Operator 6: Sub-space Spectral Sieve and Splicing Operator for Localized Complexity Clamping in SRE Dynamics

Abstract

This manuscript is final peer‑review specification for Operator 6 ($\mathcal{P}_{\text{sieve}} \cup \mathcal{O}_{\text{splice}}$), the final convergence component of Phase 1 Homogeneous‑Metric operator suite within open‑source SRE Operator 1‑6 suite. It streams spectral‑prior invariants to Operator 4 and Operator 5. Operator 6 applies Algebraic Rayleigh‑Ritz Splicing Kernels to extract global Laplacian eigen‑information purely from local sub‑domains, reducing computational cost from $\mathcal{O}(n^3)$ down to $\mathcal{O}(m_g \cdot k_{\text{rank}})$ and removing global synchronous stalls in distributed Actor clusters. Theorem 6.1 proves the Rayleigh‑Ritz approximation error bound; Theorem 6.2 establishes the sparse sub‑domain complexity clamping bound. Pre‑Pruning Operator 2‑Batch guarantees positive‑definite Fiedler eigenvalue $\lambda_2(n)>0$, eliminating floating‑point singularities for downstream operators. Operators 7‑10 are closed‑source commercial‑core modules outside this scope. 本文为状态‑关系熵(SRE)动力学算子套件6号算子($\mathcal{P}_{\text{sieve}} \cup \mathcal{O}_{\text{splice}}$)终审定稿规范,属于开源算子1‑6套件Phase1同质度量算子集群的收敛算子,向下游算子4、算子5输出谱先验。算子6采用代数Rayleigh‑Ritz拼接核,完全依靠局域子域提取全局拉普拉斯本征信息,将计算开销从$\mathcal{O}(n^3)$降至$\mathcal{O}(m_g \cdot k_{\text{rank}})$,消除分布式Actor集群的全局强同步挂起死锁。定理6.1给出Rayleigh‑Ritz逼近误差界;定理6.2证明稀疏子域复杂度钳制定理。前置剪枝算子2‑Batch保证Fiedler代数连通度$\lambda_2(n)>0$,消除下游算子浮点奇点。算子7‑10为闭源商业核心组件,不在本文档范围内。