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# A Two-Thirds Hyperbolicity Wedge for Jensen Polynomials of Riemann's Xi-Function
- URL: https://www.beyond-plausible.com/a-two-thirds-hyperbolicity-wedge-for-jensen-polynomials-of-riemanns-xi-function/
- Published: 2026-09-05T20:38:46.000Z
- Updated: 2026-09-05T20:38:46.000Z
- Author: John Savva
- Tags: AI in Mathematics

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Mathematics · Feature

# Widening the Wedge

Six carefully matched coefficients may widen a small window into the Riemann hypothesis. An AI-assisted proof candidate shows how the mathematics works, and how much checking a new result needs.

**By John Savva** · Independent researcher · September 2026

### About this article and how the work was made

This is an author's account of my preprint, *A Two-Thirds Hyperbolicity Wedge for Jensen Polynomials of Riemann's Xi-Function* ([Version 1.1](https://doi.org/10.5281/zenodo.22293642?ref=beyond-plausible.com)). It was developed with substantial AI assistance, primarily **OpenAI GPT-5.6 Sol Pro**, and checked using Lean 4, computer algebra and AI review. The Lean theorem retains explicit hypotheses for three groups of literature results. Mathematica corroborated selected calculations. No independent human mathematical review or peer review is recorded, and the work does not prove the Riemann hypothesis. I directed, reviewed and edited the work and accept responsibility for it.

In May 2019, a paper in the Proceedings of the National Academy of Sciences lit up the science press. "Mathematicians Edge Closer to Solving a 'Million Dollar' Math Problem," ran one headline.[1](#note-1) The result came with a careful qualification. Enrico Bombieri, writing the companion commentary in the same journal, said the work "remains far away from proving the Riemann hypothesis," but represents "a big step forward."[2](#note-2) Michael Griffin, Ken Ono, Larry Rolen and Don Zagier had found a pattern in a family of polynomials attached to the Riemann zeta function.[3](#note-3) For each fixed degree, sufficiently far along the family, all the roots become real. A problem about an infinite function had yielded to a surprisingly familiar collection of finite objects.

That result energized an old program. Work by the Danish mathematician Johan Jensen in the 1910s, extended by George Pólya in 1927, translated the Riemann hypothesis into a question about polynomials.[4](#note-4) In its modern form the dictionary reads: the Riemann hypothesis is true if and only if every polynomial in a certain infinite family, built from the coefficients of Riemann's xi-function, is *hyperbolic*, meaning all of its roots are real. The 2019 theorem verified an infinite part of that family, and it made the whole enterprise feel suddenly current again.

This article is about the next chapter, and about a result that widens the known territory in a direction mathematicians care about: letting the degree of the polynomials grow. It is also about how the result was made. The proof was developed by an independent researcher, myself, in sustained collaboration with an AI system, and then subjected to an unusually elaborate battery of checks: formal verification in the Lean proof assistant, an independent clean-room recomputation in Mathematica, interval arithmetic, deliberate sabotage tests, and adversarial review by rival AI models that were invited to find mistakes. They found several. The surviving argument, described below, is offered as a proof candidate, not a certified theorem.

## 1The function behind the primes

Why the zeta function matters, and why anyone would build polynomials out of its coefficients.

Start with a deceptively simple infinite series. For a real or complex number $s$ with real part greater than one, define

$$\\zeta(s) \\;=\\; 1+\\frac{1}{2^s}+\\frac{1}{3^s}+\\frac{1}{4^s}+\\cdots$$

Euler discovered in the eighteenth century that this series secretly encodes the prime numbers: the reciprocal of the zeta function factors as a product over all primes, a fact as startling as learning that a sum over all whole numbers knows which of them are indivisible. In his short memoir of 1859, Riemann outlined a connection between the distribution of primes and the *zeros* of $\\zeta(s)$: the points in the complex plane where the function vanishes.[5](#note-5)

Riemann's memoir contained a conjecture about where those zeros live. Some sit at negative even integers and are called trivial. The rest, the nontrivial zeros, lie inside the *critical strip*, the vertical band of the complex plane between real parts 0 and 1\. Riemann conjectured that every one lies on its central line, with real part exactly one-half. A modern computation has rigorously verified this up to an imaginary height of three trillion; that is a height bound, not a count of zeros.[6](#note-6) The assertion for all heights is the Riemann hypothesis. It would sharply limit the fluctuations in how many primes lie below a given number. Proving it remains one of mathematics' most famous challenges.

Re s Im s 0 1 2 critical line: Re s = 1/2 pole at s = 1 critical strip Riemann hypothesis: every nontrivial zero of ζ(s) lies on the red line 

**Figure 1.** The critical strip of the zeta function, between real parts 0 and 1\. Nontrivial zeros (blue dots, heights not to scale) all appear to sit on the central line with real part one-half. The Riemann hypothesis asserts that none ever stray. The polynomials studied in this article form a finite test of the infinite geometry behind that assertion.

