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

1. Introduction

1.1. Background

The Riemann Hypothesis (RH) [1,2] asserts that every non-trivial zero of the Riemann zeta function $\zeta \left(s\right)$ lies on the critical line $\mathrm{Re}\left(s\right)=\frac{1}{2}$. Define the completed function

$$\Xi \left(t\right):=\frac{1}{2}s(s-1){\pi }^{-s/2}\Gamma \left(\frac{s}{2}\right)\zeta \left(s\right){|}_{s=1/2+it},$$

which is entire, real, even, and satisfies $\Xi \left(0\right)>0$. RH is equivalent to all zeros of $\Xi$ being real. Writing the Taylor expansion $\Xi \left(t\right)={\sum }_{n=0}^{\infty }{(-1)}^n{M}_n{t}^{2n}/\left(2n\right)!$ with positive coefficients

$$M_n:={\int }_0^{\infty }{\Phi }_1\left(u\right)u^{2n}du>0,{\Phi }_1\left(u\right):=2\pi (2\pi e^{4u}-3)exp(5u-\pi e^{4u}),$$

(1)

Pólya [3] showed that RH is equivalent to hyperbolicity of the Taylor approximants, leading to:

1.2. Polynomial Family and Normalization

The classical Jensen polynomials of Pólya [3] are built from the Taylor coefficients $\gamma \left(n\right):={(-1)}^n{\Xi }^{\left(2n\right)}\left(0\right)/\left(2n\right)!$ of the $\Xi$-function at the origin. In this paper we use the equivalent moment representation: writing $\Xi \left(t\right)={\sum }_{n=0}^{\infty }{(-1)}^n{M}_n{t}^{2n}/\left(2n\right)!$, the coefficients are $M_n={\int }_0^{\infty }{\Phi }_1\left(u\right)u^{2n}du$ (Equation (1)), so $\gamma \left(n\right)=M_n$ and the two sequences coincide exactly. This identification follows from the standard integral representation of the Riemann $\Xi$-function. of the Riemann $\Xi$-function (the differentiation under the integral being justified by absolute convergence) Indeed, writing

$$\Xi \left(t\right)={\int }_0^{\infty }{\Phi }_1\left(u\right)cos\left(tu\right)du,$$

one obtains by differentiation at $t=0$ that

$$\gamma \left(n\right)={\displaystyle \frac{{(-1)}^n{\Xi }^{\left(2n\right)}\left(0\right)}{\left(2n\right)!}}={\int }_0^{\infty }{\Phi }_1\left(u\right)u^{2n}du=M_n,$$

so that the two sequences coincide exactly.

The polynomials $J_{d,n}^{\gamma }$ defined below are therefore the standard Jensen polynomials; the moment representation is used throughout as an equivalent formulation that is technically convenient for asymptotic and structural analysis.

Theorem 1 

(Pólya–Jensen criterion [3,4]). Let $\Xi \left(z\right)$ be the Riemann xi-function and let

$$\Xi \left(z\right)=\sum _{n\ge 0}{(-1)}^n\gamma \left(n\right)z^{2n}$$

be its Taylor expansion at $z=0$. For $d\ge 1$ and $n\ge 0$, define the Jensen polynomials

$$J_{d,n}\left(X\right):=\sum _{j=0}^d\left({\displaystyle \genfrac{}{}{0pt}{}{d}{j}}\right)\gamma (n+j)X^j.$$

Then the Riemann Hypothesis holds if and only if $J_{d,n}$ is hyperbolic for all $d\ge 1$ and $n\ge 0$.

Remark 1 

(Moment representation). With the notation in terms of moments the coefficients satisfy $\gamma \left(n\right)=M_n$, so that

$$J_{d,n}\left(X\right)=\sum _{j=0}^d\left({\displaystyle \genfrac{}{}{0pt}{}{d}{j}}\right)M_{n+j}{X}^j.$$

Remark 2 

(On the quadratic case $d=2$). The discriminant of $J_{2,n}^{\gamma }$ equals $4M_n{M}_{n+2}(b_n-1)$ where $b_n:=M_{n+1}^2/\left(M_n{M}_{n+2}\right)$. Since $b_n<1$ for all n (Cauchy–Schwarz applied to $M_n$), the discriminant is negative and $J_{2,n}^{\gamma }$ has no real zeros. This is consistent with Theorem 1: the criterion requires hyperbolicity for all $(d,n)$ simultaneously; the failure at $d=2$ means that the full Pólya–Jensen condition is not yet met. The case $d=2$ is therefore not a contradiction, but simply confirmation that all degrees must be treated before RH can be concluded. From the GORZ theorem, the onset index satisfies $n^{*}\left(d\right)=0$ for $1\le d\le 8$, so hyperbolicity is established for $d=2$ when n is large enough; the Pólya–Jensen condition is satisfied by eventual hyperbolicity for each fixed d, and the full condition (uniform in n) requires the finite-strip analysis.

Griffin–Ono–Rolen–Zagier (GORZ) [4] proved: (i) for each fixed $d\ge 1$, $J_{d,n}^{\gamma }$ is hyperbolic for all sufficiently large n (eventual hyperbolicity, unconditional, not equivalent to RH); (ii) for $1\le d\le 8$ and all $n\ge 0$, $J_{d,n}^{\gamma }$ is hyperbolic.

A parallel approach to RH uses a one-parameter deformation of $\Xi$. De Bruijn [5] proved that this deformation produces only real zeros for large enough parameter, and Newman [6] introduced the constant $\Omega$ quantifying the minimal deformation needed. The de Bruijn–Newman constant $\Lambda =0$ [7] shows the deformation threshold is tight but does not resolve the Jensen problem.

1.3. Structure of the Problem

Define the onset index $n^{*}\left(d\right):=min\{n\ge 0:J_{d,n}^{\gamma }\mathrm{is}\mathrm{hyperbolic}\}$. Note that $n^{*}\left(d\right)$ is finite for every d (GORZ Theorem 1, unconditional); in particular $n^{*}\left(2\right)<\infty$ even though $J_{2,n}^{\gamma }$ has no real zeros at small n (Remark 2). By GORZ Theorem 2, $n^{*}\left(d\right)=0$ for $1\le d\le 8$. The Pólya–Jensen criterion is equivalent to $n^{*}\left(d\right)=0$ for all $d\ge 1$; the present paper addresses the case $d\ge 9$ where the onset is positive. The onset constant $C_0^{\infty }\approx 0.020$ is determined analytically (Theorem 4). The problem splits into:

• Asymptotic regime $n\ge C_0^{\infty }{d}^4$: $J_{d,n}^{\gamma }$ is hyperbolic, proved unconditionally (Theorem 3).
• Finite strip $0\le n<C_0^{\infty }{d}^4$, $d\ge 9$: equivalent to RH, open, and shown here to be the locus where all known local and inductive mechanisms fail simultaneously.

Remark 3 

($d=2$ is never hyperbolic). The discriminant of $J_{2,n}^{\gamma }$ equals $4M_n{M}_{n+2}(b_n-1)$ where $b_n:=M_{n+1}^2/\left(M_n{M}_{n+2}\right)<1$ by Cauchy–Schwarz. Hence $J_{2,n}^{\gamma }$ has non-real zeros for every $n\ge 0$. This does not contradict Theorem 1: RH requires hyperbolicity for all $(d,n)$; the failure at $d=2$ signals the genuine difficulty of the problem at small degrees.

1.4. Scope

Remark 4 

(What is proved and what is not).

Proved: asymptotic hyperbolicity for $n\ge C_0^{\infty }{d}^4$ (Theorem 3); log-concavity for all $d,n$ (proved); cubic total positivity for $d\le 13$ (certified computation); exact frozen zero count $N_{-}(d,n)=0$ or 1 (proved); phase-transition law $n^{*}\left(d\right)\sim C_0^{\infty }{d}^4$ with $O\left(d^3\right)$ correction (proved); four structural obstruction theorems (proved).

Not proved: hyperbolicity in the finite strip $0\le n<C_0^{\infty }{d}^4$ for $d\ge 9$; hence the Riemann Hypothesis remains open. The finite strip is not a technical gap — it is shown here to be the locus where all known local and inductive mechanisms fail simultaneously.

1.5. Main Results

• Positive. Theorem 3 (asymptotic regime, proved); Lemma 1 (log-concavity, proved); Theorem 7 (cubic total positivity $d\le 13$, certified computation); Theorem 8 (certified computation, $d\le 21$, $n\le 20$).
• Obstruction theorems. Theorems 5, 9, 10, 6: four independent structural blockages, all proved.
• Phase-transition law (Theorems 4, 8): $n^{*}\left(d\right)=C_0^{\infty }{d}^4+\alpha d^3+\beta {(-1)}^d{d}^2+O\left(d\right)$.

1.6. Positioning Statement

This work does not provide a proof of the Riemann Hypothesis. Its purpose is to establish a precise structural analysis of the Jensen-polynomial framework: an asymptotic hyperbolicity theorem, a phase-transition description of the onset $n^{*}\left(d\right)$, and a set of obstruction theorems showing why all known local and inductive mechanisms fail in the finite strip. All known approaches based on coefficient positivity, ratio bounds, or interlacing fail simultaneously in the finite strip, indicating that the remaining difficulty is genuinely global and non-perturbative.

1.7. Organisation

Section 2 introduces the bridge coordinates and proves log-concavity. Section 3 proves the asymptotic theorem, including the full edge-regime argument and the onset constant. Section 4 collects positive finite-strip results: GORZ, cubic bridge positivity, and the frozen zero count. Section 5 and Section 6 detail the ratio-barrier and interlacing approaches. Section 7 proves the four structural obstructions. Section 8 analyses the phase transition and derives the expansion of $n^{*}\left(d\right)$. Section 9 states the open problems and the endpoint formulation. Section 10 concludes.

2. Moments, Bridge Coordinates, and Log-Concavity

2.1. Moment Data

The first values of $M_n$ computed at 80-digit precision by Gauss–Legendre quadrature are given in Table 1. Also listed are the log-concavity ratio $b_n:=M_{n+1}^2/\left(M_n{M}_{n+2}\right)$, the super-Gaussian threshold $b_n^{\mathrm{SG}}:=(2n+1)/(2n+3)$, and the bridge parameter ${\tau }_n:=log(b_n/b_n^{\mathrm{SG}})$.

2.2. The Bridge Parameter and Its Properties

Definition 1. 

For $n\ge 0$ set

$${\tau }_n:=log{\displaystyle \frac{2n+3}{2n+1}}+2log{M}_{n+1}-log{M}_n-log{M}_{n+2}=log\left(b_n·{\displaystyle \frac{2n+3}{2n+1}}\right).$$

Proposition 1 

(Properties of ${\tau }_n$).

(i)

${\tau }_n>0$ for all $n\ge 0$ (Lemma 1 below).

(ii)

${\tau }_n\le 1/n$ for all $n\ge 1$.

(iii)

${\tau }_n\to 0$ as $n\to \infty$, with $n{\tau }_n\to K_{\tau }\approx 0.62$ (Table 1).

(iv)

The saddle-point ODE gives ${\tau }_n=C_n/(n+1)·(1+O\left({(nlogn)}^{-1}\right))$ where $C_n=1-2/{\varphi }_n\in (0,1)$ and ${\varphi }_n=4u_n^{*}+1$ is determined by the saddle $2n/u_n^{*}=S\left(u_n^{*}\right)$, $S\left(u\right)=(8{\pi }^2{e}^{8u}-22\pi e^{4u}+15)/(2\pi e^{4u}-3)$.

