AI 中文总结
研究高阶戴森秩，发现其单位根特化有隐藏椭圆结构，通过有限\(q\)差分递推及Appell - Lerch修正项得出修正函数，满足扭曲椭圆律，平移后全纯有限部分有有限\(\theta\)分解，揭示其类似物非模拟模性而是特定有限\(\theta\)分解。
AI 中文摘要
当\(m = 1\)时，戴森秩生成函数是分拆理论、拉马努金拟\(\theta\)函数、调和马阿斯形式理论和非全纯雅可比形式理论之间的经典桥梁。秩是分拆上的一个统计量，对于\(m\geq2\)的高阶戴森系统是其天然的多变量细化，结合了\(m\)个分级秩贡献。与经典情况不同，这些高阶系统预计不适合模拟模框架，这就引出了其受何种解析结构支配的问题。我们表明它们的单位根特化具有隐藏的椭圆结构。有限\(q\)差分递推对预期的指标\(m\)椭圆变换律产生明确的多项式障碍，由于该障碍是有限的，其部分分式规范地确定有限多个Appell - Lerch修正项以消除它。修正后的函数满足扭曲的指标\(m\)椭圆律；自然平移消除扭曲，其全纯有限部分允许有限\(\theta\)分解。因此，在更高\(m\)时戴森模拟模现象的自然类似物不是模拟模性，而是由指标\(m\)椭圆变换律支配的有限\(\theta\)分解。这些结果源于人机合作，关键新公式在Lean/Mathlib中由AxiomProver形式化并机器验证。
英文摘要
When $m = 1$, the Dyson rank generating function is a classical bridge between partition theory, Ramanujan's mock theta functions, and the theory of harmonic Maass forms and nonholomorphic Jacobi forms. The rank is a statistic on partitions, and the higher Dyson systems, for $m \geq 2$, are a natural multivariable refinement of it, combining $m$ graded rank contributions. Unlike the classical case, these higher systems are not expected to fit the mock-modular framework, which raises the question of what analytic structure governs them. We show that their root-of-unity specializations carry a hidden elliptic structure. A finite $q$-difference recurrence produces an explicit polynomial obstruction to the expected index $m$ elliptic transformation law, and because the obstruction is finite, its partial fractions canonically determine finitely many Appell--Lerch correction terms that remove it. The corrected functions satisfy a twisted index $m$ elliptic law; a natural translation removes the twist, and their holomorphic finite parts admit finite theta decompositions. Thus, the natural analogue of Dyson's mock-modular phenomenon at higher $m$ is not mock modularity but a finite theta decomposition governed by an index $m$ elliptic transformation law. These results grew out of a human--AI collaboration, and the key new formulas were formalized and machine-verified in Lean/Mathlib by AxiomProver.
Comments33 pages; comments welcome
