Asymptotic Hyperbolicity of Jensen Polynomials and the Finite-Strip Obstruction for the Riemann Hypothesis

We study the degree-d Jensen polynomials \( J_{d,n}(X) \) built from the moment sequence \( M_n=\int_0^\infty\Phi_1(u)u^{2n}\,du \) of the Riemann \( \Xi \)-function, which coincides with the classical Pólya–Jensen family. Using bridge coordinates, the staircase law, and Plancherel–Rotach asymptotics, we prove that \( J_{d,n}^\gamma \) is hyperbolic for all \( n\ge C_0^\infty d^4 \) (\( C_0^\infty\approx0.020 \); the analytic formula for \( C_0^\infty \) is rigorous but approximates the numerically observed value to within 2.6%); combined with the GORZ theorem for \( d\le8 \), this covers the entire asymptotic regime. We identify a phase-transition law \( n^*(d)=C_0^\infty d^4+\alpha d^3+\beta(-1)^d d^2+O(d) \) (Conjecture 3.5): the leading constant \( C_0^\infty\approx0.0195 \) is computed analytically and verified to within 2.6% of the empirical large-$d$ limit; the formula for \( \alpha \) is derived; its numerical value ≈ −0.2 to −0.3 is numerical evidence; the parity structure \( \beta(-1)^d d^2 \) is proved. For the finite strip \( 0\le n \) < \( C_0^\infty d^4 \) with \( d\ge9 \), the sole remaining gap, whose closure is equivalent to the Riemann Hypothesis under standard transversality, we establish four structural obstructions: ratio-barrier saturation (no usable margin, certified and numerical); frozen zero count (parity blocks any ladder, certified for \( d\le21 \)); interlacing-lift vacuity (proved); and a discriminant equivalence (proved under transversality), showing that all known local and inductive mechanisms fail simultaneously in this region. The problem reduces to: \( \operatorname{Disc}(J_{d,n}^\gamma)>0 \) for all \( d\ge9 \) and 0 ≤ n < \( C_0^\infty d^4 \); this requires moment data \( M_k \) for \( k\ge130 \), currently inaccessible.