AI 中文总结
该研究开发了机器引导的递推边界理论，结合符号搜索与Q学习发现关键结果，证明了Shi和Wang的猜想3.8，得到Zagier第十二个三阶秩示例对偶纳姆和的二阶递推与五单元证明。
AI 中文摘要
我们为正定纳姆和（Nahm sums）的仿射族开发了一种携带证明的递推边界方法。一种通用的坐标邻接关系为有限证明提供了精确的代数单元；热带面极限定理确定了参数方向，在此方向上高阶纳姆和退化为低阶θ或θ超几何边界值；而递推边界原理可从足够多的独立边界极限中恢复初始和。机器学习和强化学习仅用于发现：进化符号搜索从精确代数数据中提出递推和渐近射线，而Q学习（Q-learning）智能体则搜索合法邻接单元恒等式的短序列。任何学习得到的输出都不被接受为证明：每个成功的候选都被替换为精确的符号证明。作为主要应用，我们证明了Shi和Wang的猜想3.8（arXiv:2607.23257），该猜想针对与Zagier的第十二个三阶秩示例对偶的纳姆和。搜索得到了单参数族的二阶递推和其五单元证明，以及两条分别指向二元和一元θ级数的渐近射线，这些为两个初始和提供了两个线性方程。求解所得的2×2方程组将猜想简化为两个广义η恒等式，由Γ₁(300)和Γ₁(100)上的价论证证明。因此，Zagier第十二个示例的对偶是模形式，且该仿射族的每个成员都是两个基础乘积的明确ℤ[q,q⁻¹]组合。论文附带完整的验证工件。
英文摘要
We develop a proof-carrying recurrence--boundary method for affine families of positive-definite Nahm sums. A universal coordinate-contiguous relation provides exact algebraic cells for finite certificates; a tropical face-limit theorem identifies parameter directions along which a higher-rank Nahm sum degenerates to a lower-rank theta or theta--hypergeometric boundary value; and a recurrence--boundary principle recovers the initial sums from sufficiently many independent boundary limits. Machine learning and reinforcement learning are used only for discovery. Evolutionary symbolic search proposes recurrences and asymptotic rays from exact algebraic data, while a $Q$-learning agent searches for short sequences of legal contiguous-cell identities. No learned output is accepted as proof: every successful candidate is replaced by an exact symbolic certificate. As the main application, we prove Shi and Wang's Conjecture~3.8 (arXiv:2607.23257) for the Nahm sums dual to Zagier's twelfth rank-three example. The search finds a second-order recurrence for a one-parameter family and a five-cell certificate for it, together with two asymptotic rays leading to binary and unary theta series. These give two linear equations for the two initial sums. Solving the resulting $2\times2$ system reduces the conjecture to two generalized-eta identities, certified by valence arguments on $Γ_1(300)$ and $Γ_1(100)$. Consequently the dual of Zagier's twelfth example is modular, and every member of the affine family is an explicit $\mathbb{Z}[q,q^{-1}]$-combination of the two base products. Complete verification artifacts accompany the paper.
CommentsComments Welcome!