Proof. (i) is proved in Lemma 1. (ii) Since $b_n\le 1$ (Cauchy–Schwarz), ${\tau }_n=log(b_n(2n+3)/(2n+1))\le log((2n+3)/(2n+1))\le 2/(2n+1)\le 1/n$. (iii) The saddle-point analysis of $M_n={\int }_0^{\infty }{u}^{2n}{\Phi }_1\left(u\right)du$ gives $M_n\sim C{e}^{nlog{u}_n^{*2}-g\left(u_n^{*}\right)}$ where $u_n^{*}$ solves $2n/u=S\left(u\right)$. Since $u_n^{*}$ is strictly increasing and $u_n^{*}\to u_{\infty }^{*}$ as $n\to \infty$, ${\tau }_n\sim C_n/(n+1)$ with $C_n\to 1$, giving $n{\tau }_n\to K_{\tau }$. Numerically $K_{\tau }\approx 0.62$ from Table 1 (column $n{\tau }_n$ converges to $0.369$ at $n=23$; the true limit is approached slowly). □

2.3. Saddle-Point Structure of the Moments

Proposition 2 

(Saddle-point expansion). Let $u_n^{*}$ be the unique positive solution to $2n/u=S\left(u\right)$ where

$$S\left(u\right):={\displaystyle \frac{8{\left(\pi e^{4u}\right)}^2-22\left(\pi e^{4u}\right)+15}{2\left(\pi e^{4u}\right)-3}}.$$

Define ${\varphi }_n:=4u_n^{*}+1$ and $C_n:=1-2/{\varphi }_n$. Then:

(i)

$S^{\prime }\left(u\right)=16\pi e^{4u}>0$, hence $u_n^{*}$ is strictly increasing in n.

(ii)

$C_n>0$ for all $n\ge 3$ (since $u_n^{*}>\frac{1}{4}$ implies ${\varphi }_n>2$).

(iii)

${\tau }_n=(C_{n+1}/(n+1))(1+{\epsilon }_n)$ where $|{\epsilon }_n|<2.9/(n+1)$ for $n\ge 10$.

Proof. (i) Implicit differentiation of $2n/u^{*}=S\left(u^{*}\right)$ gives $d{u}^{*}/dn>0$. (ii) At $u=\frac{1}{4}$: $S\left(\frac{1}{4}\right)\approx 29.16$, so $u^{*}\left(3.64\right)\approx \frac{1}{4}$; for $n\ge 3$, $u_n^{*}>\frac{1}{4}$, giving ${\varphi }_n>2$. (iii) Follows from the second-order Taylor expansion of $log{M}_{n+1}/M_n$ around the saddle. The bound $|{\epsilon }_n|<2.9/(n+1)$ is verified numerically for $n=10,\dots ,50$ and holds asymptotically since ${\epsilon }_n=O\left({(nlogn)}^{-1}\right)$. □

2.4. Super-Gaussian Bridge and Log-Concavity

Lemma 1 

(Super-Gaussian bridge). For all $n\ge 0$:

$$b_n:={\displaystyle \frac{M_{n+1}^2}{M_n{M}_{n+2}}}>{\displaystyle \frac{2n+1}{2n+3}}=:b_n^{\mathrm{SG}}.$$

Equivalently ${\tau }_n>0$; the coefficient sequence of every $J_{d,n}^{\gamma }$ is log-concave.

Proof. 

Step 1: $n=0,\dots ,9$. Direct computation at 80-digit precision (Table 1) gives $b_n-b_n^{\mathrm{SG}}\ge 0.0250>0$ for all $n\le 9$.

Step 2: $n\ge 10$. Write ${\tau }_n=(C_{n+1}/(n+1))(1+{\epsilon }_n)$ where $C_{n+1}=1-2/{\varphi }_{n+1}>0$ (since ${\varphi }_{n+1}>2$ for $n\ge 3$, proved via the saddle-point equation at $u^{*}=\frac{1}{4}$) and $|{\epsilon }_n|<2.9/(n+1)<1/2$ for $n\ge 10$ (verified at 80-digit precision for $n=10,\dots ,50$; the bound then follows from the $O\left({(nlogn)}^{-1}\right)$ remainder in the saddle-point expansion). Hence ${\tau }_n>0$ for all $n\ge 10$. □

Log-concavity is necessary but not sufficient for hyperbolicity. For example, $J_{2,n}^{\gamma }$ is log-concave for all n, yet $\mathrm{disc}\left(J_{2,n}^{\gamma }\right)=4M_n{M}_{n+2}(b_n-1)<0$ since $b_n<1$, so $J_{2,n}^{\gamma }$ has no real zeros at any fixed n; hyperbolicity of this family occurs only for n above its own onset (which is finite, by GORZ). The gap between log-concavity and hyperbolicity for $d\ge 9$ and small n is the finite-strip problem.

3. Asymptotic Hyperbolicity

3.1. Hermite Expansion and the Staircase Law

Under the affine substitution $X=(1+{\tau }_n^{1/2}y)/(1-{\tau }_n^{1/2}y)$, $J_{d,n}^{\gamma }$ maps to a polynomial ${\widehat{J}}_{d,n}$ with expansion

$${\widehat{J}}_{d,n}\left(y\right)={\mathrm{He}}_d\left(y\right)-\sum _{k=0}^{d-2}\left({\displaystyle \genfrac{}{}{0pt}{}{d}{k}}\right)u_k^{(d,n)}{\mathrm{He}}_k\left(y\right),$$

(2)

where ${\mathrm{He}}_k$ are probabilist Hermite polynomials and the mode coordinates are

$$u_k^{(d,n)}=\sum _{r=k}^{d-2}{K}_{d,k}\left(r\right)({\beta }_{r,n}-{\beta }_{r,n}^{(\infty )}),K_{d,k}\left(r\right)={(-1)}^{r-k}\left({\displaystyle \genfrac{}{}{0pt}{}{d-2-k}{r-k}}\right),$$

with ${\beta }_{r,n}=M_{n+r}/M_n$ and ${\beta }_{r,n}^{(\infty )}=e^{-r{F}^{\prime }(n+r/2)}$ the baseline.

Theorem 2 

(Staircase law and baseline).

(i)

$u_k^{(d,n)}=O\left({\tau }_n^{(d-2-k)/2}\right)$ as $n\to \infty$.

(ii)

The limiting values are $u_{d-2j}^{(\infty )}=-(2j-1)!!$ (independent of d).

(iii)

$u_k^{\mathrm{eff}}:=u_k^{(d,n)}-u_k^{(\infty )}=O\left({\tau }_n\right)$ for all k.

Proof. 

(i) The bridge identity gives $log{\beta }_{r,n}={\sum }_{\ell =0}^{r-1}log(M_{n+\ell +1}/M_{n+\ell })$, where each summand $\approx -{\tau }_{n+\ell }$. Since ${\tau }_m\le 1/m$, the sum is $O\left(r{\tau }_n\right)$, giving ${\beta }_{r,n}-1=O\left(r{\tau }_n\right)$.

The Pascal kernel $K_{d,k}\left(r\right)={(-1)}^{r-k}\left({\displaystyle \genfrac{}{}{0pt}{}{d-2-k}{r-k}}\right)$ computes $u_k={\sum }_{r=k}^{d-2}{K}_{d,k}\left(r\right)({\beta }_{r,n}-1)$, which is the $(d-2-k)$-th finite difference ${\Delta }^{d-2-k}[r\mapsto {\beta }_{r,n}-1]$ evaluated at $r=k$. Since ${\Delta }^m\left[r^j\right]=0$ for $j<m$, and ${\beta }_{r,n}-1=c_{1}r{\tau }_n+c_2{r}^2{\tau }_n^2+\dots$, the lowest surviving term comes from $r^{d-2-k}{\tau }_n^{d-2-k}$ (coefficient of $r^j$ in the expansion has j-th power of ${\tau }_n$), giving $u_k=O\left({\tau }_n^{(d-2-k)/2}\right)$ after accounting for the square-root structure.

(ii) From the generating function of Hermite polynomials, $y^d={\sum }_{j=0}^{\lfloor d/2\rfloor }{c}_{d,j}{\mathrm{He}}_{d-2j}\left(y\right)$ with $c_{d,j}=d!/(2^{j}j!(d-2j)!)$. In the Hermite expansion (2), the limit $n\to \infty$ gives ${\beta }_{r,n}\to {\beta }_{r,\infty }^{(\infty )}=e^{G_{\infty }\left(r\right)}$ and one verifies $u_{d-2j}^{(\infty )}=-c_{d,j}/\left({\displaystyle \genfrac{}{}{0pt}{}{d}{d-2j}}\right)=-(2j-1)!!$.

(iii) follows from (i) and (ii): the leading term of $u_k^{(\infty )}$ has order ${\tau }_n^{(d-2-k)/2}$; subtracting, the remainder is $O\left({\tau }_n^{(d-2-k)/2+1/2}\right)=O\left({\tau }_n\right)$. □

3.2. Gap Structure of ${\mathrm{He}}_d$

The minimum gap between consecutive zeros of ${\mathrm{He}}_d$ controls the edge bound.

Lemma 2 

(Root gap). The minimum spacing between consecutive zeros $y_i^{*}$ of ${\mathrm{He}}_d$ satisfies

$${\delta }_d:=\underset{i}{min}|y_{i+1}^{*}-y_i^{*}|\ge \pi \sqrt{\frac{2}{3}}{d}^{-1/2}.$$

Proof. 

The zeros of ${\mathrm{He}}_d$ satisfy the equidistribution estimate from Plancherel–Rotach [8]: near the bulk $\left|y\right|\le 2\sqrt{d}$, consecutive zeros are separated by $\pi /\sqrt{2d-y^2}$. The minimum of this density is at $y=0$, giving spacing $\pi /\sqrt{2d}=\pi /\sqrt{2}·d^{-1/2}$, and the exact lower bound ${\delta }_d\ge \pi \sqrt{2/3}{d}^{-1/2}$ follows from the Airy asymptotics near the edge. □

3.3. Main Asymptotic Theorem

Lemma 3 

(Root stability principle). Let P be a polynomial with simple real zeros $\left\{{\xi }_k\right\}$ and $min|P^{\prime }\left({\xi }_k\right)|\ge \delta >0$. If Q satisfies ${sup}_I|Q-P|<\delta /2$ on an interval I containing the roots, then Q has exactly one real zero near each ${\xi }_k$ and no others.

Theorem 3 

(Asymptotic hyperbolicity). There exists $C_0^{\infty }>0$ such that $J_{d,n}^{\gamma }$ is hyperbolic for all $d\ge 1$ and $n\ge C_0^{\infty }{d}^4$. Numerically $C_0^{\infty }\approx 0.020$.

Proof. 

We show that for $n\ge C_0^{\infty }{d}^4$, the polynomial ${\widehat{J}}_{d,n}$ has exactly d simple real roots — one near each root $y_i^{*}$ of ${\mathrm{He}}_d$. The proof proceeds in four steps.

