Radiative Near-Field 6G XL-MIMO: Projective Wavefront Geometry, Visibility-Conditioned Cramér-Rao Bounds, and Variational Hybrid Focusing

Extremely large-scale MIMO (XL-MIMO) arrays in the 6G FR3 (7–24 GHz) and FR2 bands place mobile users well inside the radiative near field, where spherical wavefront curvature, partial visibility regions (VRs), sub-array mutual coupling, and wideband beam squint occur simultaneously. Conventional near-field transceivers rely on second-order Fresnel Taylor approximations that break down at short ranges, 2D/3D polar-domain dictionary searches requiring 8x–16x pilot overhead, heuristic blockage masking that ignores cross-gap phase jumps, or costly analog true-time-delay (TTD) hardware. This paper resolves these four problems without extra pilots or analog delay lines through a joint wave-optics and variational Karush-Kuhn-Tucker (KKT) formulation (CSFI-VR-NF). First, a nonlinear projective Legendre tangent-slope transformation (tau = u / sqrt(1 - ||u||_2^2)) of multi-tone phase gradients algebraically cancels the 3D Euclidean distance radical without Fresnel Taylor truncation; circular and 3D spherical wavefronts consequently become exact projective lines and planes (Theorem 1). When nuisance phase jumps across disjoint visible segments are removed through the Fisher Schur complement, the Fisher information matrix yields closed-form visibility-conditioned Cramér-Rao lower bounds (CRLB_VR, Lemmas 1–2) inversely proportional to the aperture's polar spatial moment of inertia. Second, the stationarity conditions of a Fermi-Dirac variational objective produce an Aperture-Observability-Conditioned VR gate (AOC-VR, 100% F1-score, 48.42% power reduction) alongside a segment-decoupled Gauss-Newton solver that reaches 14.56 cm (1D 256-ULA) and 3.38 cm (3D 1024-UPA) range MAE from a single 1.0x angular pilot sweep. Third, above a closed-form sub-array partition bound (P >= P_crit), TTD-free digital phase twists combined with a contractive Polyak-Banach mutual-coupling operator (L_MC = 0.142 < 1, Theorem 2) and an oblique VR null-space projector (Proposition 1) deliver 93.27%–99.28% wideband gain (98.05% of genie capacity). Cross-validation under 16-point Pocklington-Hallén Method-of-Moments electromagnetics and bit-accurate synthesis of a square-root-free 16-bit dual-mode CORDIC processor on a Xilinx XCZU28DR RFSoC confirm 42.6 us execution latency at 1.38 W.