Context. Metamorphic Testing is recognised in IEEE/ISO software-testing standards and increasingly recommended for AI systems, but its progress is bottlenecked by metamorphic relation (MR) identification: existing approaches (structured frameworks, mining and evolutionary pipelines, LLM-assisted methods, MetaPattern catalogues) share an inductive grounding that leaves three foundational questions open: origin, closure, and transferability. Objective. We propose a framework whose downstream step from program-induced operator algebra to MetaPattern set is mechanical and provable, while the upstream curation of the algebra is a stated empirical hypothesis with explicit scope precondition. Method. NOETHER is a two-layer framework. The upstream layer is an eight-block decomposition over recurrent mathematical structures (symmetry, order, self-adjoint, time-reversal, limit, qualitative-dynamics, method-comparison, relational equivalence). The downstream CONSTRUCT-MP algorithm produces a MetaPattern set with algebraic-closure (Theorem 1) and polynomial-time decidability (Theorem 2) guarantees. We test the framework on three operator-algebraic domains. Results. On Boltzmann reactor physics NOETHER systematises a prior inductive catalogue; on equivariant ML it derives executable MRs for rotation invariance, adjoint duality, and training-trajectory reversibility; on relational query optimisers it exercises the relational-equivalence block. The central falsifiable prediction (L*-blindness on homogeneity-preserving mutators) holds on the in-scope substrate. The absolute-completeness conjecture (Theorem 1') is falsified on PWR core diffusion via two pairwise-independent counterexamples that identify five Translate-extension dimensions. Conclusion. Induction is relocated from per-program MR sampling to a per-domain algebraic layer; the downstream step is deductive and mechanical.

翻译：背景：蜕变测试已被IEEE/ISO软件测试标准认可，并日益推荐用于人工智能系统，但其发展受困于蜕变关系识别这一瓶颈：现有方法（结构化框架、挖掘与进化流程、大语言模型辅助方法、MetaPattern目录）共享归纳性根基，由此遗留三个基础性问题：起源、封闭性与可迁移性。目标：我们提出一种框架，其从程序诱导算子代数到MetaPattern集合的下游步骤具有机械性与可证明性，而上游代数的精心构造则是带有明确适用范围前提的实证假设。方法：NOETHER是一种双层框架。上游层由对递归数学结构（对称性、序、自伴随性、时间反演、极限、定性动力学、方法比较、关系等价）的八块分解构成。下游CONSTRUCT-MP算法可生成具有代数封闭性（定理1）与多项式时间可判定性（定理2）保证的MetaPattern集合。我们在三个算子代数领域测试该框架。结果：在玻尔兹曼反应堆物理学中，NOETHER系统化了先前的归纳性目录；在等变机器学习中，它推导出旋转不变性、伴随对偶性及训练轨迹可逆性的可执行蜕变关系；在关系查询优化器中，它验证了关系等价块。核心可证伪预测（关于保持同质性变异算子的L*盲性）在适用范围基底上成立。绝对完备性猜想（定理1'）在压水堆堆芯扩散场景中被两个成对独立的反例所证伪，从而识别出五个Translate扩展维度。结论：归纳过程从每个程序的蜕变关系采样迁移至每个领域的代数层；下游步骤则是演绎性与机械性的。