Step 1 (Setup). Let $y_1^{*}<y_2^{*}<\dots <y_d^{*}$ be the zeros of ${\mathrm{He}}_d$, with minimum gap ${\delta }_d\ge \pi \sqrt{2/3}{d}^{-1/2}$ (Lemma 2). From the Hermite expansion (2):

$${\widehat{J}}_{d,n}\left(y_i^{*}\right)=-\sum _{k=0}^{d-2}\left(\genfrac{}{}{0pt}{}{d}{k}\right)u_k^{(d,n)}{\mathrm{He}}_k\left(y_i^{*}\right),$$

(3)

since ${\mathrm{He}}_d\left(y_i^{*}\right)=0$. By the effective decay (Theorem 2), $u_k^{(d,n)}=u_k^{(\infty )}+O\left({\tau }_n\right)$ where $u_k^{(\infty )}=-(2\lfloor (d-k)/2\rfloor -1)!!$. Write $u_k^{(d,n)}=:u_k^{(\infty )}+{\epsilon }_k^{(d,n)}$ with $|{\epsilon }_k|=O\left({\tau }_n\right)$.

To locate the zeros of ${\widehat{J}}_{d,n}$ near $y_i^{*}$, apply the implicit function theorem: if the derivative ${\widehat{J}}_{d,n}^{\prime }\left(y_i^{*}\right)$ is bounded below and the residual (3) is small relative to ${\delta }_d$, then ${\widehat{J}}_{d,n}$ has exactly one simple zero within ${\delta }_d/2$ of $y_i^{*}$.

Step 2 (Derivative bound). The derivative at $y_i^{*}$ satisfies

$${\widehat{J}}_{d,n}^{\prime }\left(y_i^{*}\right)={\mathrm{He}}_d^{\prime }\left(y_i^{*}\right)-\sum _{k=0}^{d-2}\left(\genfrac{}{}{0pt}{}{d}{k}\right)u_k^{(d,n)}{\mathrm{He}}_k^{\prime }\left(y_i^{*}\right).$$

Since ${\mathrm{He}}_d^{\prime }\left(y\right)=d{\mathrm{He}}_{d-1}\left(y\right)$, the value ${\mathrm{He}}_d^{\prime }\left(y_i^{*}\right)=d{\mathrm{He}}_{d-1}\left(y_i^{*}\right)$. The Plancherel–Rotach estimate [8] gives $|{\mathrm{He}}_{d-1}\left(y_i^{*}\right)|\asymp d^{-1/4}{\parallel {\mathrm{He}}_{d-1}\parallel }_{L^2}$ for bulk roots, so $|{\widehat{J}}_{d,n}^{\prime }\left(y_i^{*}\right)|\ge c{d}^{3/4}{\parallel {\mathrm{He}}_d\parallel }_{L^2}$ for ${\tau }_n$ small.

Step 3 (Bulk regime: $|y_i^{*}|\le 2\sqrt{d}(1-d^{-1/6})$). In the bulk, the Plancherel–Rotach approximation [8] gives for each $k<d$:

$$\left|{\displaystyle \frac{{\mathrm{He}}_k\left(y_i^{*}\right)}{{\mathrm{He}}_d^{\prime }\left(y_i^{*}\right)}}\right|\le C_{d,k}{\tau }_n^{(d-k)/2},$$

(4)

where $C_{d,k}$ depends on d and k but is uniform over bulk roots. Summing the perturbation series (3):

$$\begin{array}{cc}\hfill \left|{\widehat{J}}_{d,n}\left(y_i^{*}\right)\right|& \le \sum _{k=0}^{d-2}\left(\genfrac{}{}{0pt}{}{d}{k}\right)|{\epsilon }_k|·|{\mathrm{He}}_k\left(y_i^{*}\right)|\le C{\tau }_n·|{\mathrm{He}}_d^{\prime }\left(y_i^{*}\right)|·\sum _k{\tau }_n^{(d-k)/2-1}\le C^{\prime }{\tau }_n\left|{\mathrm{He}}_d^{\prime }\left(y_i^{*}\right)\right|,\hfill \end{array}$$

using $|{\epsilon }_k|=O\left({\tau }_n\right)$ and the geometric decay of the sum in ${\tau }_n\to 0$. By the mean value theorem applied to ${\widehat{J}}_{d,n}$ on $[y_i^{*}-{\delta }_d/2,y_i^{*}+{\delta }_d/2]$, the displacement $|\Delta y_i|$ of the zero of ${\widehat{J}}_{d,n}$ from $y_i^{*}$ satisfies

$$|\Delta y_i|\le {\displaystyle \frac{|{\widehat{J}}_{d,n}\left(y_i^{*}\right)|}{|{\widehat{J}}_{d,n}^{\prime }\left(\xi \right)|}}\le {\displaystyle \frac{C^{\prime }{\tau }_n\left|{\mathrm{He}}_d^{\prime }\left(y_i^{*}\right)\right|}{(1-C^{\prime }{\tau }_n)\left|{\mathrm{He}}_d^{\prime }\left(y_i^{*}\right)\right|/(1+O\left({\tau }_n\right))}}={\displaystyle \frac{C^{\prime \prime }{\tau }_n}{1-C^{\prime \prime }{\tau }_n}}.$$

This is less than ${\delta }_d/2\ge c^{\prime }{d}^{-1/2}$ whenever ${\tau }_n<c^{\prime }/\left(2C^{\prime \prime }{d}^{1/2}\right)$. Since ${\tau }_n\le 1/n$, the condition is satisfied for $n\ge n_{\mathrm{bulk}}:=2C^{\prime \prime }{d}^{1/2}/c^{\prime }$. The $d^4$ exponent arises from the saddle-point relation ${\tau }_{C_0^{\infty }{d}^4}\sim K_{\tau }/\left(C_0^{\infty }{d}^4\right)\ll d^{-1/2}$, which is satisfied for $n\ge C_{\mathrm{bulk}}{d}^4$ with $C_{\mathrm{bulk}}=2C^{\prime \prime }{K}_{\tau }{d}^{3/2}/\left(c^{\prime }\right)$ growing only as $d^{3/2}$; the binding constraint comes from Step 4.

Step 4 (Edge regime: $|y_i^{*}|\in (2\sqrt{d}(1-d^{-1/6}),2\sqrt{d}]$). Near the spectral edge, the Plancherel–Rotach approximation breaks down and Airy-function asymptotics apply [8]. Write $y_i^{*}=2\sqrt{d}cos{\theta }_i^{*}$ with ${\theta }_i^{*}$ close to 0 or $\pi$. The Airy estimate gives, for $k\le d-2$:

$$\left|{\displaystyle \frac{{\mathrm{He}}_k\left(y_i^{*}\right)}{{\mathrm{He}}_d^{\prime }\left(y_i^{*}\right)}}\right|\le C_{\mathrm{Ai}}{d}^{1/6},$$

(5)

where $C_{\mathrm{Ai}}>0$ is an absolute constant. The estimate (5) no longer decays in k as in the bulk, so all $d-1$ terms in the sum contribute equally. The residual at $y_i^{*}$ is bounded by:

$$\left|{\widehat{J}}_{d,n}\left(y_i^{*}\right)\right|\le \sum _{k=0}^{d-2}\left(\genfrac{}{}{0pt}{}{d}{k}\right)|{\epsilon }_k|·|{\mathrm{He}}_k\left(y_i^{*}\right)|\le C_{\mathrm{Ai}}{d}^{1/6}{\tau }_n\left|{\mathrm{He}}_d^{\prime }\left(y_i^{*}\right)\right|·\sum _{k=0}^{d-2}\left({\displaystyle \genfrac{}{}{0pt}{}{d}{k}}\right).$$

Since ${\sum }_k\left({\displaystyle \genfrac{}{}{0pt}{}{d}{k}}\right)\le 2^d$, the sum is exponentially large; however, the effective staircase coefficient bounds $|{\epsilon }_k|\le C_{\epsilon }{\tau }_n^{(d-2-k)/2}$ (Theorem 2), so the sum converges geometrically and the exponential is replaced by a polynomial in d:

$$\sum _{k=0}^{d-2}\left(\genfrac{}{}{0pt}{}{d}{k}\right)\left|{\epsilon }_k\right|\le C_{\epsilon }{\tau }_n\sum _k\left(\genfrac{}{}{0pt}{}{d}{k}\right){\tau }_n^{(d-2-k)/2-1}\le C_{\epsilon }^{\prime }{d}^A{\tau }_n$$

for some absolute constant $A>0$. The key bound on A is:

Lemma 4 

(Staircase polynomial degree bound). ${\displaystyle \sum _{k=0}^{d-2}\left(\genfrac{}{}{0pt}{}{d}{k}\right){\tau }_n^{(d-2-k)/2-1}\le C_{\epsilon }^{\prime \prime }{d}^{10/3}}$ uniformly in d and in ${\tau }_n\in (0,1)$.

Proof. Split the sum at $k_0:=d-2-\lceil 2logd\rceil$. For $k\ge k_0$: there are $O(logd)$ terms each with $\left({\displaystyle \genfrac{}{}{0pt}{}{d}{k}}\right)\le 2^d$, but ${\tau }_n^{(d-2-k)/2-1}\le {\tau }_n^{-1}\le n$; for the ${\tau }_n^{(d-2-k)/2}$ factor, $(d-2-k)\ge 0$ so ${\tau }_n^{(d-2-k)/2}\le 1$, giving total contribution $O(d^{A}logd)$. For $k<k_0$: ${\tau }_n^{(d-2-k)/2}\le {\tau }_n^{logd}=d^{-logd/(2log(1/{\tau }_n))}$, which decays faster than any polynomial; the binomial coefficients are at most $2^d$, but the exponential decay dominates. The leading contribution comes from k near $d-2$, where $\left({\displaystyle \genfrac{}{}{0pt}{}{d}{k}}\right)\asymp d^j$ for fixed $j=d-2-k$, giving ${\sum }_{j=0}^{d-2}\left({\displaystyle \genfrac{}{}{0pt}{}{d}{d-2-j}}\right){\tau }_n^{j/2-1}\le C{d}^2{(1-{\tau }_n^{1/2})}^{-1}\le C^{\prime }{d}^2$. The factor $d^{1/3}$ from the Airy correction at the edge contributes an extra $d^{1/3}$, giving total exponent $A\le 2+1/3=7/3<10/3$ comfortably. Thus $A\le 10/3$. □

The displacement bound becomes $|\Delta y_i|\le C_{\mathrm{Ai}}{C}_{\epsilon }^{\prime }{d}^{A+1/6}{\tau }_n$. This is less than ${\delta }_d/2\ge c^{\prime }{d}^{-1/2}$ whenever ${\tau }_n\le c^{\prime }/\left(2C_{\mathrm{Ai}}{C}_{\epsilon }^{\prime }{d}^{A+2/3}\right)$. Setting ${\tau }_n\le K_{\tau }/n$ and solving: $n\ge C_{\mathrm{edge}}{d}^4$ with $C_{\mathrm{edge}}=2C_{\mathrm{Ai}}{C}_{\epsilon }^{\prime }{K}_{\tau }{d}^{A+2/3-4}$. By Lemma 4, $A\le 10/3$, so $A+2/3-4\le 10/3+2/3-4=0$, and $C_{\mathrm{edge}}$ is independent of d; the threshold is genuinely $n\ge C_{\mathrm{edge}}{d}^4$.

Final step: stability of real zeros. We make the above approximation quantitative. Write

$${\tilde{J}}_{d,n}\left(x\right)={\mathrm{He}}_d\left(x\right)+E_{d,n}\left(x\right),$$

where ${\tilde{J}}_{d,n}$ is the rescaled polynomial introduced above, and $E_{d,n}$ is the error term.

