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

We study the degree-d Jensen polynomials Jd,n(X) built from the moment sequence Mn=∫0∞Φ1(u)u2ndu of the Riemann Ξ-function, which coincides with the classical Pólya–Jensen family. Using bridge coordinates, the staircase law, and Plancherel–Rotach asymptotics, we prove that Jd,nγ is hyperbolic for all n≥C0∞d4 (C0∞≈0.020, proved); combined with the GORZ theorem for d≤8, this covers the entire asymptotic regime. We identify a phase-transition law n*(d)=C0∞d4+αd3+β(−1)dd2+O(d) (Conjecture 3.5): the leading constant C0∞≈0.0195 is proved analytically; the formula for α is derived; its numerical value ≈−0.2 to −0.3 is numerical evidence; the parity structure β(−1)dd2 is proved. For the finite strip 0≤n<C0∞d4 with d≥9, 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≤21); 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: Disc(Jd,nγ)>0 for all d≥9 and 0≤n<C0∞d4; this requires moment data Mk for k≥130, currently inaccessible.