Multi-modal medical image synthesis is pivotal for alleviating clinical data scarcity, yet existing methods fail to reconcile global anatomical consistency with high-fidelity local detail. We propose FermatSyn, which addresses three persistent limitations: (1) SAM2-based Prior Encoder that injects domain-aware anatomical knowledge via LoRA$^{+}$ efficient fine-tuning of a frozen SAM2 Vision Transformer; (2) Hierarchical Residual Downsampling Module (HRDM) coupled with a Cross-scale Integration Network (CIN) that preserves high-frequency lesion details and adaptively fuses global--local representations; and (3) continuity constrained Fermat Spiral Scanning strategy within a Bidirectional Fermat Scan Mamba (BFS-Mamba), constructing an approximately isotropic receptive field that substantially reduces the directional bias of raster or spiral serialization. Experiments on SynthRAD2023, BraTS2019, BraTS-MEN, and BraTS-MET show FermatSyn surpasses state-of-the-art methods in PSNR, SSIM, FID, and 3D structural consistency. Downstream segmentation on synthesized images yields no significant difference from real-image training ($p{>}0.05$), confirming clinical utility. Code is available at https://github.com/exsinger-hub/ FermatSyn.

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