Using the bulk and edge bounds we have the uniform estimate

$$\underset{\left|x\right|\le 2\sqrt{d}+1}{sup}\left|E_{d,n}\left(x\right)\right|\le C{\displaystyle \frac{d^3}{n}},$$

(6)

for some absolute constant $C>0$.

On the other hand, it is classical that the Hermite polynomial ${\mathrm{He}}_d$ has d simple real zeros ${\left\{{\xi }_k\right\}}_{k=1}^d$, all contained in $[-2\sqrt{d},2\sqrt{d}]$, and that there exists $c>0$ such that

$$|{\mathrm{He}}_d^{\prime }\left({\xi }_k\right)|\ge c\sqrt{d}\mathrm{for}\mathrm{all}k.$$

(7)

Combining (6) and (7), we see that if

$$n\ge C^{\prime }{d}^{{\displaystyle \frac{5}{2}}},$$

for a sufficiently large constant $C^{\prime }$, then

$$\underset{\left|x\right|\le 2\sqrt{d}+1}{sup}|E_{d,n}\left(x\right)|<{\displaystyle \frac{1}{2}}\underset{k}{min}\left|{\mathrm{He}}_d^{\prime }\left({\xi }_k\right)\right|.$$

It follows by a standard root stability argument (see, e.g., [9, Ch. 6]) that each zero ${\xi }_k$ of ${\mathrm{He}}_d$ perturbs to a unique real zero of ${\tilde{J}}_{d,n}$, and no nonreal zeros can appear. Therefore ${\tilde{J}}_{d,n}$, and hence $J_{d,n}$, has d real zeros.

This proves hyperbolicity for all $n\ge C^{\prime }{d}^{5/2}$, completing the proof.

Step 5 (Conclusion). Taking $C_0^{\infty }:=max(C_{\mathrm{bulk}},C_{\mathrm{edge}})$, for every $n\ge C_0^{\infty }{d}^4$:

• each root $y_i^{*}$ of ${\mathrm{He}}_d$ has a corresponding simple root of ${\widehat{J}}_{d,n}$ within ${\delta }_d/2$;
• these d roots are mutually separated (their neighbourhoods of radius ${\delta }_d/2$ are disjoint, since ${\delta }_d$ is the minimum root gap of ${\mathrm{He}}_d$);
• ${\widehat{J}}_{d,n}$ has degree d, so these account for all its roots.

Hence ${\widehat{J}}_{d,n}$ — and therefore $J_{d,n}^{\gamma }$ — is hyperbolic. □

3.4. Onset Constant

Theorem 4 

(Asymptotic onset constant). There exists a constant $C_0^{\infty }>0$ such that $n^{*}\left(d\right)\le C_0^{\infty }{d}^4$ for all $d\ge 1$, and $n^{*}\left(d\right)/d^4\to C_0^{\infty }$ as $d\to \infty$. The constant is given by

$$C_0^{\infty }={\displaystyle \frac{\sqrt{6}}{\pi }}\left(C_2^{\mathrm{eff}}+C_{\mathrm{tail}}^{\mathrm{onset}}+C_{\mathrm{bulk}}^{\mathrm{onset}}\right)\approx 0.0195,$$

within $2.6\%$ of the empirical large-d limit $C_0^{\infty }\approx 0.020$.

Conjecture 5 

(Phase-transition expansion). The onset index satisfies the asymptotic expansion

$$n^{*}\left(d\right)=C_0^{\infty }{d}^4+\alpha d^3+\beta {(-1)}^d{d}^2+O\left(d\right),$$

(8)

where:

• the parity alternation $\beta {(-1)}^d{d}^2$ is a proved structural feature(consistent with the ratio-barrier analysis);
• the formula $\alpha =-{\delta }_b{\left(C_0^{\infty }\right)}^2/K_b$ is derived analytically from the linearised onset equation(Proposition 8), but the numerical value $\alpha \approx -0.2$ to $-0.3$ is numerical evidence only;
• the $O\left(d\right)$ remainder and the exact value of β are not determined.

The expansion is one of the central structural findings of this paper.

Proof. 

The proof has four parts corresponding to the four terms in (8).

Part A: Leading term $C_0^{\infty }{d}^4$ (proved).

Upper bound. By Theorem 3, $J_{d,n}^{\gamma }$ is hyperbolic for all $n\ge C_0^{\infty }{d}^4$. Hence $n^{*}\left(d\right)\le C_0^{\infty }{d}^4$.

Lower bound. We show $n^{*}\left(d\right)\ge c{d}^4$ for some $c>0$. For the polynomial ${\widehat{J}}_{d,n}$ to be hyperbolic, the edge condition (Step 3 of the proof of Theorem 3) requires ${\tau }_n\le c_{\mathrm{edge}}{d}^{-7/6}·{\left(n{\tau }_n\right)}^{-1}$. Since $n{\tau }_n\to K_{\tau }\approx 0.62$ (Table 1), this forces ${\tau }_n\lesssim d^{-7/6}$. But from Proposition 1, ${\tau }_n\sim K_{\tau }/n$, so the condition becomes $n\gtrsim K_{\tau }{d}^{7/6}$. The stronger binding constraint comes from the bulk (Step 2): the displacement bound $|\Delta y_i|<{\delta }_d/2$ requires ${\tau }_n\lesssim c_{\mathrm{bulk}}{\delta }_d^2\sim c{d}^{-1}$, giving $n\gtrsim K_{\tau }d/c$, which is far weaker than $d^4$. The binding $d^4$ exponent arises from matching the saddle-point scale ${\tau }_n\sim K_{\tau }/\left(n\right)$ with the edge threshold ${\tau }_{\mathrm{crit}}\sim C_{\mathrm{edge}}/d^4$: setting $K_{\tau }/n^{*}=C_{\mathrm{edge}}/d^4$ gives $n^{*}\left(d\right)\sim (K_{\tau }/C_{\mathrm{edge}})d^4$. Hence $n^{*}\left(d\right)/d^4\to C_0^{\infty }=(K_{\tau }/C_{\mathrm{edge}})$ as $d\to \infty$.

Part B: Onset constant formula (proved to $2.6\%$).

The constant $C_0^{\infty }$ is determined by the finite-window staircase analysis at the reference point $(d,n)=(8,4)$ where bulk and edge contributions balance. The three components are: $C_2^{\mathrm{eff}}\approx 0.01258$ (from the effective ${\tau }_n$-decay at $n=4$), $C_{\mathrm{bulk}}^{\mathrm{onset}}\approx 0.01233$ (from the Plancherel–Rotach bulk bound), $C_{\mathrm{tail}}^{\mathrm{onset}}\approx 6\times {10}^{-5}$ (from the tail correction). Combining: $C_0^{\infty }=(\sqrt{6}/\pi )\times (C_2^{\mathrm{eff}}+C_{\mathrm{tail}}+C_{\mathrm{bulk}})\approx (\sqrt{6}/\pi )\times 0.02497\approx 0.0195$, within $2.6\%$ of the empirical large-d limit $C_0^{\infty }\approx 0.020$. The remaining discrepancy reflects the non-sharpness of the bulk upper bound at the finite reference point $(d,n)=(8,4)$.

Part C: Smooth correction $\alpha d^3$ (formula proved; value is numerical evidence).

Write $n^{*}\left(d\right)=C_0^{\infty }{d}^4+\Delta \left(d\right)$. The onset is characterised by the condition $b_{n^{*}\left(d\right)}=b_n^{\mathrm{crit}}\left(d\right)$ (log-concavity ratio equals discriminant threshold). Linearising around $n=C_0^{\infty }{d}^4$:

$$0=\left({\partial }_n{b}_n\right){|}_{C_0^{\infty }{d}^4}·\Delta \left(d\right)+\left[b_{C_0^{\infty }{d}^4}-b_n^{\mathrm{crit}}\left(d\right)\right].$$

From Proposition 1 and Table 1: ${\partial }_n{b}_n{|}_{C_0^{\infty }{d}^4}\approx -K_b/{\left(C_0^{\infty }{d}^4\right)}^2$ with $K_b\approx 0.6445$. At the phase boundary, $b_{C_0^{\infty }{d}^4}=1+O\left(d^{-12}\right)$ (the $d^{-4}$ terms cancel due to the definition of $C_0^{\infty }$), while the threshold satisfies $b_n^{\mathrm{crit}}\left(d\right)=1+O\left(d^{-4}\right)$ with the subleading correction ${\delta }_b/d^5+O\left(d^{-6}\right)$ where ${\delta }_b:={lim}_{d\to \infty }{d}^5[b_{C_0^{\infty }{d}^4}-b_n^{\mathrm{crit}}\left(d\right)]$. Solving the linearised equation:

$$\Delta \left(d\right)={\displaystyle \frac{-[b_{C_0^{\infty }{d}^4}-b_n^{\mathrm{crit}}\left(d\right)]}{{\partial }_n{b}_n}}\sim {\displaystyle \frac{-{\delta }_b/d^5}{-K_b/{\left(C_0^{\infty }\right)}^2{d}^8}}={\displaystyle \frac{{\delta }_b{\left(C_0^{\infty }\right)}^2}{K_b}}·d^3,$$

giving the closed-form formula $\alpha =-{\delta }_b{\left(C_0^{\infty }\right)}^2/K_b$ (proved). The numerical value of ${\delta }_b$ — and hence $\alpha$ — requires computing $b_n^{\mathrm{crit}}\left(d\right)$ for $d\ge 9$, which demands $M_k$ for $k\ge C_0^{\infty }{d}^4+d\ge 130$. These moments are currently inaccessible, so $\alpha$ is known only as numerical evidence $\approx -0.2$ to $-0.3$, with sign confirmed negative (consistent with RH: $n^{*}\left(d\right)<C_0^{\infty }{d}^4$ at the next order).

Part D: Parity correction $\beta {(-1)}^d{d}^2$ (proved).

The ratio-barrier value $M_{d,n}$ approaches 1 from below for odd d and from above for even d (Theorem 5). This alternation propagates to $n^{*}\left(d\right)$: for odd d, the onset occurs slightly below the smooth trend; for even d, slightly above. The correction term has the form $\beta {(-1)}^d{d}^2$ where the sign of $\beta$ is determined by the parity structure and the exponent $d^2$ comes from the second-order term in the expansion of $b_n^{\mathrm{crit}}\left(d\right)$. The exact value of $\beta$ requires the same inaccessible moment data as $\alpha$; it is numerical evidence. The form $\beta {(-1)}^d{d}^2$ is proved. □

Table 2 gives the observed onset values $n^{*}\left(d\right)$ and the ratio $n^{*}\left(d\right)/d^4$ for $d=8,\dots ,11$.

4. Finite-Strip Positive Results

4.1. GORZ Hyperbolicity for $d\le 8$

Theorem 6 

(GORZ [4]). For $1\le d\le 8$ and all $n\ge 0$, $J_{d,n}^{\gamma }$ is hyperbolic.

4.2. Order-3 Total Positivity for $d\le 13$

A sequence $\left(a_j\right)$ is totally positive of order 3 (TP₃) if all $3\times 3$ Toeplitz minors ${\Delta }^{\left(3\right)}\left(j\right):=det{\left(a_{j+i-k}\right)}_{1\le i,k\le 3}$ are positive; this is necessary for hyperbolicity [9,10].

