OpenAI Solves 722 Math Problems, Quasi-Riemann Hypothesis Proven

Today, the global mathematics and AI communities were stunned by this news—

Without warning, without peer review, and even disregarding long-standing academic etiquette, OpenAI has released a series of new mathematical results generated by its internal state-of-the-art models.

They simply and crudely released the GitHub repository math.

Link: https://github.com/openai/math/

It contains 722 mathematical manuscripts covering 372 previously unsolved families of major mathematical problems.

Link: https://github.com/openai/math/blob/main/overview.pdf

Among them, OpenAI's unreleased AI model has proven the Riemann Hypothesis and simultaneously released a formal verification in Lean. If confirmed, this would be a historic breakthrough in number theory and a milestone in the history of AI development!

According to OpenAI's disclosure, the proof of the vast majority of challenges required only an average of 3 hours of ChatGPT Pro thinking compute power from an internal model they have not yet released!

Ultraman posted on X: We are entering a brand-new era of discovery.

Mathematicians were furious about this.

This list of solutions to the problem is overwhelming.

Behind this "academic massacre," OpenAI and mathematicians have long been at odds.

According to Wired, as early as August this year, OpenAI secretly convened 40 of the world’s top mathematicians for a closed-door meeting, posing a chilling question: “How should we respond if AI surpasses humans entirely in pure mathematics?”

At the time, OpenAI subtly indicated that its internal models had already solved over a hundred unsolved cases.

Northwestern University renowned mathematician Bryna Kra recalled that the atmosphere on site was one of "extreme excitement and extreme fear coexisting."

Scholars urgently advised OpenAI: Do not merely post on Twitter or brief blogs like a social media influencer—always publish rigorous, peer-reviewed papers to give the academic community sufficient time to digest and verify your work.

However, OpenAI asserted its dominance in the most aggressive manner, even preemptively surpassing academic breakthroughs on the Navier-Stokes equations.

New York University visiting professor Nestor Guillen angrily denounced—

To mathematicians, the actions of these AI giants resemble those of a mafia! People are deeply alarmed—not just by AI itself, but because humanity’s highest intellectual power is being ruthlessly monopolized by a tiny handful of tech oligarchs.

Someone revealed that some OpenAI engineers have privately reached a consensus: "Classical mathematics is dead today; AI will inevitably bring an end to the careers of most professional mathematicians."

Su Weijie, a Peking University mathematics alumnus, recipient of one of statistics’ highest honors—the COPSS Presidents’ Award—and a researcher at OpenAI, bluntly stated: “This is like the beginning of a Copernican paradigm shift in humanity’s understanding of intelligence.”

Among all the fortified positions breached, the first to send the entire number theory community into frenzy was the groundbreaking Result 003—it forcibly opened the door to the Riemann Hypothesis, mathematics’ ultimate holy grail.

The Riemann Hypothesis is widely regarded in mathematics as the "jewel in the crown." Hundreds of theorems in modern number theory are built upon the assumption that the Riemann Hypothesis is true. It asserts that all non-trivial zeros of ζ(s) lie on the line where the real part ℜs = 1/2. For over 160 years, humanity has struggled even to rule out the existence of zeros far from this line.

Moreover, a ghost lingers—the Landau-Siegel zero—where certain Dirichlet L-functions may have exceptional zeros extremely close to 1 on the real axis, thwarting hope.

In this publicly released manuscript, the OpenAI model has fully solved the "Quasi-Riemann Hypothesis": proving that all Dirichlet L-functions have absolutely no zeros in the half-plane where the real part ℜs > 7/8!

Moreover, it completely eliminated the Landau-Siegel zero.

OpenAI acknowledged on its GitHub page that the vast majority of issues were generated automatically by the model, except for the work on the zero-free region of the Riemann zeta function, which underwent extremely rigorous manual review and readability refinement by the research team.

Although this has not yet fully reached the final ℜs = 1/2, pushing the zero-free region all at once to a fixed constant bound (7/8 and 11/12) and consistently excluding Siegel zeros represents an unprecedented breakthrough in analytic number theory in the past half-century!

In computer science, if P vs NP is the ultimate crown, then "General NP-hardness under the basic semidefinite threshold" is the undisputed king that determines the limits of human algorithms.

This is also the most disruptive study in this OpenAI results library (Result 102).

Link: https://github.com/openai/math/blob/main/reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf

What is NP-Hard?

In the real world, massive optimization problems—such as chip routing, logistics scheduling, flight path planning, and graph coloring—are classified as NP-Hard problems.

Humans cannot compute the optimal solution in polynomial time and must instead settle for approximate solutions. Semidefinite programming relaxation (Basic-SDP) is the most powerful known approximation tool.

In 2008, computer scientist Prasad Raghavendra published a landmark paper proving a remarkable result: for any fixed finite constraint language (Max-CSP), the approximation ratio achievable by Basic-SDP is the absolute theoretical limit for any polynomial-time algorithm!

Link: https://dl.acm.org/doi/epdf/10.1145/1374376.1374414

However, this great theorem rests on a fatal assumption—it must be based on the validity of the Unique Games Conjecture (UGC).

UGC is a century-old problem proposed by Subhash Khot in 2002.

If the UGC is false, Raghavendra’s theoretical framework would collapse instantly—this has been the Achilles’ heel of theoretical computer science for nearly two decades.

For the past two decades, a long-standing goal among theoretical computer scientists has been: Can one directly prove that the gap problem corresponding to the Basic-SDP threshold is NP-Hard under the classical framework, without relying on the UGC assumption, and solely based on P≠NP?