For this story the raw zeta function is not the most convenient object. Mathematicians prefer to package it, multiplying by a gamma factor and two elementary terms to obtain Riemann's *xi-function*:

$$\\xi(s) \\;=\\; \\tfrac{1}{2}\\, s (s-1)\\, \\pi^{-s/2}\\, \\Gamma(s/2)\\, \\zeta(s).$$

The packaging removes a pole, makes the function *entire* (defined and well-behaved everywhere in the complex plane), and, crucially, gives it a mirror symmetry: $\\xi(s) = \\xi(1-s)$. The xi-function is symmetric about the point $s = 1/2$, the midpoint of the critical strip. Shift the origin to that point by writing $s = 1/2 + w$, and only even powers of $w$ survive:

$$\\xi\\!\\left(\\tfrac{1}{2}+w\\right) \\;=\\; \\sum\_{n \\geq 0} \\frac{\\gamma(n)}{n!}\\, w^{2n}.$$

The positive numbers $\\gamma(0), \\gamma(1), \\gamma(2), \\ldots$ encode the Taylor expansion; the coefficient of $w^{2n}$ is $\\gamma(n)/n!$. They are not the function's zeros. Know every $\\gamma(n)$ and you can rebuild $\\xi$ exactly. Riemann's integral representation supplies another way to calculate them: apart from a factorial prefactor, they are *moments* of a positive, rapidly decaying kernel built from the theta function, an object also used to describe heat flowing around a ring. In the normalization of this work,

$$\\gamma(n) \\;=\\; \\frac{8\\, n!}{(2n)!}\\int\_0^\\infty \\omega(e^{2u})\\, e^{u/2}\\, u^{2n}\\, du,$$

where $\\omega$ is an explicit combination of theta functions. A coefficient of the most storied function in mathematics turns out to be the average of a concrete, positive, exponentially decaying quantity against the powers $u^{2n}$. That innocuous-looking factor of 8 in front will return later as a genuine trap.

## 2Finite shadows of an infinite function

Jensen's polynomials, Pólya's equivalence, and why one more power in an exponent is worth years of work.

From the coefficient sequence, build a polynomial. For a degree $d$ and a starting position $n$, the *Jensen polynomial* is

$$J^{d,n}(X) \\;=\\; \\sum\_{j=0}^{d} \\binom{d}{j}\\, \\gamma(n+j)\\, X^{j}.$$

It grabs a window of $d+1$ consecutive coefficients and mixes them with the binomial coefficients familiar from Pascal's triangle. It is an ordinary, finite polynomial, and so it can be asked an ordinary geometric question: where are its roots?

A polynomial with only real roots is called *hyperbolic*; the definition permits repeated roots. Here all the coefficients are positive, so any real roots must be negative. The new theorem goes further by showing that its roots are distinct. Jensen introduced these polynomials while studying when entire functions can be approximated by polynomials with real roots. Pólya, extending Jensen's work and unpublished notes in 1927, established the equivalence used here: the Riemann hypothesis holds if and only if $J^{d,n}$ is hyperbolic for every degree $d$ and every shift $n$.

That equivalence reframes the most intimidating problem in analysis as an infinite tower of finite, checkable statements. It does not make it easy. The tower has no top, and any finite number of verified floors settles nothing. But it explains why hyperbolicity theorems for Jensen polynomials are prized: each one certifies a floor of the tower outright.

The 2019 jolt supplied a new pattern across every fixed degree. Griffin, Ono, Rolen and Zagier proved that $J^{d,n}$ is hyperbolic once $n$ is large enough. Their method also proved hyperbolicity through degree eight for every shift.[7](#note-7) After a suitable rescaling, the polynomials converge to the Hermite polynomials, the classical family appearing in the quantum harmonic oscillator. Hermite roots are real and simple, though not evenly spaced. A follow-up by Griffin, Ono, Rolen, Jesse Thorner, Zachary Tripp and Ian Wagner made the threshold effective and developed sharper asymptotics for xi.[8](#note-8)

A fixed-degree theorem lets the degree stand still while the coefficients march to infinity. The harder question is how fast the degree may grow along with them.

The original limit fixes $d$ first, then sends $n$ to infinity. The follow-up went further: its bound $n \\geq c e^{d/2}$, for a constant $c>0$, already allows the degree to grow logarithmically with $n$. Separate results using verified zeta zeros also cover enormous finite ranges of degrees for every shift. Jonathan Holland's 2026 preprint moved the growing-degree comparison to a power of $n$.[9](#note-9) His three-fifths result, the starting point for this project, guarantees hyperbolicity under a condition of the form

$$n^{3}\\,\\log^{2}(n+2) \\;\\ge\\; K\\, d^{5}, \\qquad\\text{that is, roughly}\\qquad d \\;\\lesssim\\; n^{3/5},$$

where $K$ is a constant and logarithmic factors have been suppressed.

The new preprint raises that boundary to two-thirds: its theorem allows $d\\lesssim n^{2/3}$, again suppressing a logarithmic factor. The difference between $3/5=0.6$ and $2/3\\approx0.667$ may look small, but an exponent has a long reach. Ignoring constants and logarithms, the ratio of the new degree scale to the old is $n^{1/15}$; multiplying $n$ by a billion multiplies that ratio by about four. No improvement in a fixed constant can produce this change of growth rate. The proposed proof needs an additional mechanism.

log n log d earlier exponent: 3/5 this work: exponent 2/3 region covered by both power scales additional asymptotic region 

**Figure 2.** Two growth rates. Each point represents a coefficient index $n$ and degree $d$, on logarithmic axes. The lines retain the exponent ratio $(2/3)/(3/5)=10/9$. Their shared origin is schematic: constants and logarithmic factors are omitted. The blue band illustrates the larger asymptotic range claimed by this preprint relative to Holland's three-fifths result; it is not a plot of certified finite cases or of all earlier hyperbolicity results.

## 3The theorem

The preprint's precise claim, and the limits of its guarantee.

Here is the main result of the new preprint, stated in full.

### Theorem (two-thirds hyperbolicity wedge)

There is an absolute constant $K > 0$ such that for all integers $d \\geq 1$ and $n \\geq 0$, the condition

$$n^{2}\\,\\log(n+2) \\;\\ge\\; K\\, d^{3}$$

implies that $J^{d,n}$ has exactly $d$ distinct negative real zeros.

