This paper presents a minimal operator-only term rewriting system with seven constructors and eight reduction rules. The main contribution is a mechanically-verified proof of strong normalization for a guarded fragment using a novel triple-lexicographic measure combining a phase bit, multiset ordering (Dershowitz-Manna), and ordinal ranking. From strong normalization, a certified normalizer with proven totality and soundness is derived. Assuming local confluence (verified through critical pair analysis), Newman's Lemma yields confluence and therefore unique normal forms for the safe fragment. Impossibility results showing that simpler measures, such as additive counters, polynomial interpretations, and single-bit flags, provably fail for rules with term duplication, are established. The work demonstrates fundamental limitations in termination proving for self-referential systems. A conjecture is stated: no relational operator-only TRS can have its full-system termination proved by internally definable methods. Here "relational" is equivalent to "capable of ordered computation": systems with a recursor enabling iteration over successors, comparison, or sequential counting. Such recursors necessarily redistribute step arguments across recursive calls, defeating all additive termination measures. This structural limitation applies to any TRS expressive enough to encode ordered data. All theorems have been formally verified in a proof assistant. The Lean formalization is available at https://github.com/MosesRahnama/OperatorKO7.

翻译：本文提出一个仅包含算子的极小项重写系统，该系统具有七个构造子和八条归约规则。主要贡献在于：通过结合相位位、多重集序（Dershowitz-Manna）和序数排名的三重字典序度量，对受保护片段给出了机械验证的强范式化证明。基于强范式化结果，导出一个经过认证的范式化器，其完全性与可靠性均获证明。在局部合流性（通过临界对分析验证）的假设下，应用纽曼引理可推出合流性，从而确保安全片段存在唯一范式。研究建立了不可能性结果：对于包含项复制的规则，可证明加法计数器、多项式解释和单比特标志等简单度量方法必然失效。这项工作揭示了自指系统终止性证明的根本局限性。文中提出一个猜想：任何纯关系算子TRS的完整系统终止性均无法通过内部可定义的方法得到证明。此处“关系性”等价于“具备有序计算能力”：即系统需包含能在后继元上进行迭代、比较或顺序计数的递归子。此类递归子必然在递归调用间重新分配步进参数，从而破坏所有加法型终止度量。该结构局限性适用于任何足以编码有序数据的TRS。所有定理均在证明助手中完成形式化验证。Lean形式化代码详见 https://github.com/MosesRahnama/OperatorKO7。