连续体中多场不稳定性的拓扑基础：第1部分：基础 第2部分：一维自旋链的解析公式 第3部分：数值粗粒化
Topological Foundations of Multi-Field Instabilities in Continua: Part 1: Foundations Part 2:Analytical Formulation for 1-D Spin Chains Part 3: Numerical Upscaling
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中文总结 AI 辅助
该系列论文建立颗粒连续体中多场不稳定性的无参数拓扑分类。通过离散VMC接触公式等方法，扩展麦克斯韦刚度计数。经数值积分等验证不同通道数特性，VMC通道映射到可观测量，可进行无参数评估及提出可证伪协议。
中文摘要 AI 辅助
本系列三部分建立了颗粒连续体中多场不稳定性的无参数拓扑分类，将麦克斯韦刚度计数扩展到动态非平衡过程。第1部分给出离散体积-力学-构型（VMC）接触公式，将接触尺度拓扑映射到宏观多物理场耦合。奇偶性定理使奇数通道数的结构零模式出现，产生时间反演对称性破缺的“通道”层。第2部分给出一维自旋链的解析公式，最小通道为N = 3的VMC接触，其斜块有持续零特征值和构型漂移。第3部分通过对三对角斜对称昂萨格链的四精度积分，证实奇数N和偶数N的不同特性，VMC通道映射到可测量的离散单元法可观测量，可对通道数进行无参数评估及四个可证伪的侧限仪协议。
英文摘要
This three-part series establishes a parameter-free, topological classification of multi-field instability in granular continua, extending Maxwell's rigidity count to dynamic, non-equilibrium processes. Part 1 (Foundations): a discrete Volumetric-Mechanical-Configurational (VMC) contact formulation maps contact-scale topology to macroscopic multiphysics coupling. A Parity Theorem, $\det(\mathsf{L})=(-1)^N\det(\mathsf{L})$, forces a structural null-mode for every odd channel count $N$, creating "Gateway" layers of broken time-reversal symmetry; once the basis-invariant Gateway number $\mathcal{G}_{\rm inv}\geq 1$, gyroscopic pumping drives non-modal transient amplification along the null direction. Part 2 (analytical, 1-D spin chains): the minimal Gateway is the $N=3$ VMC contact, whose skew block $\mathsf{L}\in\mathfrak{so}(3)$ carries a persistent zero eigenvalue and an unresisted configurational drift that operates even without friction. In an acyclic chain (first Betti number $β_1=0$) this isolates dilatancy; closed-form solutions give secular drift for $N=3$ and harmonic confinement for $N=4$. Part 3 (numerical upscaling): quad-precision integration of tridiagonal skew-symmetric Onsager chains ($N=3$ to $50$) confirms the contrast between odd-$N$ secular drift and even-$N$ confinement on invariant tori, with even-chain frequencies scaling as $|λ_{\min}^{\rm even}|\simγπ/N$. VMC channels map to measurable DEM observables, enabling parameter-free evaluation of $\mathcal{G}$ and four falsifiable oedometer protocols.
