Parity Bifurcation, P_III(D6) Topology, and a Stieltjes Framework to Jensen Polynomial Hyperbolicity

We investigate the onset of hyperbolicity in Jensen polynomials $J_{d,n}$ associated with the Riemann $\Xi$-function and identify a robust parity-driven bifurcation with a natural geometric interpretation. Numerical analysis for degrees $5\le d\le 16$ reveals two distinct regimes. For even $d$, the roots form a compact complex cluster whose imaginary extent decreases smoothly, and the transition to hyperbolicity is governed by a single complex-conjugate pair, consistent with a low-dimensional (tame) geometric structure. For odd $d$, a hierarchy of more intricate onset mechanisms emerges, including single-event transitions ($d=11$) and intermittent regimes ($d\ge 13$) with decoupled geometric invariants, suggestive of dynamics on decorated (wild) character varieties. We interpret this dichotomy through a connection with the $P_{\mathrm{III}}(D_6)$ tau-function arising in the Painlevé confluence diagram. Defining $\tau(t)=\Xi(\frac{1}{2}+\sqrt{-t})/\Xi(\frac{1}{2})$, we construct a generating function $B(w)=\sum_{j\ge0} b_j w^j$ from the cumulants of $\log \Xi(\frac{1}{2}+z)$ using high-precision Cauchy/DFT methods (280--400-digit arithmetic), without explicit use of the zero expansion. Two independent numerical diagnostics indicate that $B$ exhibits Stieltjes-type behavior: (i) positivity of Hankel determinants up to order $N=30$, and (ii) Padé approximants whose poles converge to $\gamma_k^2$ (squares of Riemann-zero ordinates) with stabilizing residues. These results provide strong evidence that the parity bifurcation observed in Jensen polynomial onset reflects a finite-dimensional manifestation of an underlying moment-based positivity structure. Motivated by this correspondence, we formulate a conjecture relating the Stieltjes nature of $B(w)$ to the eventual hyperbolicity of Jensen polynomials. This conjecture suggests a bridge between finite-dimensional root geometry and an infinite-dimensional kernel-based positivity framework, while leaving open the problem of establishing such positivity independently of the zero expansion.