Second Operator (Local Metric and Probabilistic Pruning Operator M_chi o E_local) Rigorous Mathematical Derivation and Specification (Standardized Edition)

Abstract

This document constitutes the full revised mathematical‑specification manuscript for the Second Operator ($\mathcal{M}_\chi \circ \mathcal{E}_{\text{local}}$), one core open‑source component within the Operator 1‑6 suite of Status‑Relational‑Entropy (SRE) Dynamics. This specification succeeds the formal parametric topology generated by the First Operator ($\mathcal{G}_{n \to n+1}$) and strictly adheres to the No‑Dimension Principle. The framework completely eradicates any dependency on background coordinate metrics, embedding spaces, or artificial spatiotemporal metrics, computing discrete evolutionary steps purely via local topological invariants including causal‑depth measures and graph‑walk interference statistics. The manuscript delivers rigorous axiomatic derivations covering graph‑to‑matrix morphic mapping, asymptotic $O(1)$ local computational complexity bounds, full un‑truncated two‑step topological path‑interference expansion, and algebraic criteria for topological frustration within arbitrary‑length graph cycles. Two competing pruning mechanisms (Paradigm A edge‑zeroing mode and Paradigm B elimination‑conduction / forced‑spin‑1 mode) are mathematically evaluated; Paradigm B is formally adopted as the standard pruning rule, proven to preserve cycle‑space frustration invariants without physically severing graph connectivity. Statistical‑mechanical derivation of local topological Hamiltonian phase‑separation is presented, alongside an endogenous proof for the global vertex‑degree saturation upper‑bound $K_0$. Critical circular dependency of coupling‑constant $\lambda(n)$ against instantaneous spectral‑radius is resolved by adopting historical‑step spectral invariants. Explicit topological‑weight mapping for overlapping multi‑cycle systems and prospective frustration‑judgement formalism over the symbolic‑polynomial ring are established. Compatibility and null‑space preservation of third‑order graph‑Laplacian structures under Boolean pruning‑mask transformations are demonstrated. Two asymptotic boundary‑limit regimes ($\lambda \to 0$ zero‑dissipation crystalline phase; $\lambda \to \infty$ infinite‑dissipation heat‑death phase) are analysed. This revised release integrates P0‑level emergency revisions and P1‑level supplementary improvements, completing ten numbered theorems and associated corollaries. Operators 7‑10 belong to closed‑source commercial‑core modules and are outside the scope of this document. 本文为状态‑关系熵(SRE)动力学开源算子1‑6套件内第二算子($\mathcal{M}_\chi \circ \mathcal{E}_{\text{local}}$)的完整修订数学规范文档。本规范承接第一算子($\mathcal{G}_{n \to n+1}$)生成的形式参数化拓扑,严格遵循无背景度规原则。框架完全不依赖背景坐标度规、嵌入空间或者人为预设时空度规,仅依靠因果深度、图游走干涉统计等局域拓扑不变量完成离散演化步计算。 文稿给出严格公理化推导:图‑矩阵态射同态映射、局域计算开销渐近$O(1)$复杂度上界、无截断二步拓扑路径干涉全展开、任意长度环路拓扑阻挫代数判据。对两套剪枝机制(范式A置零模式、范式B消除‑传导/强制自旋‑1模式)做数学评估;范式B被确立为标准化剪枝规则,证明该模式可以保全圈空间阻挫不变量,同时不会物理破坏图连通性。给出局域拓扑哈密顿量相分离的统计力学推导,内生证明全局顶点度数饱和上界$K_0$;采用历史步谱不变量解决耦合常数$\lambda(n)$与瞬时谱半径之间关键循环依赖;建立多交叠环路系统的拓扑权重显式映射,以及符号多项式域的前瞻阻挫判定形式体系;证明布尔剪枝掩码变换下三阶图拉普拉斯结构的相容性与零空间保全性质。同时分析两套渐近边界极限工况($\lambda \to 0$零耗散结晶相;$\lambda \to \infty$无穷耗散热寂相)。 本次修订整合P0级紧急修订、P1级补充完善内容,包含十条正式定理及配套推论。算子7‑10属于闭源商业核心模块,不在本文档覆盖范围内。