用最大广义原理推导图谱核动力学的闭式几何泛函，揭示固定点与相变预警机制。
Spectral Kernel Dynamics via Maximum Caliber: Fixed Points, Geodesics, and Phase Transitions
- 基于最大广义原理，将谱核动力学解耦为可解析求解的独立一维问题。
- 提出谱熵作为网络结构相变的O(N)早期预警信号，验证于8节点路径图。
- 方法具物理类比性，适合研究复杂网络动态与相变的科研人员。
我们通过将最大广义（MaxCal）变分原理应用于图拉普拉斯特征基下的谱转移函数 h(λ)，推导出有限图上核动力学的闭式几何泛函。核心结果是：MaxCal驻定条件可解耦为N个一维问题，其显式解为 h*(λ_l) = h_0(λ_l) exp(-1 - T_l[h*])，从而通过指数倾斜获得自洽（固定点）核，实现对数线性Fisher-Rao测地线（推论2）、对角海森稳定性判据（推论3），以及谱核空间的l^2_+同构（命题3）。谱熵 H[h_t] 提供了计算开销为O(N)的网络结构相变早期预警信号（备注7）。所有结论均在路径图P_8上，以高斯互信息源进行数值验证，使用开源库kernelcal。该框架基于与爱因斯坦场方程的结构类比，仅作引导而非等价关系；具体极限条件见第6节。
原文摘要 · Abstract (English)
We derive a closed-form geometric functional for kernel dynamics on finite graphs by applying the Maximum Caliber (MaxCal) variational principle to the spectral transfer function h(lambda) of the graph Laplacian eigenbasis. The main result is that the MaxCal stationarity condition decouples into N one-dimensional problems with explicit solution: h*(lambda_l) = h_0(lambda_l) exp(-1 - T_l[h*]), yielding self-consistent (fixed-point) kernels via exponential tilting (Corollary 1), log-linear Fisher-Rao geodesics (Corollary 2), a diagonal Hessian stability criterion (Corollary 3), and an l^2_+ isometry for the spectral kernel space (Proposition 3). The spectral entropy H[h_t] provides a computable O(N) early-warning signal for network-structural phase transitions (Remark 7). All claims are numerically verified on the path graph P_8 with a Gaussian mutual-information source, using the open-source kernelcal library. The framework is grounded in a structural analogy with Einstein's field equations, used as a guiding template rather than an established equivalence; explicit limits are stated in Section 6.
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