The argument makes the constant *effective*: in principle, it can be computed from the estimates. But it does not supply a useful numerical cutoff. The geometric part of the ledger alone is about $5.22 \\times 10^{44}$. The final sufficient constant is the maximum of that number and further terms from the analytic error and the finite cutoff; $5.22 \\times 10^{44}$ by itself is not the final $K$. These deliberately generous bounds establish a growth regime rather than a practical threshold. Numerical examples in the package are diagnostics. An observed remainder of about $10.7/L$, for a saddle quantity $L$, is explicitly excluded from the proof's constant ledger.

### What the theorem does not say

- It does not prove the Riemann hypothesis, and it does not locate a single zero of the zeta function.
- It does not say every Jensen polynomial is hyperbolic; its sufficient condition allows degrees up to a constant times $n^{2/3}\\log^{1/3}(n+2)$.
- It does not give a practical numerical cutoff. Small examples can be explored by root-finding, but rigorous certification needs error control.
- It has not yet been checked by human mathematicians. Every review it has received was conducted by AI systems, and is reported as such.

## 4Six matches

The heart of the method: build a model polynomial you understand completely, then prove the real one cannot escape it.

Almost nobody tries to find the roots of $J^{d,n}$ directly. The winning strategy, used in some form by all of the recent work, is comparison. Construct a second polynomial whose roots you can control by classical theory, make it agree with the true polynomial to high order, and then prove that the leftover error is too small to disturb the root pattern.

Think of two curves meeting at a point. If they share only a value, they can separate immediately. If they share a value and a slope, they stay close a little longer. Share curvature and the next few derivatives too, and separation is postponed to still higher order. The entire exponent in the theorem, the difference between three-fifths and two-thirds, is the visible trace of engineering one additional order of contact.

The model draws on a classical family. In their usual orthogonal parameter range, *Jacobi polynomials* have real, simple roots in $(-1,1)$. A change of variable moves those roots to a positive interval.[10](#note-10) The proof combines two such scaled factors, controlled by four positive parameters $A,B,C,D$:

$$p\_F(y) \\;=\\; {}\_3F\_2\\!\\left(\\begin{matrix}-d,\\; A,\\; C\\\\\[2pt\] B,\\; D\\end{matrix};\\; \\frac{D}{AC}\\,y\\right) \\;=\\; Q\_{A,B}^{(1)} \\boxconv Q\_{C,D}^{(D)}.$$

The compact symbol $\\,{}\_3F\_2$ names a hypergeometric series; the parameter $-d$ makes it stop after degree $d$. The operation $\\boxconv$ is *finite free multiplicative convolution*, a classical construction also called Schur-Szegő composition, studied in a modern framework by Adam Marcus, Daniel Spielman and Nikhil Srivastava.[11](#note-11) It combines two degree-$d$ polynomials into another degree-$d$ polynomial. Ordinary multiplication would double the degree. For these positive-root inputs, classical results preserve positive real roots. A stronger separation result, recorded by Andrei Martínez-Finkelshtein, Rafael Morales and Daniel Perales, controls ratios between neighboring roots and keeps them from merging.[12](#note-12) A bound due to Szegő, stated as Theorem 1.6 by Marcus, Spielman and Srivastava, limits the largest root by the product of the two input maximum roots. Together these results supply the model's distinct positive roots and the bounds needed to locate them.

Before comparing, rescale and reflect the xi polynomial: write $P(y)=J^{d,n}(-y/S)/\\gamma(n)$ with a suitable $S>0$. This puts it in the model's coordinate system. Two normalizations match the first two coefficient positions. Four further equations tune $A,B,C,D$ by matching logarithmic second differences of coefficient ratios. The counting is elementary: if a sequence starts with two zeros and its next four second differences vanish, its first six entries are zero. The matching uses the underlying coefficient sequences before the common signed binomial factors impose degree $d$. They agree at positions 0 through 5; when $d\\leq5$, these matches already identify the entire transformed polynomial.

After rescaling the four parameters into coordinates $(\\alpha,t,w,\\delta)$, the limiting tuning equations have an exact solution:

$$(\\alpha, t, w, \\delta) \\;=\\; \\left(3,\\; 2,\\; \\tfrac{16}{3},\\; \\tfrac{1}{3}\\right).$$

Exact rational numbers, no decimals anywhere. The Jacobian determinant of the system at this point is exactly $-1/144$, which certifies that the solution is isolated and robust; a quantitative contraction argument then keeps a unique nearby solution on the hook for every large $n$. All of this, the matching adapter included, is verified line by line in the Lean formalization.

ANALYTIC STREAM ALGEBRAIC STREAM Riemann theta / Mellin integral for the coefficients γ(n) sectorial saddle branch N = L(πe^L + 3/4) coefficient asymptotic with holomorphic error (Theorem 7.1) logarithmic derivatives through order six two Jacobi factors with positive simple roots finite free convolution → model polynomial p\_F root and critical-point localization recurrence → derivative radius ρ of order √(nd) SIX COEFFICIENT MATCHES sixth-order residual bound sign transfer → d negative roots Literature hypotheses for the algebraic stream: Jacobi roots · MMP root separation · MSS product-root bound 

**Figure 3.** The architecture. An analytic stream (left) produces a uniform description of the true xi coefficients through six derivatives; an algebraic stream (right) produces a comparison polynomial with perfectly understood roots. Six exact coefficient matches splice the streams together, a sixth-order residual bound prices the remaining error, and a sign-transfer argument converts everything into the stated roots. The classical results at the bottom enter as explicitly typed assumptions rather than hidden axioms.

## 5The arithmetic of an exponent

Where the number two-thirds comes from: a short calculation linking the error bound to the degree.