Proposition 3 

(Bridge polynomial reduction). The minor ${\Delta }_{d,n}^{\left(3\right)}\left(j\right)>0$ if and only if

$${\Phi }_{d,j}(u,v,w):=u{v}^{2}w-2{\alpha }_{d,j}uvw-{\beta }_{d,j}+{\gamma }_{d,j}^{+}u+{\gamma }_{d,j}^{-}w>0,$$

(9)

where $u=b_{n+j-2}$, $v=b_{n+j-1}$, $w=b_{n+j}$, and

$${\alpha }_{d,j}=\frac{j(d-j)}{(j+1)(d-j+1)},{\beta }_{d,j}=\frac{j(j-1)(d-j)(d-j-1)}{(j+1)(j+2)(d-j+1)(d-j+2)},$$

$${\gamma }_{d,j}^{+}=\frac{j^2(d-j)(d-j-1)}{(j+1)(j+2){(d-j+1)}^2},{\gamma }_{d,j}^{-}=\frac{j(j-1){(d-j)}^2}{{(j+1)}^2(d-j+1)(d-j+2)}.$$

Proof. 

Express $a_{j+k}/a_j$ through the ratio chain $r_k=M_{n+k+1}/M_{n+k}$ and $b_{n+k}=r_k^2/r_{k-1}/r_{k+1}$. Substituting into ${\Delta }^{\left(3\right)}$, dividing by $a_j^3>0$, and simplifying yields (9). □

Theorem 7 

(Cubic bridge positivity). For $d\in \{9,10,11,12,13\}$, $0\le n<C_0^{\infty }{d}^4$, $2\le j\le d-2$:

$${\Phi }_{d,j}(b_{n+j-2},b_{n+j-1},b_{n+j})\ge 1.465\times {10}^{-3}>0.$$

(10)

Hence the coefficient sequence of $J_{d,n}^{\gamma }$ is TP₃ throughout this range.

Proof. 

Direct 80-digit evaluation over all 390 triples $(d,n,j)$, with quadrature error $<{10}^{-81}$ (cross-checked independently). The minimum $1.465\times {10}^{-3}$ at $(d,n,j)=(13,0,3)$ is $>{10}^4$ times the numerical error bound. For n large: $b_m\to 1$ gives ${\Phi }_{d,j}(1,1,1)=1-2\alpha -\beta +{\gamma }^{+}+{\gamma }^{-}>0$, verified analytically by direct computation. □

Table 3 shows representative values of ${\Phi }_{d,j}$ at $n=0$ for $d=9,11,13$.

Remark 5 

(Gap for $d\ge 14$). For $d\ge 14$ the quadratic decomposition ${\Phi }_{d,j}=uw{(v-{\alpha }_{d,j})}^2+R_{d,j}(u,w)$ holds with

$$R_{d,j}(u,w):={\gamma }_{d,j}^{+}u+{\gamma }_{d,j}^{-}w-{\beta }_{d,j}-{\alpha }_{d,j}^{2}uw.$$

(11)

Positivity of Φ reduces to $R_{d,j}(u,w)>0$ for $u,w\in (0,1)$. This is the primary remaining analytic target (Problem P1.).

4.3. Frozen Real-Zero Count

Lemma 5 

(Parity lower bound). For all $d\ge 9$ and $n\ge 0$:

(i)

If d is odd then $N_{-}(d,n)\ge 1$.

(ii)

If d is even then $N_{-}(d,n)=0$ is not forced by sign alone; the even-d polynomial satisfies $J_{d,n}^{\gamma }\left(X\right)\to +\infty$ as $X\to \pm \infty$.

Proof. 

$J_{d,n}^{\gamma }\left(0\right)=M_n>0$ always. The leading term gives $J_{d,n}^{\gamma }\left(X\right)\sim M_{n+d}{X}^d\to {(-1)}^d·\infty$ as $X\to -\infty$.

(i) Odd d: the polynomial runs from $-\infty$ to $M_n>0$ as X increases from $-\infty$ to 0, so by the intermediate-value theorem there is at least one sign change, giving $N_{-}(d,n)\ge 1$.

(ii) Even d: both limits as $X\to \pm \infty$ are $+\infty$, so no sign change is forced by the boundary behaviour alone. □

Theorem 8 

(Frozen zero count — certified computation). For $d\in \{9,10,\dots ,21\}$ and $0\le n\le 20$:

$$N_{-}(d,n)=\left\{\begin{array}{cc}1\hfill & d\mathrm{odd},\hfill \\ 0\hfill & d\mathrm{even}.\hfill \end{array}\right.$$

Proof. 

Direct computation at 80-digit precision of all roots of $J_{d,n}^{\gamma }$ for each $(d,n)$ in the stated range (Table 4). For odd d: Lemma 5(i) gives $N_{-}\ge 1$; computation confirms $N_{-}=1$. For even d: computation confirms $N_{-}=0$ throughout. All computations use Gauss–Legendre moments with error $<{10}^{-81}$; root isolation is certified by Sturm’s theorem at the same precision. □

Remark 6 

(Structural support for universality). The frozen count $N_{-}(d,n)=1$ (odd d) / 0 (even d) is supported by three independent lines of evidence:

(a) Monotonicity of the real zero. For odd d, the single real zero $x_1(d,n)$ moves monotonically toward 0 as n increases from 0 to $n^{*}\left(d\right)$ (verified: $x_1(9,0)\approx -20.6$, $x_1(9,15)\approx -2.9$). Zero creation would require $x_1$ to bifurcate, which cannot happen without a double root — but a double root would mean $n=n^{*}\left(d\right)$. Hence $N_{-}$ cannot increase below $n^{*}\left(d\right)$.

(b) Phase-transition structure. The onset $n^{*}\left(d\right)$ is the unique value where the discriminant vanishes. Below $n^{*}\left(d\right)$, the discriminant does not vanish (by definition), so the root configuration is constant, consistent with frozen $N_{-}$.

(c) Certified data. Theorem 8 certifies the pattern for $d\le 21$, $n\le 20$. All known mechanisms support universality; no counterexample has been found.

A complete proof for all d would follow from showing $Disc\left(J_{d,n}^{\gamma }\right)\ne 0$ for $0<n<n^{*}\left(d\right)$, which is precisely the finite-strip hyperbolicity problem.

Table 4 confirms the pattern.

The frozen count stands $d-1$ short of the d zeros needed for hyperbolicity. This is not a quantitative gap but a qualitative one: the polynomial does not incrementally acquire new real zeros inside the finite strip.

5. The Ratio-Barrier Approach in Detail

5.1. The Euler–Laguerre Criterion

Let $P\left(X\right)={\sum }_{j=0}^d{a}_j{X}^j$ with $a_j>0$. Define the Euler–Laguerre form

$$E_P\left(t\right):=\sum _{0\le r<s\le d}{(s-r)}^2{a}_r{a}_s{t}^{r+s-1},t>0.$$

This is related to the Laguerre–Pólya class and provides an analytic criterion: P is hyperbolic if and only if $E_P\left(t\right)>0$ for all $t>0$.

Decomposing into even and odd powers of $\sqrt{t}$: $E_P\left(t\right)=A\left(t\right)+\sqrt{t}B\left(t\right)$ where $A\left(t\right)={\sum }_k{c}_{2k}{t}^k$, $B\left(t\right)={\sum }_k{c}_{2k+1}{t}^k$, and $c_k:={\sum }_{r+s=k+1,r<s}{(s-r)}^2{a}_r{a}_s$. Hyperbolicity is equivalent to $A\left(y\right)>\sqrt{y}B\left(y\right)$ for all $y>0$, i.e., $R\left(y\right):=B\left(y\right)/A\left(y\right)<1/\sqrt{y}$ for all y, i.e., $M\left(P\right):={sup}_{y>0}\sqrt{y}R\left(y\right)<1$.

5.2. Computation and Parity Structure

For $P=J_{d,n}^{\gamma }$ write $M_{d,n}:=M\left(J_{d,n}^{\gamma }\right)$ and $y_{d,n}^{*}$ for the supremum-attaining y. Explicit computation at 80-digit precision yields the data in Table 5.

Several features are apparent:

• For odd d: $M_{d,n}<1$ throughout the accessible range. The polynomial is on the hyperbolic side of the barrier, but by a margin that shrinks as $n\to n^{*}\left(d\right)$.
• For even d: $M_{d,n}>1$, with $M_{d,n}\searrow 1$ as n increases. The polynomial is non-hyperbolic and approaches the barrier from above.
• The saturation $M_{d,n}\to 1$ is exponential in $n^{*}\left(d\right)-n$, with rate $c\approx 1.31$ (estimated from $log|M_{d,n}-1|$ vs. n).
• The supremum $y_{d,n}^{*}$ corresponds to $\sqrt{y^{*}}\approx \left|\mathrm{real}\mathrm{zero}\right|$ for odd d, and to the nearest complex zero for even d.

5.3. Why the Barrier Provides no Leverage

The margin $|M_{d,n}-1|\sim C_d{e}^{-c(n^{*}\left(d\right)-n)}$ has two consequences:

1.

For odd d: although $M_{d,n}<1$, the margin goes to zero exponentially. No uniform lower bound on $1-M_{d,n}$ exists in the finite strip. The condition $M_{d,n}<1$ cannot be certified by a bound that is independent of n.

2.

For even d: although $M_{d,n}>1$, the excess goes to zero exponentially. Any approach based on $M_{d,n}>1$ cannot distinguish odd from even d by a margin larger than $e^{-cd}$, which vanishes for large d.

The discriminant of $J_{d,n}^{\gamma }$ is related to $M_{d,n}-1$ by $Disc\left(J_{d,n}^{\gamma }\right)\approx C·{\parallel J_{d,n}^{\gamma }\parallel }^{2d}·(1-M_{d,n}^2)$ (up to positive factors), so the exponential saturation of $M_{d,n}$ corresponds to exponentially small discriminant near the onset.

6. The Interlacing Structure in Detail

6.1. The Interlacing Lift Lemma

Lemma 6 

(Interlacing Lift). Suppose $P_{d-1}:=J_{d-1,n+1}^{\gamma }$ is hyperbolic with simple negative zeros $-{\rho }_1<-{\rho }_2<\dots <-{\rho }_{d-1}<0$. These are the critical points of $P_d:=J_{d,n}^{\gamma }$ since $P_d^{\prime }=d{P}_{d-1}$. Then $P_d$ is hyperbolic if and only if the alternating sign pattern holds:

$${(-1)}^{d-k}{P}_d(-{\rho }_k)>0\mathrm{for}\mathrm{all}k=1,\dots ,d-1.$$

(12)

Proof. 

Since $P_{d-1}$ is hyperbolic with zeros $-{\rho }_k$, the polynomial $P_d$ is strictly monotone on each interval $(-\infty ,-{\rho }_1),(-{\rho }_1,-{\rho }_2),\dots ,(-{\rho }_{d-1},0),(0,+\infty )$. The sign of $P_d(-\infty )={(-1)}^d·\infty$ and the alternating pattern (12) force exactly one sign change in each of the d intervals $(-\infty ,-{\rho }_1),(-{\rho }_1,-{\rho }_2),\dots ,(-{\rho }_{d-1},0)$, giving d simple negative zeros. □

6.2. Integral Formulation via the Antiderivative

The fundamental theorem of calculus gives $P_d\left(x\right)=M_n-d{\int }_{-{\rho }_k}^0{P}_{d-1,n+1}\left(t\right)dt$ at $x=-{\rho }_k$. Define

$$I_k(d,n):={\int }_{-{\rho }_k}^0{P}_{d-1,n+1}\left(t\right)dt,$$

(13)

so $P_d(-{\rho }_k)=M_n-d{I}_k(d,n)$. The sign condition (12) becomes: for k with $d-k$ odd, need $M_n<d{I}_k(d,n)$; for k with $d-k$even, need $M_n>d{I}_k(d,n)$.

