Flexible-antenna systems, which use a small number of radio frequency (RF) chains to dynamically access a large set of candidate antenna locations, have emerged as a hardware-efficient architecture for 6G networks. Acquiring accurate channel state information (CSI) is critical for these systems, but it typically incurs a prohibitive pilot overhead that scales with the massive number of candidate locations. To address this bottleneck, we propose a unified sensing-assisted channel estimation framework tailored for flexible-antenna systems. It reduces the full CSI reconstruction problem to a consistent two-stage process: it first resolves the dominant DOAs from the uplink data symbols by exploiting the spatial geometry, requiring no dedicated sensing pilot, and then calibrates the associated path gains using a minimal number of calibration pilots. Building on this pipeline, we develop two Newton-MUSIC algorithms tailored to different propagation environments. For line-of-sight (LOS)-dominant environments with uncorrelated sources, we propose SOC-Newton-MUSIC, which leverages second-order covariance (SOC) for low-complexity DOA sensing. For non-line-of-sight (NLOS) environments with coherent multipath, where the number of sources may exceed the number of activated RF chains, we propose FOC-Newton-MUSIC, which exploits fourth-order cumulants (FOC) to restore source identifiability and structurally expand the available spatial degrees of freedom (DOFs) through a continuous difference co-array. In both cases, by reformulating the spatial spectrum search as a continuous optimization problem, we replace exhaustive dense grid searches with parallelized Newton refinements.

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