Let $E(z)$ be the logarithm of the ratio between the true coefficients and the model coefficients along the chosen branch. The six matches say that $E$ vanishes at the six integer points:

$$E(0) \\;=\\; E(1) \\;=\\; E(2) \\;=\\; E(3) \\;=\\; E(4) \\;=\\; E(5) \\;=\\; 0.$$

A classical piece of interpolation theory, the Hermite-Genocchi formula, then controls $E$ everywhere in a neighborhood: if the sixth derivative of $E$ is bounded by $M$ on a convex region containing the six nodes and the point $z$, then

$$|E(z)| \\;\\le\\; \\frac{M}{720}\\,\\prod\_{j=0}^{5} |z - j|.$$

The fraction $1/720$ is exactly $1/6!$, the volume of the six-dimensional simplex in the integral remainder formula. A five-node formula instead carries $1/120$. Using the wrong one would invalidate the displayed calculation and its constant ledger; the verification suite includes a mutation designed to catch that substitution.

Two quantities now compete. The analytic engine described in the next section bounds the sixth derivative at scale $M \\approx C/(n^{5}\\log n)$, and the algebraic engine bounds how far $z$ can stray from the six nodes: within a tube of radius $\\rho$ of order $\\sqrt{nd}$, a scale that comes from the geometry of the model's critical points. Six factors $|z-j|$ each contribute up to about $3\\rho$, so the product is of order $\\rho^{6} \\approx (nd)^{3}$. Multiplying:

$$\\frac{\\rho^{6}}{n^{5}\\log n} \\;\\asymp\\; \\frac{(nd)^{3}}{n^{5}\\log n} \\;=\\; \\frac{d^{3}}{n^{2}\\log n}.$$

And there it is. This bound keeps the comparison error small whenever $d^{3}/(n^{2}\\log n)$ is sufficiently small. Absorbing the constants and small-index restrictions gives the wedge condition $n^{2}\\log(n+2) \\geq Kd^{3}$. This is the calculation that produces the third power of $d$. With only five matches, the next uncontrolled term would be of fifth order and the balance would return to the older scale. The estimate is sufficient: it does not say that hyperbolicity fails outside the wedge.

## 6A saddle in the complex plane

To match six coefficients, the true coefficients must be known six derivatives deep, uniformly. Enter steepest descent.

None of the matching machinery would function without a sharp, uniform description of the numbers $\\gamma(n)$ and of their logarithmic derivatives through order six. The delivery vehicle is the oldest trick in asymptotic analysis: the saddle point method, known to physicists as steepest descent.

Recall the integral representation of $\\gamma(n)$. Its integrand is a tug-of-war: $u^{2n}$ favors large $u$, while the decaying kernel suppresses it. For large $n$, almost the entire integral comes from a narrow window where the two effects balance, the saddle. Locate it and zoom in, and the integrand takes approximately the shape of a Gaussian bell curve. This is the idea behind Laplace's method on real integrals and its complex counterpart, steepest descent.

After rewriting the moment integral in a logarithmic coordinate, the leading saddle is defined by the equation

$$N \\;=\\; L\\left(\\pi\\, e^{L} + \\tfrac{3}{4}\\right),$$

where $N \\approx 2n$ is the large parameter. On the positive real axis there is a unique solution $L\_N$, and a pleasant mountain-pass picture holds: the integral concentrates near $L\_N$, the phase bends quadratically away from it, tails die exponentially.

branch cut of the logarithm Re u Im u 0 1 original integration ray shifted ray saddle LN \= α + ib Gaussian window: nearly all the mass tails: exponentially small connector higher theta modes (m ≥ 2): exponentially smaller at every step 

**Figure 4.** The contour deformation (schematic). The proof splits the positive real integration ray at $u=1$ and shifts the infinite part to a horizontal ray through the complex saddle $L\_N$. Near the saddle, the real part of the phase decreases along this path. The central window gives a Gaussian approximation; the remaining pieces are controlled separately. Splitting the contour keeps the translated part away from the logarithm's branch cut (red), repairing a gap found in an early draft.

A real-axis picture is not enough. The sixth-derivative bound must hold in a sector of the complex plane because the interpolation tube sends $z$, and hence the asymptotic parameter, off the axis. Rouché's theorem, which counts zeros of analytic functions, establishes a unique saddle near a predicted location throughout a narrow sector. The resulting *holomorphic branch* varies analytically with $N$. The proof then shifts the contour to a horizontal ray through that saddle, controlling the real part of the phase and keeping the principal logarithm well defined. It bounds the tails, connectors and higher theta modes, then combines the result with Stirling's formula. The coefficient estimate has a holomorphic relative error of order $\\log|n|/|n|$, uniformly in the sector. Cauchy's integral formula on small discs controls all six derivatives of that error.

What comes out the other end is disarmingly tidy. The leading constants of the second through sixth logarithmic derivatives of the coefficient moment are

$$2,\\quad -2,\\quad 4,\\quad -12,\\quad 48.$$

The sixth derivative of the leading logarithmic moment $G\_0$ is less tidy. After scaling, it becomes a rational function whose numerator has total degree 13 and 82 nonzero terms. A term-by-term bound, combined with a lower bound on the denominator, puts the scaled rational function below $10^{4}$ on the certified parameter region. With the remaining factors, this gives $|G\_0^{(6)}(N)| \\leq 20000/(|N|^{5}\\log|N|)$ for the leading logarithmic moment.[13](#note-13) An expression this large is an inviting hiding place for a sign error, making it a priority for independent calculation.

The factor of eight mentioned earlier is another caution. The source audit identifies an eightfold normalization discrepancy between the coefficient definition and a printed integral formula. A constant factor disappears from logarithmic derivatives, so the derivative calculations alone would not expose it. The project rederives the Mellin identity from the theta kernel to fix the normalization, then tests whether a deliberate change to the factor is detected.

## 7From coefficients to roots

Small coefficient errors do not automatically preserve roots. Alternating signs provide the final test.