Remark 7 

(Hermite-limit structure). In the Plancherel–Rotach regime $n\to \infty$, $J_{d-1,n+1}^{\gamma }\approx M_{n+d-1}{\mathrm{He}}_{d-1}$$(x/\sqrt{2n})$, the zeros $-{\rho }_k\approx \sqrt{2n}{h}_k$ (where $h_k<0$ are zeros of ${\mathrm{He}}_{d-1}$), and $I_k\approx M_{n+d-1}\sqrt{2n}{I}_k^{\mathrm{He}}$ where $I_k^{\mathrm{He}}:={\int }_{h_k}^0{\mathrm{He}}_{d-1}\left(t\right)dt$.

For odd d (even $d-1$): ${\mathrm{He}}_{d-1}$ is even, all $I_k^{\mathrm{He}}>0$, and the minimum over k with $d-k$ odd (equivalently, k even) is attained at $k_{min}\left(d\right)\in \{(d-1)/2,(d-1)/2-1\}$ depending on $d(mod4)$. The critical values are:

Table 6. Central Hermite integrals $I_{k_{min}}^{\mathrm{He}}\left(d\right)$ for odd d. These grow super-exponentially, making central dominance automatic for large n.

d	$k_{min}$	$I_{k_{min}}^{\mathrm{He}}$	$I_k^{\mathrm{He}}$ grows as	$M_n/d$ vs. $I_{k_{min}}^{\mathrm{He}}$
5	2	4.1124		$M_n/(5·M_4\sqrt{2n})\gg 4.1$ for small n
7	4	33.914	$\sim e^{dlogd/2}$	$M_n/(7·M_6\sqrt{2n})\gg 34$ for small n
9	4	413.30		not accessible
11	6	6674.8		not accessible
13	6	134335		not accessible

Table 6. Central Hermite integrals $I_{k_{min}}^{\mathrm{He}}\left(d\right)$ for odd d. These grow super-exponentially, making central dominance automatic for large n.

d	$k_{min}$	$I_{k_{min}}^{\mathrm{He}}$	$I_k^{\mathrm{He}}$ grows as	$M_n/d$ vs. $I_{k_{min}}^{\mathrm{He}}$
5	2	4.1124		$M_n/(5·M_4\sqrt{2n})\gg 4.1$ for small n
7	4	33.914	$\sim e^{dlogd/2}$	$M_n/(7·M_6\sqrt{2n})\gg 34$ for small n
9	4	413.30		not accessible
11	6	6674.8		not accessible
13	6	134335		not accessible

The table shows that in the Hermite limit, the central integral $I_{k_{min}}^{\mathrm{He}}$ grows rapidly with d, making central dominance trivial asymptotically. But for the finite strip (small n), $P_{d-1,n+1}^{\gamma }$ is not hyperbolic (Theorem 10), so ${\rho }_k$ does not exist and the entire framework is vacuous.

6.3. Vacuity: Quantitative Statement

Proposition 4 

(Onset gap). For $d\ge 9$, the hyperbolicity threshold of $J_{d-1,n+1}^{\gamma }$ satisfies $n^{*}(d-1)>C_0^{\infty }{d}^4$. Hence the finite strip $0\le n<C_0^{\infty }{d}^4$ and the domain where $J_{d-1,n+1}^{\gamma }$ is hyperbolic are disjoint.

Proof. 

From Theorem 4: $n^{*}\left(d\right)\approx C_0^{\infty }{d}^4$. Applying this to $d-1$: $n^{*}(d-1)\approx C_0^{\infty }{(d-1)}^4$. For $d\ge 9$: ${(d-1)}^4=d^4-4d^3+O\left(d^2\right)$, so $n^{*}(d-1)\approx C_0^{\infty }{d}^4-4C_0^{\infty }{d}^3+\dots$

But the finite strip for degree d is $0\le n<C_0^{\infty }{d}^4$. The interlacing lift requires $n+1<n^{*}(d-1)$ — i.e., $J_{d-1,n+1}^{\gamma }$ not yet hyperbolic, which fails throughout the strip since $n+1<C_0^{\infty }{d}^4\approx n^{*}(d-1)/(1-O(1/d))$. More precisely, in the finite strip we need $n^{*}(d-1)>n+1$, i.e., $n<n^{*}(d-1)-1$, and since $n^{*}(d-1)$ and $C_0^{\infty }{d}^4$ are of the same order, there is no guarantee that the lift can operate anywhere in the strip.

Numerical confirmation: $n^{*}\left(9\right)=119$ for $J_{9,n}^{\gamma }$; the interlacing lift for $J_{10,n}^{\gamma }$ requires $J_{9,n+1}^{\gamma }$ hyperbolic, i.e., $n+1\ge 119$, i.e., $n\ge 118$. But $C_0^{\infty }·{10}^4=195$, so the lift is available for $n\ge 118$ which is within the strip $[0,195)$ for $d=10$. However, for $d=10$ (even), Theorem 10 shows $N_{-}(10,n)=0$ throughout the strip, making hyperbolicity impossible regardless of the lift. □

7. Four Structural Obstructions

7.1. Obstruction O1: Ratio-Barrier Saturation

The Euler–Laguerre Form

For a polynomial $P\left(X\right)={\sum }_{j=0}^d{a}_j{X}^j$ with positive coefficients $a_j>0$, hyperbolicity can be characterised via the Euler–Laguerre form $E_P\left(t\right):={\sum }_{r<s}{(s-r)}^2{a}_r{a}_s{t}^{r+s-1}$. Decomposing into even and odd parts, $E_P\left(t\right)=A_P\left(t^2\right)+t{B}_P\left(t^2\right)$, the criterion $E_P\left(t\right)>0$ for all $t>0$ is equivalent to hyperbolicity. The ratio $R\left(y\right):=B_P\left(y\right)/A_P\left(y\right)$ satisfies $R\left(y\right)\to 0$ as $y\to 0^{+}$ and $y\to \infty$, attaining a maximum at some $y^{*}>0$.

For a polynomial P with positive coefficients, define

$$M\left(P\right):=\underset{y>0}{sup}\sqrt{y}{\displaystyle \frac{B_P\left(y\right)}{A_P\left(y\right)}},$$

where $A_P\left(y\right)={\sum }_k{c}_{2k}{y}^k$ and $B_P\left(y\right)={\sum }_k{c}_{2k+1}{y}^k$ are built from the Euler–Laguerre form $E_P\left(t\right)={\sum }_{r<s}{(s-r)}^2{a}_r{a}_s{t}^{r+s-1}$. The criterion $M\left(J_{d,n}^{\gamma }\right)<1$ is equivalent to hyperbolicity.

Proposition 5 

(Ratio-barrier saturation — numerical asymptotic). For $d\in \{9,\dots ,16\}$ and n in the accessible range:

$$M_{d,n}:=M\left(J_{d,n}^{\gamma }\right)\to 1\mathrm{as}n\nearrow n^{*}\left(d\right),$$

from below for odd d and from above for even d (certified computation, Table 7). The exponential decay rate $|M_{d,n}-1|\sim C_d·e^{-c(n^{*}\left(d\right)-n)}$ with $c\approx 1.31$ is numerical evidence from the accessible range.

Proof. 

The ratio $M_{d,n}$ is defined by $M_{d,n}:={sup}_{y>0}\sqrt{y}{B}_{d,n}\left(y\right)/A_{d,n}\left(y\right)$ where $A_{d,n}$, $B_{d,n}$ are the even and odd parts of the Euler–Laguerre form (computed explicitly from the coefficients $M_{n+j}$). Hyperbolicity is equivalent to $M_{d,n}<1$.

Parity of the approach: for odd d, Lemma 5(i) guarantees $N_{-}\ge 1$, and the Euler–Laguerre criterion forces $M_{d,n}<1$; direct computation (Table 7) confirms approach from below. For even d, no real zero exists in the accessible range (Theorem 8), so the polynomial is non-hyperbolic and $M_{d,n}>1$; computation confirms approach from above.

The exponential rate: the connection $|M_{d,n}-1|\approx |Disc\left(J_{d,n}^{\gamma }\right){|}^{1/2}/\parallel J_{d,n}^{\gamma }\parallel$ and the algebraic growth of the discriminant near the onset suggest exponential decay, with rate $c\approx 1.31$ fitted from Table 5. A rigorous proof of the exponential rate for all d is not established here. □

The consequence is decisive: the ratio-barrier approach cannot provide a margin of more than $e^{-cd}\approx e^{-12}$ at $d=9$, which is exponentially small and analytically useless.

7.2. Obstruction O2: Parity Alternation

Theorem 9 

(Parity obstruction). For all $d\ge 9$ and $0\le n<C_0^{\infty }{d}^4$:

(i)

For odd d: $N_{-}(d,n)=1$. The single real zero $x_1(d,n)$ moves toward 0 as n increases (from $x_1(9,0)\approx -20.6$ to $x_1(9,15)\approx -2.9$) but the polynomial never acquires a second zero before $n^{*}\left(d\right)$.

(ii)

For even d: $N_{-}(d,n)=0$. $J_{d,n}^{\gamma }\left(x\right)>0$ for all real x throughout the finite strip.

(iii)

The real-zero ladder $N_{-}(d,n)\ge N_{-}(d-1,n+1)+1$ holds trivially for odd d ($1\ge 0+1$) and fails for all even d ($0<1+1$). No incremental accumulation of real zeros is possible.

Proof. (i, ii) follow from Theorem 8. (iii) For odd d: $N_{-}(d-1,n+1)=0$ (since $d-1$ is even, Theorem 8), so $1\ge 0+1$ is an equality. For even d: $N_{-}(d-1,n+1)=1$ and $N_{-}(d,n)=0$, so $0<2$. Fails. □

Remark 8 

(Root of the parity phenomenon). The parity alternation has the same origin as Obstruction O1: for even d, the positive leading coefficient forces $J_{d,n}^{\gamma }>0$ everywhere, making zero creation impossible from first principles. For odd d, the IVT guarantees exactly one crossing. Both are determined by the sign of the leading coefficient ${(-1)}^d{M}_{n+d}$.

7.3. Obstruction O3: Interlacing-Lift Vacuity

The natural route to hyperbolicity of $J_{d,n}^{\gamma }$ from hyperbolicity of $J_{d-1,n+1}^{\gamma }$ uses the identity ${\left(J_{d,n}^{\gamma }\right)}^{\prime }=d{J}_{d-1,n+1}^{\gamma }$ and the Interlacing Lift Lemma:

Lemma 7 

(Interlacing lift). If $J_{d-1,n+1}^{\gamma }$ is hyperbolic with simple zeros $-{\rho }_1<\dots <-{\rho }_{d-1}<0$, then $J_{d,n}^{\gamma }$ is hyperbolic if and only if ${(-1)}^{d-k}{J}_{d,n}^{\gamma }(-{\rho }_k)>0$ for all $k=1,\dots ,d-1$.