If this conclusion holds, it means that under the pure P≠NP assumption, any polynomial-time deterministic algorithm attempting to outperform Basic-SDP is mathematically impossible!

How is AI positively tearing down this barrier? Below is the step-by-step reasoning.

First, the AI reviewed Raghavendra's original framework and confirmed that duplicate variables and local probability distributions do not provide loopholes for constructing counterexamples.

AI realized that the core obstacle, if bypassing UGC, is that in classical PCP constructions, tensor representations "leak" projection coordinates, allowing cheaters to easily pass.

To suppress information leakage without compromising completeness, the AI abandoned the smooth function approach and introduced an algebraic core over the finite field of characteristic 2:

Then, the AI designed a nonlinear decoder with shift equivariance.

Extremely insensitive to minor noise, yet consistently captured by high-rank linear features, this resolves the information leakage dilemma.

Then, the AI uses a probability of only

An extremely sparse projection, combined with the innovative "Row Fiber Richness Lemma," rapidly eliminates statistical error while preserving ample decoding coordinates, completely closing off any possibility of fraud on local slices.

Ultimately, the AI divided the entire elaborate proof into two precise stages:

Step 1: Unconditionally construct Unique Games hardness with near-perfect completeness (1−ε) and arbitrarily small soundness (δ);

Step 2: Integrate the Dictatorship Testing framework, using low-influence Gaussian variables to losslessly transfer this gap to any finite-constraint Basic-SDP threshold.

As a result, AI established for the first time the ordinary NP-hardness of the Basic-SDP threshold purely based on the standard P≠NP assumption, without any reliance on UGC, thereby definitively closing the theoretical physical boundary for human-effective approximation algorithms!

In Result 01 manuscript, AI has conquered a major stronghold of the Hodge Conjecture: providing a complete proof of the "rational Hodge conjecture" for abelian varieties with complex multiplication (CM) over the complex numbers, in all dimensions and codimensions!

Official statement from OpenAI:

Most results were automatically generated by standard models, but the proof of the Hodge conjecture for abelian varieties with complex multiplication represents a special breakthrough that deviates from conventional processes.

Moreover, AI extended this result to arbitrary finite products of projective complex K3 surfaces and, in the process, proved the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture in arbitrary characteristic.

Link: https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026/paper.pdf

The AI's problem-solving logic is as follows:

1. Transformation and Projection: The core challenge in proving the Hodge conjecture lies in demonstrating that abstract "Hodge classes" are fundamentally algebraic. Rather than attempting to tackle all manifolds directly, AI focused on highly symmetric CM abelian varieties and K3 surfaces.

2. Kuga–Satake Algebraization: AI leverages the sophisticated Kuga–Satake correspondence to embed the transcendental cohomology of a K3 surface into the second cohomology of an abelian variety. It successfully proves that this correspondence is induced by a rational algebraic cycle.

3. Degeneration and Variational Extension: Subsequently, the AI constructs an algebraic path from a special curve cover to the overall self-power variety, leveraging Lie algebra symmetries and the variational rigidity of Hodge generic points, proving that these Hodge classes are necessarily spanned entirely by algebraic cycles over the field of rational numbers.

This step has opened a massive breach right at the heart of the Hodge Conjecture!

In addition, the manuscript from OpenAI contains numerous groundbreaking advances in number theory, convex geometry, and analytic geometry.

Ordinary two-point correlation of the multiplication function (Result 007)

This is a central problem in number theory, related to the famous Chowla conjecture and Elliott conjecture. The core issue is whether the average value of the product of a bounded multiplicative function under different shifts tends to zero.

AI proved the general two-point Chowla conjecture and achieved logarithmic power savings in the error term at every scale.

Link: https://github.com/openai/math/blob/main/reasoning_traces/ordinary-two-point-correlations.pdf

Symmetric and General Mahler Conjecture (Result 087)

The Mahler conjecture, unsolved for decades in the field of convex geometry.

It asserts that, in n-dimensional real space, the product of the volumes of a convex body and its polar body attains its minimum at the simplex (for general convex bodies) or at the cube/cross-polytope (for symmetric convex bodies).

AI simultaneously solved the symmetric and asymmetric geometric Mahler conjecture in all dimensions and provided a complete classification of equality conditions for Hanner polytopes and simplices.

Link: https://github.com/openai/math/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf

After reading these, the math community felt a profound sense of helplessness and awe.

Previously, we thought that AI proving mathematics was merely performing pattern matching on vast amounts of text.

But the manuscripts released today are filled with human-understandable concepts such as "intuitive transfer," "constructive counterexamples," "Laplace expansion," and "physical intuition (such as heat flow simulation and Hamiltonian systems)."

It not only learned human mathematical frameworks, but also developed its own mathematical intuition.

Returning to the sobering question at the beginning of the article: What should human mathematicians do in the face of a model that can produce top-tier research findings in just three hours on average?

Bryna Kra said: “We must adapt in this space. It has changed the way we operate, but it’s also a moment where we can think more long-term… This is a frightening time, but absolutely an incredibly exciting one.”

When the answers to these more than a hundred unresolved century-old problems lie in GitHub repositories like cold streams of data, the era of classical mathematics may have come to an end.

But humanity’s silicon-based pursuit of truth has only just begun. In this new era, powered by silicon-based intelligence, humanity’s quest for truth has only just set sail toward the stars and oceans.

Tonight, the mathematics community is destined to be sleepless.

Reference materials:

https://github.com/openai/math/tree/main

This article is from the WeChat public account "New Intelligence Yuan" (ID: AI_era), authored by ASI Revelation.