Knowing that two polynomials have nearly equal coefficients does not, by itself, mean their roots are alike. A polynomial whose roots are bunched together reacts violently to tiny perturbations. Before any comparison can be trusted, the model's roots must be shown to be not just real but well separated, and the model must be shown to be insensitive to the specific perturbation at hand. Three tools close the gap.

**Localization.** The Jacobi roots can be represented as eigenvalues of symmetric tridiagonal matrices. Gershgorin's theorem traps those eigenvalues in explicit intervals. The maximum-root product inequality then gives an upper bound for the convolution's roots. Applying the inequality again to reciprocal polynomials supplies the lower bound; positive interval endpoints are essential to this step. After the appropriate scaling, every model root lies within $(12+8\\sqrt6)\\sqrt{Bd}$ of $B$, and $12+8\\sqrt6<32$. The critical points lie between the roots, so they obey the same bound. A lemma in the earlier approach could not be reused: it required a parameter ratio below one quarter, while the new branch sends that ratio toward one. The new proof therefore needs a localization argument without that restriction.

**The recurrence.** The model is a terminating hypergeometric polynomial, which means it satisfies an explicit differential equation. That equation converts into a four-term recurrence among the scaled derivatives $T\_k = y^{k} p^{(k)}(y)/p(y)$ of the model:

$$P\_{3,m}\\,T\_{m+3} \\;+\\; P\_{2,m}\\,T\_{m+2} \\;+\\; P\_{1,m}\\,T\_{m+1} \\;+\\; P\_{0,m}\\,T\_{m} \\;=\\; 0.$$

At a critical point the first derivative vanishes, so $T\_1 = 0$ and the recurrence links every $T\_k$ to its neighbors. A maximality argument, applied to the globally largest normalized derivative rather than the first one to misbehave, then forces every $T\_k$ to obey a radius $K\_r\\sqrt{Bd}$ with $K\_r = 4096$. That is the quantity called $\\rho$ in Section 5\. One subtlety deserves note: a parameter that the old architecture treated as small, because it tended to zero, now tends toward one-half. The recurrence had to be run non-perturbatively, with that term carried along at full size, or the whole edifice collapses.

**Sign transfer.** The finish uses a familiar fact: a continuous function that changes sign must cross zero. The degree-$d$ model $p$ has $d-1$ critical points with alternating nonzero values. Newton interpolation of the coefficient correction, together with Cauchy's estimate and the derivative recurrence, proves $|P(y)/p(y)-1|<1/16$ at each critical point. The six matches remove the low-order terms; this final stability step needs only the first five. Thus $P$ and $p$ have the same signs at the critical points. Add the point $y=0$ and a sufficiently large positive endpoint, where their signs also agree. These $d+1$ test points create $d$ disjoint intervals, each containing a root of $P$. Its degree rules out any additional roots. Finally, $y=-SX$ with $S>0$ moves those distinct positive roots to the negative real axis.

y = 0 large y λ₃ λ₂ λ₁ λ₀ crit crit crit + − + − + five test points: two endpoints (squares) and three critical points (open circles) alternating signs force four roots, one in each interval between test points 

**Figure 5.** Sign transfer for degree four. Filled circles mark model roots; open circles and squares mark the five test points. At each critical point, the polynomial ratio $P/p$ differs from one by less than $1/16$, preserving the sign. Matching signs at zero and sufficiently far to the right complete four intervals with opposite endpoint signs. Each interval must contain a root of $P$. The $1/16$ bound concerns polynomial values at critical points, not individual coefficients.

## 8Checking the proof

Different checks reveal different mistakes, including mistakes in the checks themselves.

Long proofs can fail at conventions that differ by a factor of eight, at contours that cross branch cuts, or at an inequality used beyond its valid domain. The project combined several checks because each tests a different part of the argument.

**Lean.** A proof assistant checks formal deductions against precise definitions and assumptions.[14](#note-14) The project's Lean 4 development covers the analytic chain from the theta integral through the saddle estimates and six differentiations, the matching construction, the recurrence, and the final sign transfer. Its global theorem gives exactly $d$ distinct negative roots *conditional on explicitly typed literature inputs*: Jacobi root theory, the MMP separation result and the MSS product-root bound. Those general theorems are not re-proved in the project. The recorded audits find only Lean's standard foundational principles and no unfinished-proof placeholders or custom axioms in the audited production proofs. Separate comparator challenge statements contain intentional placeholders; they do not justify those proofs. Kernel checking establishes the formal implication. It does not, by itself, establish that every imported hypothesis faithfully states an applicable theorem from the literature.

**Mathematica, clean room.** The large symbolic expressions were first derived and checked in SymPy. A separate Mathematica 15.0.1 calculation reconstructed selected objects from definitions without importing those expressions. I entered and ran cells supplied by AI: this was an independent computer-algebra path, not an independent human rederivation. The exact checks recovered the derivative constants $2,-2,4,-12,48$, the parameter solution and Jacobian determinant, and the full 82-term numerator and its coefficientwise bound. They also checked the recurrence at three test degrees. Agreement between two algebra systems gives useful corroboration of those calculations; it does not prove the entire theorem. The detailed results and provenance are in the release ledger.

**Interval arithmetic.** Arb/ACB calculations enclose selected coefficients, contour integrals and root computations in intervals or complex balls that carry explicit error bounds.[15](#note-15) When these bounds establish a sign or isolate a root, they provide a rigorous certificate for that particular claim. They do not replace the estimates required for every pair $(n,d)$ in the theorem.

**Mutation testing.** A verification gate is only meaningful if it can fail. So the suite deliberately corrupts load-bearing statements and requires the gates to reject each one.