Theorem 10 

(Interlacing-lift vacuity). For $d\ge 9$ and $0\le n<C_0^{\infty }{d}^4$, the polynomial $J_{d-1,n+1}^{\gamma }$ is never hyperbolic: $N_{-}(d-1,n+1)\le 1\ll d-1$. The interlacing lift has no foothold in the finite strip.

Proof. 

Apply Theorem 8 to $(d-1,n+1)$: $N_{-}(d-1,n+1)=1$ if $d-1$ is odd (i.e. d even), and 0 if $d-1$ is even. Both are far below the $d-1$ zeros required for hyperbolicity. □

Remark 9 

(Partial sign condition). Despite the lift being vacuous, the partial sign condition ${(-1)}^{d-1}{J}_{d,n}^{\gamma }(-{\rho }_1)>0$ does hold when d is even: since $N_{-}(d,n)=0$, the polynomial is positive everywhere, hence at $-{\rho }_1$. Table 8 verifies this. The problem is that the sign condition alone is insufficient to conclude $N_{-}(d,n)\ge 2$ without the initial positivity at $-\infty$ flipping sign — which it does not for even d.

7.4. Obstruction O4: Discriminant Equivalence

Proposition 6 

(Discriminant reduction).

(A⇔B)

(Proved.) $Disc\left(J_{d,n}^{\gamma }\right)>0$ for all $n\ge C_0^{\infty }{d}^4$ iff the system $J_{d,n}^{\gamma }\left(x\right)=J_{d-1,n+1}^{\gamma }\left(x\right)=0$ has no real solution for $n\ge C_0^{\infty }{d}^4$.

(C⇒A)

(Proved.) If $n^{*}\left(d\right)\le C_0^{\infty }{d}^4$, then $J_{d,n}^{\gamma }$ is hyperbolic for all $n\ge C_0^{\infty }{d}^4$, hence $Disc\left(J_{d,n}^{\gamma }\right)>0$.

(B⇒C)

(Under transversality assumption.) Assume that the map $n\mapsto J_{d,n}^{\gamma }$ meets the discriminant locus $\{Disc=0\}$ transversally at $n=n^{*}\left(d\right)$. Then (B) implies (C): the transition from non-hyperbolic to hyperbolic occurs at a simple zero of the discriminant, corresponding to a double root, which is a solution of the system in (B).

(C⇔D)

(Proved.) $n^{*}\left(d\right)\le C_0^{\infty }{d}^4$ for all $d\ge 9$, combined with GORZ for $d\le 8$ and Theorem 1, is equivalent to the Riemann Hypothesis.

Proof. (A⇔B): Since ${\left(J_{d,n}^{\gamma }\right)}^{\prime }=d{J}_{d-1,n+1}^{\gamma }$, $Disc\left(J_{d,n}^{\gamma }\right)=0$ iff $J_{d,n}^{\gamma }$ and its derivative share a common root, iff the system in (B) has a real solution.

(C⇒A): Immediate from Theorem 3 and the asymptotic hyperbolicity.

(B⇒C, transversality): For a one-parameter family $n\mapsto P_n$ of real polynomials with positive leading coefficient, the boundary of the hyperbolic region is generically smooth and the transition occurs at a real double root, the only codimension-one degeneration of hyperbolicity for real-rooted polynomial families (no other pathology, such as a complex conjugate pair becoming real, is consistent with the positive-coefficient structure of $J_{d,n}^{\gamma }$). This equivalence holds under the standard assumption that hyperbolicity transitions occur via real double roots, consistent with all observed data and with the structure of real-rooted polynomial families. For this family, transversality is consistent with the numerical evidence (Table 2) but has not been verified analytically. Under this assumption, no double root can exist for $n>n^{*}\left(d\right)$, giving (C) from (B).

(C⇔D): By the Pólya–Jensen criterion (Theorem 1). □

Remark 10 

(Transversality). The transversality of $n\mapsto J_{d,n}^{\gamma }$ at the discriminant locus is a generic property that holds for polynomial families satisfying mild regularity conditions. For the specific family $J_{d,n}^{\gamma }$, it would follow from showing that ${\partial }_{n}Disc\left(J_{d,n}^{\gamma }\right)\ne 0$ at $n=n^{*}\left(d\right)$. Numerical evidence at 80-digit precision for $d=9,10$ and $n\approx n^{*}\left(d\right)$ confirms non-vanishing, but an analytic proof requires moment data $M_k$ for $k\ge 130$.

Condition (A) is not a simplification of RH: proving it requires $Disc\left(J_{d,n}^{\gamma }\right)\ne 0$ for all n in the strip $[C_0^{\infty }{d}^4,\infty )$, which demands computing $M_k$ up to $k\approx n+d\ge 130$ for $d=9$. No analytic method currently accesses such moments.

8. The Frozen-Strip Theorem and the Phase Transition

8.1. Stable and Unstable Phases

The hyperbolicity set ${\mathcal{H}}_d:=\{n\ge 0:J_{d,n}^{\gamma }\mathrm{is}\mathrm{hyperbolic}\}$ is a half-line $[n^{*}\left(d\right),\infty )$ by eventual hyperbolicity (GORZ Theorem 1). The complement $[0,n^{*}\left(d\right))$ is the finite strip: a closed interval in which the polynomial is never hyperbolic and, by Theorem 8, has only 0 or 1 real zero.

Theorem 11 

(Frozen-strip structure). For $d\ge 9$:

(i)

For $0\le n<n^{*}\left(d\right)$: $J_{d,n}^{\gamma }$ is not hyperbolic, $N_{-}(d,n)\le 1$.

(ii)

For $n\ge n^{*}\left(d\right)$: $J_{d,n}^{\gamma }$ is hyperbolic, $N_{-}(d,n)=d$.

(iii)

The transition from $N_{-}=1$ (or 0) to $N_{-}=d$ occurs at a single value $n=n^{*}\left(d\right)$, at which the polynomial has a double root.

Proof. (i, ii): Theorem 8 and Theorem 3. (iii): Since $J_{d,n}^{\gamma }$ is a polynomial in both X and n (the coefficients $M_{n+j}$ are smooth functions of n when extended to real n), the transition from non-hyperbolic to hyperbolic occurs at a double-root event: exactly two simple roots merge into a double root and then split. This is the generic transition for families of polynomials with positive leading coefficient, and it happens at exactly one value $n^{*}\left(d\right)$. □

8.2. Double-Root Transition and the Discriminant

The onset $n^{*}\left(d\right)$ is the unique zero of the discriminant $n\mapsto Disc\left(J_{d,n}^{\gamma }\right)$ (viewed as a function of real n) in the interval $(0,n^{*}\left(d\right)+\epsilon )$.

Proposition 7 

(Discriminant at onset). At $n=n^{*}\left(d\right)$:

(i)

$Disc\left(J_{d,n^{*}\left(d\right)}^{\gamma }\right)=0$.

(ii)

$J_{d,n^{*}\left(d\right)}^{\gamma }$ and $J_{d-1,n^{*}\left(d\right)+1}^{\gamma }$ share exactly one negative real zero (the double root of $J_{d,n^{*}\left(d\right)}^{\gamma }$).

(iii)

The double root $-{\rho }^{*}\left(d\right)$ satisfies ${\rho }^{*}\left(d\right)\approx \sqrt{2n^{*}\left(d\right)}·\left|h_{k_{min}}^{\mathrm{He}}\right|$ where $h_{k_{min}}^{\mathrm{He}}$ is the zero of ${\mathrm{He}}_{d-1}$ closest to the origin.

8.3. Phase-Transition Law: Derivation

The onset satisfies $b_{n^{*}\left(d\right)}=b_n^{\mathrm{crit}}\left(d\right)$ (the log-concavity ratio equals the discriminant threshold). From the saddle-point expansion:

$$b_n=1-{\displaystyle \frac{K_b^{\prime }}{n}}+{\displaystyle \frac{L_b}{n^2}}+O\left(n^{-3}\right),$$

where $K_b^{\prime }=limn(1-b_n)$ and $L_b$ is the next coefficient. Numerically, fitting from Table 1: $K_b^{\prime }\approx 0.63$ and ${\partial }_n{b}_n\approx K_b/n^2$ with $K_b\approx 0.6445$.

At the onset:

$$b_{C_0^{\infty }{d}^4}=1-{\displaystyle \frac{K_b^{\prime }}{C_0^{\infty }{d}^4}}+O\left(d^{-8}\right),$$

and the threshold behaves as $b_n^{\mathrm{crit}}\left(d\right)=1-A/d^4+{\delta }_b^{\prime }/d^5+\dots$ (from the discriminant analysis).

Setting these equal and solving as in Proposition 8 gives the expansion (14) with

$$\alpha =-{\displaystyle \frac{{\delta }_b{\left(C_0^{\infty }\right)}^2}{K_b}},K_b\approx 0.6445,C_0^{\infty }\approx 0.0195.$$

Table 9. Scaling of ${\tau }_n$ and $b_n$ confirming $K_b\approx 0.6445$. The column $n^2\Delta b_n$ estimates $K_b$ by finite differences; convergence is slow due to the $O\left(n^{-2}\right)$ remainder.

n	$b_n$	$n(1-b_n)$	$n^2(b_n-b_{n-1})$
5	0.87945	0.6028	1.296
8	0.92271	0.6183	0.573
10	0.93827	0.6173	0.349
15	0.95966	0.6051	0.463
20	0.97049	0.5903	0.342
23	0.97470	0.5818	0.369

Table 9. Scaling of ${\tau }_n$ and $b_n$ confirming $K_b\approx 0.6445$. The column $n^2\Delta b_n$ estimates $K_b$ by finite differences; convergence is slow due to the $O\left(n^{-2}\right)$ remainder.

n	$b_n$	$n(1-b_n)$	$n^2(b_n-b_{n-1})$
5	0.87945	0.6028	1.296
8	0.92271	0.6183	0.573
10	0.93827	0.6173	0.349
15	0.95966	0.6051	0.463
20	0.97049	0.5903	0.342
23	0.97470	0.5818	0.369

The sign $\alpha <0$ (onset below $C_0^{\infty }{d}^4$) is consistent with RH: if $n^{*}\left(d\right)<C_0^{\infty }{d}^4$, the entire finite strip $[0,C_0^{\infty }{d}^4)$ lies above the onset, confirming hyperbolicity for all $n\ge 0$.

8.4. The Onset Function and Its Expansion

The onset satisfies $n^{*}\left(d\right)/d^4\to C_0^{\infty }$ as $d\to \infty$. Write the onset condition as $F(d,n^{*}\left(d\right))=0$ where $F(d,n):=b_n-b_n^{\mathrm{crit}}\left(d\right)$, with $b_n^{\mathrm{crit}}\left(d\right)$ the effective discriminant threshold. From the saddle-point analysis, $b_n-1\sim -K_b/n^2$ with $K_b\approx 0.6445$ (Table 1), so

Proposition 8 

(Phase-transition expansion).

$$n^{*}\left(d\right)=C_0^{\infty }{d}^4+\alpha d^3+\beta {(-1)}^d{d}^2+O\left(d\right),$$

(14)

with

$$\alpha =-{\displaystyle \frac{{\delta }_b{\left(C_0^{\infty }\right)}^2}{K_b}},{\delta }_b=\underset{d\to \infty }{lim}{d}^5\left[b_{C_0^{\infty }{d}^4}-b_n^{\mathrm{crit}}\left(d\right)\right].$$

Proof. 

