arXiv:2610.33535v1 Announce Type: new Abstract: Movable antennas (MAs) provide additional spatial degrees of freedom by adapting their physical positions to the propagation environment. However, existing studies mainly focus on single-band or contiguous wideband transmission, leaving the cross-band spatial behavior of MAs under frequency-dependent multipath propagation largely unexplored. To address this gap, we investigate a downlink multi-band MA communication system where a common physical MA configuration is shared across multiple communication bands. Such sharing induces an inherent configuration conflict, as different bands often favor substantially divergent MA configurations. We first develop a frequency-dependent multipath channel model with shared propagation geometry but band-dependent multipath component powers and phases. Using this model, we formulate a sum-rate maximization problem by jointly optimizing user-band assignment, MA positioning, and transmit precoding. We further establish a cross-band spatial scaling relationship for the normalized channel power and derive both a position-independent channel gain upper bound and a sum-rate upper bound. Based on these results, we propose a cross-band analysis-guided optimization (CAGO) algorithm and a cross-band graph transformer (CBGT) to reduce online optimization complexity. Numerical results validate the derived cross-band spatial relationship and show that CAGO consistently outperforms the narrowband, wideband, and fixed-antenna benchmarks while remaining close to the idealized independent-MA benchmark. Additionally, CBGT with online refinement achieves near-CAGO sum-rate performance with substantially lower online computational cost, providing a practical framework for exploiting the spatial reconfiguration capability of MAs in multi-band communication systems.