Higher-Order Hankel Obstructions to Free Infinite Divisibility for Beta Distributions

We study free infinite divisibility in the two-parameter family of beta distributions $\{{\beta }_{p,q}:p,q>0\}$. 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 $$2s^3(s+1)+pq(s^3-7s^2-16s-12)\ge 0,s=p+q\mathrm{.}$$ 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: ${\beta }_{1\mathrm{/}2,q}$ is freely infinitely divisible if and only if $q\ge 3\mathrm{/}2$. The $3\times 3$ determinant is also obtained explicitly in the symmetric variables $s=p+q$ and $u=pq\mathrm{/}{s}^2$. Finally, exact-rational $LD{L}^{\mathsf{T}}$ certificates show that each leading Hankel test from $3\times 3$ through $12\times 12$ strictly enlarges the exclusion supplied by all preceding leading tests. In particular, the $4\times 4$ test already detects an open set with $p+q>3$, beyond the range accessible to the $2\times 2$ 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.