莫比乌斯拓扑电磁场中的标量绝缘与几何通道锁合:从矢量骨架到有效光学度规(Scalar vs. Geometric Locking in Möbius EM Fields)

Abstract The near-field electromagnetic structure generated by a biconical Möbius coil driven by a 10 MHz, 4 A continuous wave exhibits highly selective, layer-resolved coupling of its topological information to different physical field quantities. Using the complex-phasor field data as the sole authoritative input, we systematically investigate the topological misalignment angles across the upper and lower cross-sections (z = ±1.1 cm) for five physical layers: energy density (scalar), drift field (vector), Maxwell stress (rank-2 tensor), Gordon effective optical metric (metric), and effective Ricci curvature (curvature). The results indicate that the scalar energy-density misalignment is only 9.9°, rendering it fully insulating to the Möbius twist. In contrast, the vector drift field exhibits a spatial-angle misalignment of 86.4° (m=2 spectral phase ±172.7°); the Maxwell-stress principal axis misalignment is 89.8°; the Gordon optical-metric principal axis misalignment is 81.1° (coinciding with the stress principal axis within <1.5° in strong-field regions); and the effective-Ricci principal axis misalignment is 79.1°. All vector/tensor/curvature channels lock onto the topological skeleton at approximately 90° (phase 180°), demonstrating a significant dichotomy: "scalar insulation vs. geometric channel following." On the waist plane (z = 0), the Ricci scalar R≈0 (+0.07 m⁻²) coexists with a spatial-curvature anisotropy as high as 16, exhibiting a traceless shear geometry (no volumetric breathing, pure shape distortion). The energy-momentum conservation check yields |∇·g|_max = 1.4×10⁻⁸ N/(m³·s), confirming strict steady-state conservation in vacuum. A key calibration correction is also reported (the local 45.9° half-angle hypothesis fails; the full-ring 89.8° value stands), providing theoretical criteria for structures such as topological deflectors. It should be particularly noted that conventional Transformation Optics (TO) manipulates local refractive indices and optical metrics by engineering complex metamaterial physical media. In contrast, our findings demonstrate that, without the intervention of any physical media, the Möbius topological structure of the electromagnetic field alone can locally generate an effective Gordon optical metric featuring traceless shear characteristics. This provides a novel 'field-topology-driven' paradigm for controlling the effective propagation geometry of photons, offering distinct conceptual value beyond traditional metamaterial designs. 摘要 双锥-莫比乌斯线圈在 10 MHz、4 A 连续波驱动下产生的近场电磁结构,其拓扑信息在不同物理场量之间呈现高度选择性的分层耦合。本文以复相量场数据为唯一权威输入,系统检验了能量密度(标量)、漂移场(矢量)、麦克斯韦应力(二阶张量)、Gordon 等效光学度规(度规)与有效 Ricci 曲率(曲率)五个物理层的上下十字架(z=±1.1 cm)拓扑错位角。结果显示:标量能量密度错位仅 9.9°,对莫比乌斯扭转完全绝缘;矢量漂移场空间角错位 86.4°(m=2 谱相位 ±172.7°);麦克斯韦应力主轴错位 89.8°;Gordon 光学度规主轴错位 81.1°(强场区与应力主轴重合误差 <1.5°);有效 Ricci 曲率主轴错位 79.1°。所有矢量/张量/曲率通道均以约 90°(相位 180°)的错位角与拓扑骨架锁合,呈现出显著的“标量绝缘与几何通道跟随”的二分现象。腰部 z=0 平面 Ricci 标量 R≈0(+0.07/m²)而空间曲率各向异性高达 16,呈现出无迹剪切几何特征(无体积呼吸、纯形状扭曲)。能量-动量守恒核验 |∇·g| 全域 max=1.4×10⁻⁸ N/(m³·s),稳态真空严格守恒。本文同时给出关键口径勘误(局部 45.9° 半角假说不成立,全环 89.8° 为准),为“拓扑偏转器”类结构提供理论判据。此外,需要特别指出的是,传统的变换光学(Transformation Optics)主要通过设计复杂的超材料物理介质来改变局部的折射率和光学度规。而本文的研究表明,无需任何实体介质的介入,单纯依靠电磁场的莫比乌斯拓扑结构,即可在局部生成具有无迹剪切特征的有效 Gordon 光学度规。这为控制光子等效传播几何提供了一种全新的‘场拓扑驱动’范式,具有区别于传统超材料设计的独特概念价值。