### Break it on purpose

A sample of the deliberate corruptions the verification gates must catch, drawn from the technical supplement:

- replace $n!$ by $2^{2n}$, or the kernel weight $e^{u/2}$ by $e^{u}$, or the factor 8 by 4;
- swap the two small saddle quantities $r = 1/L$ and $\\sigma = L/N$;
- substitute the fifth derivative for the sixth, or replace the simplex mass $1/720$ by $1/120$;
- permute the coordinates of the parameter system $(\\alpha, t, w, \\delta)$;
- restore the formerly unguarded call to the imported root inequality, the one missing its positivity hypotheses.

Each listed mutation must make its associated check fail. This demonstrates sensitivity to those particular errors; it does not establish that every possible error would be caught.

**Adversarial AI review.** Other AI systems examined the analytic, algebraic and reproducibility arguments, including Claude Opus 5, Kimi K3 and Qwen3.8-Max. The record distinguishes separated first passes from follow-up reviews that retained earlier context. The Claude reports in the nonseparated review folder are explicitly labeled correlated re-reviews. This was a useful source of criticism, though it was not human peer review. The findings included several concrete errors:

- One draft called a contour segment horizontal when it was parameterized vertically. At a saddle, the quadratic part of the phase changes sign between those two directions, so the supposed Gaussian descent was secretly ascent. The corrected proof shifts the contour through a real increment and only then compares against the full Gaussian.
- A later hostile pass noticed that the shifted contour, as written, crossed the branch cut of the principal logarithm, making the deformation illegal. The repair splits the original integral at $u = 1$ and translates only the safe part, a geometry now printed explicitly in the paper.
- Another review found that the manuscript had defined its logarithmic object as the log of a gamma-normalized coefficient while the calculation differentiated an auxiliary moment, a mismatch that would have double-counted a term. The exact bridge between the two is now displayed before anything is differentiated.
- The algebraic review rejected a first-failure maximality argument that left one recurrence neighbor uncontrolled, forcing the global-maximum version now in the proof.

The most instructive catch came late. A clean-packet review found that the formal interface to the imported Marcus-Spielman-Srivastava inequality permitted negative interval endpoints. Outside the result's valid domain, those choices exposed a contradiction: the combined assumptions could not be satisfied for positive degrees. The headline implication was therefore *vacuously true*, like a promise about a situation that can never occur. The repair added positive-root and degree certificates for both factors and required positive lower endpoints. Lean now derives the endpoint inequalities from $B,D\\geq256d$ before invoking the imported result. A regression check rejects the old unguarded call. The review recorded one blocking defect, two major findings and six minor ones. The follow-up, by the same reviewer instance, checked the repairs and reported no release-blocking finding; its two final advisories were also repaired.

Paper proof the argument, its definitions and estimates Lean 4 kernel audited production proofs; explicit literature hypotheses Two independent computer-algebra systems SymPy derivation + clean-room Mathematica 15.0.1 reconstruction Rigorous interval arithmetic Arb/ACB ball enclosures for coefficients, contours, roots Mutation tests deliberately damaged constants and formulas must make gates fail Adversarial AI review separated first passes and documented follow-up reviews typed literature inputs: Szegő (Jacobi roots) · MMP (log-mesh) · MSS (product root) 

**Figure 6.** The evidence layers. The manuscript gives the argument; Lean checks its formal deductions under stated hypotheses; computer algebra and interval calculations corroborate selected steps; mutation tests check particular alarms; AI review searches for conceptual mistakes. The dashed base lists the literature hypotheses. These methods provide different kinds of evidence, with possible shared blind spots. Independent human mathematical review remains absent from the record.

The released referee packet contains 1,004 manifested files, per-file cryptographic hashes, manuscript sources and replay scripts.[16](#note-16) These make the recorded calculations and formal deductions available for inspection. Hashes establish which files were checked; they do not establish that the mathematics is correct.

## 9The humans and the machines

How the work was actually made, and where responsibility sits.

The mathematics was developed in extended dialogue with OpenAI's GPT-5.6 Sol Pro, called Sol in the project. The model explored strategies, drafted arguments and Lean code, and revised the manuscript under my direction. The work proceeded through dozens of tracked phases. I set the scope and verification requirements, reviewed and edited the work, ran the AI-prepared Mathematica cells, and authorized the repository changes. AI systems are not listed as authors; responsibility remains mine.

Other models were recruited specifically to distrust the result. Codex deployments from the GPT-5 family, Claude Opus 5 from Anthropic, Kimi K3 from Moonshot AI, and Qwen3.8-Max via OpenRouter performed source comparisons, independent recalculations, and the adversarial reviews described above, including the one that found the vacuity defect. Mathematica 15.0.1 supplied the independent computer-algebra channel. None of these systems is listed as an author, and the repository states plainly that the AI reviews, being correlated with AI-generated work, cannot substitute for human judgment.

AI reviewers can sound authoritative and still share blind spots. Formal proof, separate calculations and repeated criticism reduce some risks, but none supplies immunity. The repository records specific errors found and repaired, automated checks that passed at the documented checkpoints, and a boundary around what those checks establish. Independent human expertise is still needed.

## 10What it would mean

If the proof survives scrutiny, three things are gained, and one thing is not.

First, if the proof survives scrutiny, its uniform hyperbolicity region grows from degrees on a three-fifths power scale to a two-thirds power scale, relative to the earlier argument used by the project. Second, the method suggests a broader strategy: combine an additional exact coefficient match with the derivative estimates and model-root control needed to exploit it. Whether that combination helps other sequences is a question for further work. Third, the record identifies specific places where the argument broke, from normalization conventions to contour geometry and missing hypotheses. Those repairs may be useful to anyone building on the method.

The two-thirds exponent comes from six exact matches. The hard part is proving that the remaining error stays small enough to preserve every root.

The Riemann hypothesis remains unresolved. This theorem's sufficient condition allows $d$ below a constant times $n^{2/3}\\log^{1/3}(n+2)$, while the Jensen-Pólya criterion requires every pair $(n,d)$. The wedge theorem does not address arbitrarily large degree at a fixed modest index. Its conservative constant ledger gives no practical cutoff. Nor does another improvement follow automatically: a seventh match would require another adjustable parameter or a new relation among the existing ones.

