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 $$2s^3(s+1)+pq\bigl(s^3-7s^2-16s-12\bigr)\geq0, \qquad s=p+q.$$ 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: $\beta_{1/2,q}$ 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.