AI 中文总结
本文通过几何插值方法，确定了 Sine$_\beta$ 过程相关函数在临界和次临界融合区域的精确首阶修正系数，解决了作者先前提出的猜想。
AI 中文摘要
我们确定了 Sine$_\beta$ 过程在临界和次临界区域 $m\beta\leq1$ 中相关函数对领先 Vandermonde 融合律的首个修正。当 $m\beta<1$ 时，归一化修正的阶为 $|\varepsilon|^{1+m\beta}$，其系数严格为负，由显式 gamma 函数比值乘以一个绝对收敛的算术-几何平均亏损积分给出。在 $m\beta=1$ 时，修正为 $-\frac{\sum_{i<j}(a_i-a_j)^2}{8m^2(2m+1)}\varepsilon^2\log(1/|\varepsilon|)+O(\varepsilon^2)$。对于两个合并点，次临界系数化简为涉及 $\sec(\pi\beta)-1$ 的 gamma 函数表达式，临界对数系数为 $-1/160$。论证始于圆 Jacobi 权重的几何插值。在插值导数中选择一个粒子会使融合电荷增加 $\beta$，并在剩余的随机 zeta 期望中留下严格正幂矩余量。这产生了一个绝对收敛的单粒子恒等式以及在碰撞尺度上的均匀控制。这些结果解决了作者早期预印本中的临界与次临界猜想，并补充了其超临界二阶展开。
英文摘要
We determine the first correction to the leading Vandermonde fusion law for the correlation functions of the Sine$*β$ process in the critical and and subcritical regimes $mβ\leq1$. When $mβ<1$, the normalized correction is of order $|\varepsilon|^{1+mβ}$, with a strictly negative coefficient given by an explicit gamma-function ratio times an absolutely convergent arithmetic-geometric mean deficit integral. At $mβ=1$, it is $-\frac{\sum*{i<j}(a_i-a_j)^2}{8m^2(2m+1)}\varepsilon^2\log(1/|\varepsilon|)+O(\varepsilon^2)$. For two merging points, the subcritical coefficient reduces to a gamma-function expression involving $\sec(πβ)-1$, and the critical logarithmic coefficient is $-1/160$. The argument starts from a geometric interpolation of circular-Jacobi weights. Selecting one particle in the interpolation derivative increases the fused charge by $β$ and leaves a strict positive-power moment margin in the remaining stochastic-zeta expectation. This yields an absolutely convergent one-particle identity and uniform control at the collision scale. The results resolve the critical and subcritical conjecture in the author's earlier preprint and complement its supercritical second-order expansion.
Comments9 pages. Companion paper to arXiv:2608.23742