There is also a quiet methodological wager on display. Formal verification is often sold as a stamp, a binary seal of correctness. Used well, it is something better: an instrument that makes an argument's anatomy visible. A reader can walk from Riemann's theta integral to the saddle branch, from the saddle to six matched coefficients, from the matches through a sixth-order remainder to the final roots, and at every step see which assertions the machine has checked and which classical results are being borrowed on typed, documented credit. For mathematics produced outside the traditional institutions, and produced in dialogue with machines, that visibility may matter as much as the exponent.

Jensen's polynomials have linked finite algebra to the Riemann hypothesis for more than a century. The 2019 work revealed a pattern at every fixed degree; later bounds allowed the degree to grow. This candidate aims to raise one such power-law range from three-fifths to two-thirds. The fraction names an exponent, not a fraction of the Riemann hypothesis solved. If the argument survives scrutiny, its contribution will be a larger region where the roots are controlled, and an inspectable record of how that control was built.

## Notes and sources

1. The May 2019 coverage cluster: R. Letzter, "Mathematicians Edge Closer to Solving a 'Million Dollar' Math Problem," *Live Science*, May 28, 2019 ([livescience.com](https://www.livescience.com/65577-riemann-hypothesis-big-step-math.html?ref=beyond-plausible.com)); "Mathematicians revive abandoned approach to Riemann Hypothesis," *ScienceDaily* (from Emory University, by Carol Clark), May 21, 2019 ([sciencedaily.com](https://www.sciencedaily.com/releases/2019/05/190521162441.htm?ref=beyond-plausible.com)), including Ken Ono's caution that the work falls short of the hypothesis; and R. J. Lipton and K. W. Regan, "A Prime Breakthrough," *Gödel's Lost Letter and P=NP*, June 29, 2019 ([rjlipton.com](https://rjlipton.com/2019/06/29/a-prime-breakthrough/?ref=beyond-plausible.com)). Emory's original release is at [esciencecommons.blogspot.com](https://esciencecommons.blogspot.com/2019/05/mathematicians-revive-abandoned.html?ref=beyond-plausible.com).
2. E. Bombieri, "New progress on the zeta function: From old conjectures to a major breakthrough," *Proceedings of the National Academy of Sciences* **116** (2019), 11085-11086 ([doi:10.1073/pnas.1906804116](https://doi.org/10.1073/pnas.1906804116?ref=beyond-plausible.com)). The quoted sentence is from the commentary's summary of the Griffin-Ono-Rolen-Zagier paper.
3. M. Griffin, K. Ono, L. Rolen and D. Zagier, "Jensen polynomials for the Riemann zeta function and other sequences," *PNAS* **116** (2019), 11103-11110 ([doi:10.1073/pnas.1902572116](https://doi.org/10.1073/pnas.1902572116?ref=beyond-plausible.com); preprint [arXiv:1902.07321](https://arxiv.org/abs/1902.07321?ref=beyond-plausible.com)).
4. G. Pólya, "Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen," *Kgl. Danske Vid. Sel. Math.-Fys. Medd.* **7** (1927), 3-33\. The equivalence is often called the Jensen-Pólya criterion; see note 3 for the modern statement used in this article.
5. B. Riemann, "Über die Anzahl der Primzahlen unter einer gegebenen Grösse," *Monatsberichte der Königlichen Preußischen Akademie der Wissenschaften zu Berlin* (1859), 671-680\. English translations appear in H. M. Edwards, *Riemann's Zeta Function* (Academic Press, 1974) and in P. Borwein et al., eds., *The Riemann Hypothesis* (Springer, 2008).
6. D. J. Platt and T. S. Trudgian, "The Riemann hypothesis is true up to 3·1012," *Bulletin of the London Mathematical Society* **53** (2021), 792-797 ([doi:10.1112/blms.12460](https://doi.org/10.1112/blms.12460?ref=beyond-plausible.com); preprint [arXiv:2004.09765](https://arxiv.org/abs/2004.09765?ref=beyond-plausible.com)). All zeros up to height 3·1012 are certified, by interval arithmetic, to lie on the critical line and to be simple.
7. Griffin-Ono-Rolen-Zagier, note 3, Theorem 1 and its corollaries. Their method proves eventual hyperbolicity at every fixed degree and hyperbolicity for every shift through degree eight. This is not a historical maximum: earlier work by Chasse used verified zeta zeros to cover much larger finite degree ranges; see the introduction of the paper in note 8.
8. M. Griffin, K. Ono, L. Rolen, J. Thorner, Z. Tripp and I. Wagner, "Jensen polynomials for the Riemann xi-function," *Advances in Mathematics* **397** (2022), 108186 ([doi:10.1016/j.aim.2022.108186](https://doi.org/10.1016/j.aim.2022.108186?ref=beyond-plausible.com); [arXiv:1910.01227v3](https://arxiv.org/abs/1910.01227v3?ref=beyond-plausible.com), revised 2020). Theorem 1.1 gives the exponential threshold in degree; its introduction discusses the distinct route using verified zeros.
9. J. Holland, "A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's xi-function," [arXiv:2608.08682](https://arxiv.org/abs/2608.08682?ref=beyond-plausible.com) (2026). The two-thirds project pins and audits its quoted statements to version 1 of this preprint; see the repository's source audit.
10. G. Szegő, *Orthogonal Polynomials*, 4th ed., American Mathematical Society Colloquium Publications **23** (AMS, 1975).
11. A. W. Marcus, D. A. Spielman and N. Srivastava, "Finite free convolutions of polynomials," *Probability Theory and Related Fields* **182** (2022), 807-848 ([doi:10.1007/s00440-021-01105-w](https://doi.org/10.1007/s00440-021-01105-w?ref=beyond-plausible.com)). The paper gives a modern framework for classical additive and multiplicative convolutions. The maximum-root product inequality appears in Theorem 1.6, credited there to Szegő.
12. A. Martínez-Finkelshtein, R. Morales and D. Perales, "Real roots of hypergeometric polynomials via finite free convolution," *International Mathematics Research Notices* **2024** (2024), 11642-11687 ([doi:10.1093/imrn/rnae120](https://doi.org/10.1093/imrn/rnae120?ref=beyond-plausible.com); [arXiv:2309.10970v3](https://arxiv.org/abs/2309.10970v3?ref=beyond-plausible.com)). Proposition 2.17 records logarithmic-mesh preservation, citing Katkova, Shapiro and Vishnyakova (2011). The project audits the exact v3 statements and their specialization to its Jacobi factors.
13. Technical supplement to the main paper: full contour estimates, the derivative tower, and the coefficientwise majorant. The clean-room Mathematica result ledger is hash-frozen at SHA-256 1faea5fcb35b504b5d0aad9391999a99e530373dce36addb496eabb133fb04cc. Both are in the release package (note 16).
14. The Lean prover and mathlib4 library: [leanprover-community.github.io](https://leanprover-community.github.io/?ref=beyond-plausible.com). The recorded axiom audit covers 1,217 declarations and reports only `propext`, `Classical.choice` and `Quot.sound`. The headline is `riemannXiJensen_twoThirds_global_headline_exactly`; it takes explicit literature-input hypotheses. The recorded Phase 33 verification includes an 8,830-job build and fresh kernel replay. These are repository records, not a new full replay performed for this article.
15. F. Johansson, "Arb: efficient arbitrary-precision midpoint-radius interval arithmetic," *IEEE Transactions on Computers* **66** (2017), 1281-1292 ([arXiv:1606.06957](https://arxiv.org/abs/1606.06957?ref=beyond-plausible.com)). ACB extends Arb's ball arithmetic to complex numbers.
16. J. Savva, *A Two-Thirds Hyperbolicity Wedge for Jensen Polynomials of Riemann's Xi-Function*, Version 1.1 ([doi:10.5281/zenodo.22293642](https://doi.org/10.5281/zenodo.22293642?ref=beyond-plausible.com)). Version 1.1 updates attribution; its mathematical content and frozen Version 1.0 reviewer evidence are unchanged. The record includes the Phase 32 findings, Phase 33 repair disposition and follow-up AI re-review. The follow-up used the same reviewer instance as Phase 32\. No human or peer review is claimed.

