AI 中文总结
研究双参数贝塔分布族的自由无限可分性，通过对自由累积量条件正定性产生的汉克尔条件进行分析，得到必要不等式及相关行列式，借助精确有理\(LDL^{\mathsf T}\)证书展示各阶汉克尔检验效果，完成部分边界族分类，给出有限阶障碍结论。
AI 中文摘要
我们研究双参数贝塔分布族\(\{\beta_{p,q}:p,q>0\}\)中的自由无限可分性。自由累积量的条件正定性产生了一系列必要的汉克尔条件。我们对第一个非平凡行列式进行因式分解，得到明确的必要不等式\[2s^3(s + 1)+pq(s^3 - 7s^2 - 16s - 12)\geq0\]，其中\(s = p + q\)。其严格反向定义了一个开放的二维非自由无限可分区域，该区域不包含在先前已知的排除区域中。作为边界结果，我们完成了一个边界族的分类：\(\beta_{1/2,q}\)自由无限可分当且仅当\(q\geq3/2\)。还明确得到了对称变量\(s = p + q\)和\(u = pq/s^2\)下的\(3\times3\)行列式。最后，精确有理\(LDL^{\mathsf T}\)证书表明，从\(3\times3\)到\(12\times12\)的每个主导汉克尔检验都严格扩大了所有先前主导检验提供的排除区域。特别是，\(4\times4\)检验已经检测到一个\(p + q>3\)的开放集，超出了\(2\times2\)行列式可及的范围。结果是有限阶障碍而非完整分类；两个极限论证解释了为什么主导汉克尔层次结构的任何固定成员\(H_N\)都不能在小参数边界上提供统一障碍。
英文摘要
We study free infinite divisibility in the two-parameter family of beta distributions $\{β_{p,q}:p,q>0\}$. Conditional positive definiteness of free cumulants yields a hierarchy of necessary Hankel conditions. We factor the first nontrivial determinant and obtain the explicit necessary inequality \[ 2s^3(s+1)+pq\bigl(s^3-7s^2-16s-12\bigr)\geq0, \qquad s=p+q. \] Its strict reverse defines an open two-dimensional non-freely-infinitely-divisible region not contained in the previously known exclusions. As a boundary consequence, we complete the classification of one boundary family: $β_{1/2,q}$ is freely infinitely divisible if and only if $q\geq3/2$. The $3\times3$ determinant is also obtained explicitly in the symmetric variables $s=p+q$ and $u=pq/s^2$. Finally, exact-rational $LDL^{\mathsf T}$ certificates show that each leading Hankel test from $3\times3$ through $12\times12$ strictly enlarges the exclusion supplied by all preceding leading tests. In particular, the $4\times4$ test already detects an open set with $p+q>3$, beyond the range accessible to the $2\times2$ determinant. The results are finite-order obstructions rather than a complete classification; two limiting arguments explain why no fixed member $H_N$ of the leading Hankel hierarchy can provide a uniform obstruction up to the small-parameter boundary.
Comments9 pages, 1 figure; exact-rational verification code and certificates included as ancillary files
