In 1917, Japanese mathematician Sōichi Kakeya posed a seemingly simple question: what is the smallest area in the plane within which a needle can rotate through 360 degrees?

Intuitively, the answer might seem to be a disk whose radius equals the length of the needle. Mathematicians soon discovered, however, that sets of arbitrarily small area can still allow a needle to rotate through every direction. Such sets are known as Kakeya sets.

The Kakeya conjecture extends this question to higher dimensions: in n-dimensional space, must every set containing a unit line segment in every direction have a “size”—measured by its Hausdorff and Minkowski dimensions—equal to n? In other words, is it impossible for Kakeya sets in higher-dimensional space to be “too thin”?

Closely connected with harmonic analysis, partial differential equations, and number theory, the conjecture is regarded as one of the central challenges in modern analysis. Major breakthroughs have been made in three dimensions, but the set version of the Kakeya conjecture in four and higher dimensions remains open. It requires researchers to construct rigorous, tightly connected chains of argument within exceptionally complex geometric and analytic structures.

With a research history spanning more than a century, the Kakeya conjecture has long attracted the attention of the mathematical community. Work continues to advance from lower-dimensional cases toward higher-dimensional generalizations. The four-dimensional set version remains an important open problem in harmonic analysis and geometric measure theory, placing demanding requirements on mathematical argument and logical reasoning.

Recently, Unisound’s proprietary U2-Flash model autonomously explored the four-dimensional set version of the Kakeya conjecture. It completed an end-to-end workflow spanning problem analysis, construction of arguments, iterative revision, and generation of a preprint manuscript, while retaining records of its reasoning and iterations.

This exercise does not mean that the four-dimensional Kakeya conjecture has been definitively proved. Instead, it places a large model in the context of a real frontier problem in mathematics to assess whether it can move beyond solving isolated questions, sustain progress on a complex task, and produce research output for further review.

From answering questions to exploring open mathematical problems, Unisound’s U2-Flash is testing the long-horizon reasoning capabilities of large models through concrete tasks.

01 Tackling a Frontier Mathematical Problem to Test Long-Horizon Reasoning

The difficulty of the Kakeya conjecture lies less in any single calculation than in building a rigorous, continuous mathematical argument. For the four-dimensional set version, researchers must identify an entry point within a complex theoretical framework, clarify relationships among different concepts, and advance through a series of interdependent deductions.

This presents a challenge for large models as well.

A model must not only understand the problem’s conditions but also organize multiple stages of reasoning, keep track of intermediate conclusions, and adjust its subsequent approach in light of interim results. An omission in a single step can affect the entire chain of argument.

Open mathematical problems therefore test more than whether a model can provide a plausible answer. They also test whether it can maintain logical coherence over a long-horizon task and continually check and revise its reasoning.

02 Exploring the Full Autonomous Reasoning Workflow from Problem Decomposition to Manuscript Generation

In this task, U2-Flash carried out sustained reasoning on the four-dimensional set version of the Kakeya conjecture, attempting to complete the full workflow from problem analysis to preprint manuscript generation.

Problem decomposition: building a reasoning path. The model examined the problem’s structure and relevant conditions, analyzed connections among mathematical concepts, and used them to organize its subsequent reasoning.

Sustained reasoning: advancing complex arguments. Across multiple interconnected reasoning stages, the model needed to keep track of key conditions and intermediate results, pursuing the overall objective rather than stopping at isolated local answers.

Iterative revision: adjusting the approach based on results. Over multiple rounds of exploration, the model revisited its deductions, attempted to identify potential issues, adjusted its reasoning path, and continued developing the argument.

Manuscript generation: organizing interim findings. After reasoning and iteration, the model compiled the material into a preprint manuscript and retained corresponding process records to support subsequent analysis and review.

By spanning problem decomposition, sustained reasoning, iterative revision, and manuscript output, this exercise extended the assessment of model capabilities from a single response to a complete task workflow. It also provided a concrete case for examining performance on complex tasks.

Completing the reasoning workflow does not establish the validity of a proof. The mathematical arguments in the manuscript still require expert review and appropriate independent verification.

03 From RSI to Long-Horizon Reasoning: Exploring New Paths for Model Improvement

Improving performance on complex tasks depends not only on the quality of a single reasoning attempt but also on whether a model can identify problems through task execution and feedback and continue to improve.

Unisound continues to explore approaches such as Recursive Self-Improvement (RSI), seeking to involve models in generating task data, analyzing execution trajectories, and identifying weaknesses. Through traceable processes and corresponding verification mechanisms, it is exploring new ways to improve model capabilities.

Long-horizon reasoning focuses further on performance in actual tasks: whether a model can sustain goal-directed analysis, manage logical dependencies across multiple stages, and adjust its subsequent path based on interim results.

U2-Flash’s autonomous exploration of the four-dimensional set version of the Kakeya conjecture provides a practical case for examining how a model advances through a complex open problem. RSI concerns mechanisms for improving model capabilities, while this mathematical exercise concerns complex task execution. Together, they reflect Unisound’s exploration of continual model improvement and complex reasoning.

04 Bringing Complex Reasoning into Verifiable Practical Applications

In mathematical research, generating a manuscript is only the first step. The essential questions remain whether the reasoning is rigorous and whether the conclusions withstand expert review. The value of long-horizon reasoning must likewise be assessed through real tasks and rigorous verification.

From frontier mathematical exploration to software development, professional analysis, and multi-step task execution, large models face growing expectations to move beyond single responses and advance complete tasks.

Unisound will continue research and development in foundation models, post-training, and agent technologies, exploring the limits of model capabilities in complex reasoning and specialized tasks. Through ongoing iteration and verification mechanisms, it aims to develop these capabilities into more reliable practical applications.

From answering questions to sustained reasoning, and from generating results to subjecting them to verification, Unisound will continue to explore how large models can contribute to solving complex problems.