conditioned Cramér-Rao Bounds, and Variational Hybrid Focusing in 1D and 3D Radiative Near-Field XL-MIMO

Extremely large scale MIMO (XL-MIMO) arrays in the 6G FR3 (7 to 24 GHz) and FR2 bands place users in the radiative near field, where spherical curvature, visibility regions (VRs), mutual coupling, and wideband beam squint degrade planar and polar transceivers. This paper establishes two exact wave optics dualities for 1D linear (ULA) and 2D planar (UPA) arrays and solves a Fermi-Dirac variational objective via its Karush-Kuhn-Tucker (KKT) conditions (CSFI-VR-NF). First, under a projective Legendre slope map, circular and spherical wavefronts map to exact projective lines and planes (Theorem 2), while Lemmas 1 and 2 derive the Visibility Conditioned Cramér-Rao Lower Bounds via the polar moment of inertia. Second, KKT stationarity yields a logistic visibility gate (100% F1, 48.42% PA power saving) and a Gauss-Newton solver achieving 14.56 cm (256-ULA) and 3.38 cm (1024-UPA) range error from one 1D sweep. Third, digital phase twist synthesis with a Polyak-Banach operator (Theorem 1) and oblique projector (Proposition 1) achieves 93.27% to 99.28% wideband gain and 98.05% genie capacity in 1.47 ms on CPU and 42.6 us on FPGA (1.38 W).