Chinese mathematicians have resolved the loose end of a “doomsday hypothesis” that had long puzzled the mathematical community.

The feat was made possible through major computational methods which could be applied to further problems in the field.

As a result of the breakthrough, the mathematical world finally has proof that manifolds of Kervaire invariant one do exist in dimension 126, ending a decades-long mystery.

The Kervaire invariant is a function that measures whether a smooth framed manifold, or a topological space or shape that can have curvature but locally appears flat, can be converted into a sphere through “surgery”, a concept introduced by American mathematician John Milnor in 1950.

If it can be converted into a sphere, the invariant evaluates to zero. The Kervaire invariant problem seeks to discover dimensions in which the answer is non-zero, or one, meaning they can host strange shapes that do not convert into a sphere.

The paper, which has not undergone peer review, was written by Wang Guozhen and Lin Weinan from the Fudan University Shanghai Centre for Mathematical Sciences and Xu Zhouli from the University of California Los Angeles (UCLA).

“We establish the existence of smooth framed manifolds with Kervaire invariant one in dimension 126, thereby resolving the final case of the Kervaire invariant problem,” the team said in a preprint paper published on the arXiv repository in December.

“We conclude that smooth framed manifolds with Kervaire invariant one exist in and only in dimensions 2, 6, 14, 30, 62, and 126.”

The problem has stumped mathematicians for decades. In 1963, work published by Michel Kervaire and Milnor led to proof that the manifolds of the Kervaire invariant one did exist in dimensions 6 and 14.

By 1984, it was proved that this was also the case in dimensions 30 and 62, following a pattern of numbers in the function 2 less than 2 to the power of n.

Mathematicians believed this should be the case for the next numbers that followed in this pattern, such as 126 and 254, but progress stopped at dimension 62 for decades.

The assumption that manifolds of Kervaire invariant one must exist in higher dimensions was used to build mathematical propositions about exotic shapes, but the fact that this assumption could be false became known as the doomsday hypothesis.

In 2009, American mathematician Michael Hopkins from Harvard University and his collaborators showed that manifolds of Kervaire invariant one “exist only in dimensions at most 126”, and do not exist in dimensions 254 and higher. This meant that the doomsday hypothesis had come true.

While their proof had solved a long-standing problem in algebraic topology, the mystery of whether it existed in dimension 126 remained over the next 15 years.

Now the mathematical world finally has proof that manifolds of Kervaire invariant one do exist in dimension 126, tying up the loose ends of a decades-long mystery.

The “jaw-dropping” proof was made possible through both theoretical insights and computer calculation methods developed by the researchers, Hopkins told Quanta Magazine, which is funded by Simons Foundation, a mathematics and basic sciences organisation.

Before their proof was released, Hopkins said that mathematicians had thought such a “heroically computational” achievement was still far out of reach.

All three authors of the study are Peking University alumni with a focus on algebraic topology. Both Xu and Wang completed their bachelor’s and master’s degrees at the university in 2011, and Lin completed his bachelor’s in 2015.

Xu went on to receive his doctorate in mathematics from the University of Chicago in 2017, after which he was an instructor at the Massachusetts Institute of Technology (MIT) until 2020.

He was a professor at the University of California San Diego from 2020 to 2024, before joining UCLA as a professor last November, according to his website.

When Xu arrived at the University of Chicago, his adviser proposed the problem of dimension 126 and introduced him to Mark Mahowald, who was a professor at Northwestern University at the time.

According to Quanta Magazine, Mahowald rejected his proposal and suggested Xu study related problems in lower dimensions instead.

Wang received his doctorate from MIT in 2015, before becoming a postdoctoral fellow at Copenhagen University until 2016.

He joined Fudan University as a postdoctoral fellow in 2016 and became a professor in 2020, according to his website.

Xu and Wang, who studied together, are long-time collaborators who have published several papers together.

Lin, who is currently a tenure-track young investigator at Fudan University, received his doctorate from the University of Chicago in 2021, after which he spent some time as an instructor at Peking University.

Solving the issue of dimension 126 involved studying the stable homotopy groups of spheres, which describe how points on high-dimensional spheres can be deformed into lower ones.

The Adams spectral sequence is a tool or atlas of dots used to help mathematicians understand this complicated world.

It has long been known that if one specific dot in the 126th column of the atlas can survive until the final page of the atlas, then there are manifolds in this dimension that do not convert into a sphere.

The issue was that there are 105 hypothetical ways for it to disappear before reaching it, which had to be eliminated, according to Quanta Magazine.

Xu and Wang developed new computational techniques and passed them to Lin, who wrote a programme to rule out 101 of the possibilities. After a year of work, the authors were able to rule out the last four.