Algebraic Construction of Localized Topological Degree Statistics Operator (No.4) and Rigorous Proof of Dirichlet Energy Positive Boundedness

Abstract

This manuscript is specification for Operator 4 ($\mathcal{M}_{\text{degree}}$), a core open‑source component of the Phase 1 Homogeneous‑Metric Operator Suite within Status‑Relational‑Entropy (SRE) Dynamics. It receives outputs from Operator 1 and Operator 2 and provides spectral priors for Operator 5. Targeting discontinuous step‑noise from distributed‑actor local‑horizon fragmentation and local zero‑degree vacuum singularities, Operator 4 constructs analytic homogeneous smoothing measures combining the 2‑Step Graph‑Walk Kernel and spectral‑bound regularisation terms. The central mathematical result is Theorem 4.1, the Rigid‑Clamping Theorem for the lower bound of the Dirichlet‑Energy Functional, which proves that under sparse or zero‑degree vacuum conditions the functional is rigidly confined to fully positive‑definite compact subspace $\mathcal{E}_D(E_s) \ge \lambda_2(n) \cdot \|E_s\|_2^2 >0$, eliminating floating‑point logarithmic‑divergence singularities and guaranteeing mathematical completeness for long‑timescale distributed simulation. The document also sketches the subsequent pipeline roadmap towards Operator 6 with Rayleigh‑Ritz boundary‑splicing kernel and Lanczos low‑rank iteration. 本文为状态‑关系熵(SRE)动力学Phase1齐次度量算子组4号算子($\mathcal{M}_{\text{degree}}$)数学规范,承接算子1、算子2输出,为算子5提供谱先验。针对分布式Actor局域视界割裂带来的阶跃噪声、局域零度数真空奇点,算子4融合2阶图行走核与谱边界正则项构造解析齐次平滑测度。核心数理成果为定理4.1狄利克雷能量泛函下界刚性钳制定理:证明在极端稀疏、零度数真空工况下狄利克雷能量泛函被刚性约束于完全正定紧致子空间$\mathcal{E}_D(E_s) \ge \lambda_2(n) \cdot \|E_s\|_2^2 >0$,消除浮点对数发散奇点,保障长周期分布式仿真数理完备性。文档同时给出流水线下一步算子6的研发路线,包含Rayleigh‑Ritz边界拼接核与Lanczos低秩迭代。