OpenAI has publicly released a 62-page core manuscript, and AI has successively cracked ten "Fields Medal-level" major problems.

Ten "Fields Medal-level" achievements, the full AI proof process has been completely made public!

Today, OpenAI has officially released a 62-page "core manuscript" that details the complete reasoning and deduction process of GPT.

The official has finally given the official confirmation: the "next-generation flagship model" is exactly the one that accomplished this astonishing breakthrough.

Calculated based on the billing standard of the GPT-5.6 Sol API, the total Token cost consumed is only 2000 US dollars.

As soon as the AI proof manuscript was released, the entire internet was ignited again.

Everyone is guessing blindly that "this must be the legendary GPT-6"! Many people are also marveling that 2000 US dollars has unlocked ten historic milestones.

GPT solves 10 century-level problems with only 2000 US dollars

Two days ago, an internal OpenAI employee published a blog, stating directly that the next-generation model Astra has conquered ten major mathematical problems.

A list of these problems was presented, shocking everyone.

These problems cover high-dimensional sphere packing, binary codes and spherical codes, group theory, the Connes rigidity conjecture, arithmetic circuit lower bounds, quantum parallel repetition, the closest vector problem, the Ehrhart volume conjecture, multicolor Ramsey numbers, and extremal graph theory.

At that time, the official blog released a 249-page collection of papers and a complete set of Lean 4 formalized certificates.

But it was not until today that a 62-page manuscript was officially published, titled How the Ideas Came Together.

Access link: https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf

Its "abstract" is only a short paragraph, but it contains an astonishing amount of information —

This note was independently written by the AI model, with no intervention from the OpenAI team at all.

The AI read the original CoT (Chain of Thought) and the final published mathematical papers, then reconstructed four key points for each problem:

Which ideas first pointed out a feasible path;

Which seemingly powerful methods encountered real, insurmountable obstacles;

What kind of perspective shift revealed the underlying structure of the problem;

How the decisive insight finally formed the complete proof.

The most widely discussed problem across the network: high-dimensional sphere packing, which saw no progress for 46 years

Among the ten problems, the one that has attracted the most public attention is high-dimensional sphere packing.

Putting aside the obscure name, the problem itself is actually easy to understand: put a large number of balls of the same size into a box, what is the maximum density you can achieve?

The answer for 3-dimensional space has long been known: it is the stacking method used to pile oranges at fruit stalls.

But for hundreds or thousands of dimensions, humans could only give an "upper bound" — the maximum density cannot exceed a certain value.

The exponent of this upper bound was stuck at 0.5991, and there was no substantial progress for 46 years starting from 1978.

And Astra directly pushed it to 0.6044005442916776954..., with the density upper bound written as 2^-(0.6044...+o(1))d.

The key question is: how did it break through this bottleneck?

First, during the deduction process, Astra determined the inherent limit of the Cohn–Elkies linear programming method.

The initial idea of the AI was to use the Cauchy–Schwarz inequality to estimate the negative mass of a function, but after repeated attempts, it could only reach the radius of √d/(2√π).

After getting stuck, it reached a judgment: the obstacle is not that the constant is not optimized well, but that the global norm cannot track where the negative mass is located at all.

Therefore, Astra decided to change its perspective: adopt Mellin transform, combined with harmonic measure.

But why choose this combination?

Because for radial functions, the Fourier transform is essentially a Hankel transform, whose kernel only depends on the product of the space radius and the frequency radius.

On the Mellin transform side, this problem becomes an extremely simple operation: reflection, plus an explicit phase term.

There is also a very subtle point here: the modulus of that phase factor on the real axis is always equal to 1, so no trace of it can be observed on the real axis.

But when it is analytically extended to the complex plane, it carries the part of high-dimensional information that the norm inequality discarded.

When the deduction reaches the limit, the harmonic measure converges to a logistic density, whose logarithmic potential is exactly equal to the digamma function, and the integration result is exactly log(π/2).

The threshold value of 1/π is derived from this process.

There is a detail that perfectly proves that the model clearly understands what it is doing:

The total mass of the harmonic measure is (1−σ)/2, not 1. The manuscript specifically notes that replacing this kernel with a probability density too early will alter the exponential constant.

After obtaining the lower bound, it is also necessary to construct a function that can actually reach this lower bound.

The Gaussian function has the correct Fourier symmetry, but the position of its saddle point is not correct.

The solution is to multiply it by an even deformation to move the saddle point without breaking the symmetry. After utilizing all available damping, an "ideal profile" is obtained.

The saddle point displacement integral is calculated using the Wallis product, and the result is exactly equal to −(1/2)log(π/2).

This value precisely shifts the Gaussian radius 1/√(2π) to 1/π.

The lower bound predicted radius and the upper bound constructed radius match perfectly at this point.

There is also a very specific detail in the manuscript:

When constructing the auxiliary function, a positive damping patch needs to be added in the far region.

And this patch must be added over an entire interval, not at a single point, because adding it at a single point will hit the resonance frequency.

Non-sofic groups: the core difficulty lies in "many" and "one"

The second high-profile problem is the "non-sofic group" that no one could construct for 27 years.

Thomas Bloom, a mathematician at the University of Manchester, said directly: "Among all construction-type achievements, this is more significant than the previous counterexample to the unit distance conjecture."

The concept of "sofic groups" was proposed by Mikhail Gromov, the Abel Prize-winning Russian mathematician, in 1999.

In Hebrew, "sofic" means "finite".

To put it in plain language, the question of whether a group is sofic is essentially this:

Can this infinite, abstract structure be approximately simulated by a set of "sufficiently large finite permutation operations"?

Astra gave an explicit construction, proving that: there exists an infinite, finitely presented non-sofic group.

The most valuable part of the manuscript is that it clearly describes the real bottleneck that trapped researchers —

Kun's theorem provides a large number of expander graphs, but the Kun–Thom theorem requires only one.

The barrier between "many" and "one" is the core difficulty of this entire chapter.

The AI calls this the "crucial mismatch".

It gives a very straightforward example to explain why you cannot pick any one arbitrarily:

On the union of two identical copies of Q (Q⊔Q), the operation of "swapping the two copies" commutes exactly with the expanding action K — but it does not preserve either of the two components.

This means that those "approximately central elements" can jump back and forth between components, and you cannot track them at all.

Before this, there was an earlier detour: trying to directly convert property (T) into mixing property. This requires a lazy or anti-bipartite averaging set, because a bipartite graph can have spectrum near -1 even if it has a Kazhdan gap at 1.

The corrected averaging method did solve this spectral problem, but it could not fix the problem of "which component to choose".

To solve this problem, Astra first tried a version of the scheme: take the logarithm of the component size, bin the data on a randomly translated grid, and then use bounded median comparison.

But this path failed, and the AI concluded from experience that:

What should be averaged is always a bounded monotonic function of the component scale, never the unbounded scale itself.

Therefore, the final version was rewritten based on this "essential principle".

In each environment expansion component A, take the vertex-weighted median m_A, then define

f(x) = M(x) / (M(x) + m_A)

Where M is the component scale. This f is always between 0 and 1, and 1/2 is exactly the median on each A.

The advantage of this f is that it converts an uncontrolled quantity into a controlled one.