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

We investigate the onset of hyperbolicity in Jensen polynomials Jd,n associated with the Riemann Ξ-function and identify a robust parity-driven bifurcation with a natural geometric interpretation. Numerical analysis for degrees 5≤d≤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≥13) with decoupled geometric invariants, suggestive of dynamics on decorated (wild) character varieties. We interpret this dichotomy through a connection with the PIII(D6) tau-function arising in the Painlevé confluence diagram. Defining τ(t)=Ξ(12+−t)/Ξ(12), we construct a generating function B(w)=∑j≥0bjwj from the cumulants of logΞ(12+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 γk2 (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.