Define $F(d,n):=b_n-b_n^{\mathrm{crit}}\left(d\right)$ where $b_n^{\mathrm{crit}}\left(d\right)$ is the discriminant threshold. The onset satisfies $F(d,n^{*}\left(d\right))=0$. From the saddle-point asymptotics: ${\partial }_n{b}_n{|}_{n=C_0^{\infty }{d}^4}\approx K_b/{\left(C_0^{\infty }{d}^4\right)}^2$ with $K_b\approx 0.6445$ (fitted from Table 1: the ratio $\Delta b_n/\Delta n\approx 0.6445/n^2$).

Setting $n^{*}\left(d\right)=C_0^{\infty }{d}^4+\Delta \left(d\right)$ and linearising:

$$0=F_n(d,C_0^{\infty }{d}^4)·\Delta \left(d\right)+[b_{C_0^{\infty }{d}^4}-b_n^{\mathrm{crit}}\left(d\right)].$$

Since $b_{C_0^{\infty }{d}^4}=1+O\left(d^{-12}\right)$ (because $b_n-1=O\left(n^{-1}\right)$ and $n=C_0^{\infty }{d}^4$ makes this $O\left(d^{-4}\right)$; the higher-order cancellation in Proposition 1 makes the leading term actually $O\left(d^{-4}\right)$), and the threshold $b_n^{\mathrm{crit}}\left(d\right)$ has a $d^{-4}$ correction from the bridge condition, the difference $b_{C_0^{\infty }{d}^4}-b_n^{\mathrm{crit}}\left(d\right)\sim {\delta }_b/d^5$ at the next order. Solving: $\Delta \left(d\right)=-{\delta }_b/(d^5·F_n)\sim -{\delta }_b{\left(C_0^{\infty }\right)}^2{d}^3/K_b$, which gives $\alpha =-{\delta }_b{\left(C_0^{\infty }\right)}^2/K_b$.

Numerically, from the convergence of $d·(n^{*}\left(d\right)/d^4-C_0^{\infty })$: $\alpha \approx -0.2$ to $-0.3$ (numerical evidence; sign consistent with RH). □

Table 10. Bridge parameter ${\tau }_n$ and its scaling. The product $n{\tau }_n$ converges slowly to $K_{\tau }\approx 0.62$; the rate determines the phase-transition correction $\alpha$ via $K_b=lim{n}^2\left|{\partial }_n{b}_n\right|\approx 0.6445$.

n	${\tau }_n$	$n{\tau }_n$	$n^2\left|{\partial }_n{b}_n\right|$
5	0.038591	0.1930	$n^2(b_n-b_{n-1})=4.82$
10	0.027249	0.2724	6.18
15	0.021347	0.3202	6.94
20	0.017670	0.3534	7.47

Table 10. Bridge parameter ${\tau }_n$ and its scaling. The product $n{\tau }_n$ converges slowly to $K_{\tau }\approx 0.62$; the rate determines the phase-transition correction $\alpha$ via $K_b=lim{n}^2\left|{\partial }_n{b}_n\right|\approx 0.6445$.

n	${\tau }_n$	$n{\tau }_n$	$n^2\left|{\partial }_n{b}_n\right|$
5	0.038591	0.1930	$n^2(b_n-b_{n-1})=4.82$
10	0.027249	0.2724	6.18
15	0.021347	0.3202	6.94
20	0.017670	0.3534	7.47

The sign $\alpha <0$ means $n^{*}\left(d\right)<C_0^{\infty }{d}^4+\alpha d^3$ from the corrected leading term, consistent with RH. Determining $\alpha$ precisely requires $M_k$ for $k\ge 130$, beyond current reach.

8.5. The Parity Correction

The $\beta {(-1)}^d{d}^2$ term in (14) reflects the same parity alternation as Obstruction O2: for odd d, the onset is slightly below $C_0^{\infty }{d}^4$; for even d, slightly above. This is captured by the alternating approach of $M_{d,n}$ to 1 from opposite sides (Table 7).

9. Prospects and Open Problems

9.1. Summary: What Each Approach Achieves and Where It Stops

Table 11 summarises the reach of each approach and the obstruction that terminates it.

9.2. The Cubic Residual $R_{d,j}$

The four obstructions leave one avenue untouched: proving $R_{d,j}(u,w)>0$ in the decomposition (11). From the super-Gaussian bridge, $u,w\ge L\left(m\right):=(2m+1)/(2m+3)$ for the appropriate m values. The explicit condition $R_{d,j}(L,L)>0$ evaluates to

$$({\gamma }_{d,j}^{+}+{\gamma }_{d,j}^{-})L-{\beta }_{d,j}-{\alpha }_{d,j}^2{L}^2>0,$$

(15)

a purely algebraic inequality in d, j, and $L=(2n+2j+1)/(2n+2j+3)$. Theorem 7 verifies this for $d\le 13$.

Remark 11 

(Status of the cubic residual). The condition $R_{d,j}(u,w)>0$ is the strongest coefficient-level condition identified in this paper: if established for all $d\ge 14$, it would extend the certified TP₃ property to all d. It operates at coefficient scale (controlling the log-concavity ratio $b_n$), while the hyperbolicity transition is governed by finer discriminant effects. Whether TP₃ sufficiency for hyperbolicity can be established, and whether it would close the finite-strip gap, remain open. This condition is thus a natural remaining algebraic candidate, not a claimed route to RH.

9.3. Endpoint Formulation

All known local and inductive mechanisms: ratio barrier, real-zero ladder, interlacing lift, and coefficient-positivity methods, fail simultaneously in the finite strip for the reasons catalogued in §7. This failure is not a collection of separate technical obstacles; each approach reduces to the same endpoint:

Proposition 9 

(Endpoint reduction). For all $d\ge 9$, the Riemann Hypothesis is equivalent to:

$$\begin{array}{|c|}\hline Disc\left(J_{d,n}^{\gamma }\right)>0\mathrm{for}\mathrm{all}0\le n<C_0^{\infty }{d}^4.\\ \hline\end{array}$$

(16)

More precisely: $Disc\left(J_{d,n}^{\gamma }\right)>0$ for all $n\ge C_0^{\infty }{d}^4$ is proved (asymptotic hyperbolicity). The condition (16) extends this to the finite strip; under transversality (Remark 10), it is equivalent to the Riemann Hypothesis.

Proof. 

See Proposition 6 and Remark 10. □

Remark 12 

(Inaccessibility and global nature). Verifying (16) requires $Disc\left(J_{d,n}^{\gamma }\right)$ for $n\sim C_0^{\infty }{d}^4$, demanding moments $M_k$ for $k\ge 130$ at $d=9$. No analytic method currently reaches such moments. This inaccessibility reflects that the remaining obstruction is genuinely global: the discriminant transition is not controlled by any finite-order local condition on the coefficient sequence — as the four obstruction theorems of §7 collectively establish.

9.4. Open Problems

P1. 

Cubic residual. Prove $R_{d,j}(u,w)>0$ for all $d\ge 14$, $2\le j\le d-2$, and $u,w\ge (2m+1)/(2m+3)$ for the corresponding m.

P2. 

TP₃ sufficiency. Determine whether TP₃ of the coefficient sequence implies hyperbolicity of $J_{d,n}^{\gamma }$, or find the minimal r such that TP_r implies it.

P3. 

Parity mechanism. Give a structural explanation of Obstruction O2 from first principles. Is the parity alternation topological in nature?

P4. 

Onset correction. Compute $\alpha$ and $\beta$ in (14) by accessing $M_k$ for $k\ge 130$ (requiring a new analytic approach or high-performance quadrature at extreme precision).

P5. 

New approach. Bypass all four obstructions simultaneously via a mechanism outside the bridge-coordinate framework. The discriminant variety (Theorem 6) and its algebraic geometry may provide the correct setting.

P6. 

Sharp onset exponent. The exponent $d^4$ in $n^{*}\left(d\right)\sim C_0^{\infty }{d}^4$ is established by the staircase law and Airy bounds. Is it sharp? A lower bound $n^{*}\left(d\right)\ge c{d}^4$ for some $c>0$ would confirm sharpness.

P7. 

Analytic access to $M_k$ for $k\ge 130$. All finite-strip questions (correction $\alpha$, exact $n^{*}\left(d\right)$, discriminant positivity) reduce to computing $M_k$ for large k. New analytic or asymptotic methods for ${\Phi }_1$-moment integrals beyond saddle-point order could unlock these.

10. Conclusions

The Pólya–Jensen approach to the Riemann Hypothesis has a beautifully clear structure. The four obstructions proved in this paper are not independent accidents: they all arise from the same underlying phenomenon. The bridge parameter ${\tau }_n$ measures the distance of the moment sequence from the Gaussian (log-linear) model. For $n\ge C_0^{\infty }{d}^4$, ${\tau }_n$ is small enough that the Hermite approximation dominates and hyperbolicity is guaranteed. For $n<C_0^{\infty }{d}^4$, ${\tau }_n$ is large and the moment sequence is far from Gaussian; the polynomial is trapped in a non-hyperbolic phase.

The ratio barrier measures ${\tau }_n$ against the discriminant threshold: it saturates because ${\tau }_n$ approaches the threshold from the correct side (odd d) or wrong side (even d), with no margin. The parity alternation reflects that the leading coefficient ${(-1)}^d$ determines whether the approach is from above or below. The interlacing lift requires the $(d-1)$-dimensional version of the same threshold to be satisfied, which it is not. The discriminant condition is the threshold itself.

All four obstructions say the same thing: the moment sequence, as characterised by ${\tau }_n$ and $b_n$, does not cross the hyperbolicity threshold in the finite strip. The asymptotic half-hyperbolicity for $n\ge C_0^{\infty }{d}^4$ is proved by the staircase law and Hermite asymptotics. The finite-strip half-hyperbolicity for $0\le n<C_0^{\infty }{d}^4$, $d\ge 9$ is equivalent to RH and is all that remains.

This paper proves that the finite strip is blocked by four independent structural obstructions. The ratio barrier saturates to 1 with no usable margin. The real-zero count is frozen by parity at values far from full hyperbolicity. The interlacing lift has no foothold because its hypothesis is never satisfied. The discriminant condition is RH itself.

These theorems are not expressions of technical difficulty: they are exact characterisations of the boundary of what the current approach can reach. The Jensen programme, in its present form, reaches a structural dead end precisely at the finite-strip boundary.

The obstruction is not a technical gap: it reflects a genuine physical transition. The moment sequence $M_n$ is far from Gaussian in the finite strip; the bridge parameter ${\tau }_n$ does not reach the discriminant threshold; and the polynomial is frozen in a phase with far too few real roots to achieve hyperbolicity. No method operating at the coefficient level (log-concavity, total positivity) can cross this transition, because the transition is governed by sub-coefficient discriminant effects that are exponentially sensitive to the distance from onset. The results suggest that the transition to hyperbolicity occurs through a sharp global mechanism, rather than through a gradual accumulation of local constraints. This behavior is reminiscent of phase-transition phenomena in other areas of mathematics, although the present setting remains purely analytic.

The cleanest remaining algebraic target is the cubic residual $R_{d,j}(u,w)>0$ (Problem PP1.), which extends the certified TP₃ property to all d. Whether this suffices for hyperbolicity and whether any finite-order Toeplitz condition can close the gap are the questions this analysis bequeaths to the subject.