## References

1. J. L. W. V. Jensen, *Recherches sur la théorie des équations*, Acta Mathematica **36** (1914), 181-195.
2. G. Pólya, *Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen*, Kgl. Danske Vid. Sel. Math.-Fys. Medd. **7** (1927), 3-33.
3. M. Griffin, K. Ono, L. Rolen, D. Zagier, [*Jensen polynomials for the Riemann zeta function and other sequences*](https://doi.org/10.1073/pnas.1902572116?ref=beyond-plausible.com), Proceedings of the National Academy of Sciences **116** (2019), 11103-11110.
4. M. Griffin, K. Ono, L. Rolen, J. Thorner, Z. Tripp, I. Wagner, [*Jensen polynomials for the Riemann xi-function*](https://doi.org/10.1016/j.aim.2022.108186?ref=beyond-plausible.com), Advances in Mathematics **397** (2022), 108186.
5. J. Holland, [*A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's xi-function*](https://arxiv.org/abs/2608.08682?ref=beyond-plausible.com), arXiv:2608.08682 (2026).
6. A. Martínez-Finkelshtein, R. Morales, D. Perales, [*Real roots of hypergeometric polynomials via finite free convolution*](https://doi.org/10.1093/imrn/rnae120?ref=beyond-plausible.com), International Mathematics Research Notices **2024** (2024), 11642-11687; [arXiv:2309.10970](https://arxiv.org/abs/2309.10970?ref=beyond-plausible.com).
7. A. W. Marcus, D. A. Spielman, N. Srivastava, [*Finite free convolutions of polynomials*](https://doi.org/10.1007/s00440-021-01105-w?ref=beyond-plausible.com), Probability Theory and Related Fields **182** (2022), 807-848; [arXiv:1504.00350](https://arxiv.org/abs/1504.00350?ref=beyond-plausible.com).
8. G. Szegő, *Orthogonal Polynomials*, 4th ed., American Mathematical Society Colloquium Publications **23**, AMS, Providence, 1975.
9. E. C. Titchmarsh, *The Theory of the Riemann Zeta-Function*, 2nd ed. (D. R. Heath-Brown, ed.), Oxford University Press, 1986.

**About the work.** *A Two-Thirds Hyperbolicity Wedge for Jensen Polynomials of Riemann's Xi-Function*, John Savva, Version 1.1, 3 September 2026\. Permanent record: [doi:10.5281/zenodo.22293642](https://doi.org/10.5281/zenodo.22293642?ref=beyond-plausible.com). The release includes the main paper, a technical supplement, a reader's synopsis, the Lean 4 formalization, the Mathematica clean-room notebook and result ledger, SymPy and Arb/ACB scripts, mutation-test suites, and the complete AI-review record with disclosures. The work does not prove the Riemann hypothesis. Development relied substantially on AI assistance, primarily OpenAI GPT-5.6 Sol Pro, with adversarial and verification contributions from additional models; Wolfram Mathematica 15.0.1 provided independent computer-algebra corroboration of selected calculations. No human mathematical review or peer review has occurred. Figures in this article are schematic illustrations prepared for this piece, not plots from the paper.