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Existence of a third self-similar solution of the Yang--Mills equation in high odd dimensions

This paper proves the existence of a third smooth self-similar solution to the Yang--Mills equation in every odd spatial dimension $d \geq 11$ by verifying a conjecture from Bizoń, Glogić, and Wasserman through a combination of endpoint sign calculations and algebraic arguments.

Original authors: Piotr Bizoń, Robert Kluszczyński

Published 2026-10-28

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CC BY 4.0

Original authors: Piotr Bizoń, Robert Kluszczyński

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of theoretical physics, where scientists try to understand how the universe holds itself together, there is a fundamental force known as the strong nuclear force. This force acts like a powerful glue, binding the tiny particles inside atoms so tightly that they cannot be pulled apart. To describe how this force behaves, physicists use a set of rules called the Yang–Mills equations. These equations are complex, but they are essential for predicting how energy and matter interact at the smallest scales. One of the most intriguing questions in this field concerns what happens when a system becomes unstable and collapses. Imagine a star running out of fuel and imploding; in the mathematical world of these equations, similar collapses can happen in a fraction of a second. Scientists are particularly interested in "self-similar" solutions to these equations. These are special patterns of collapse that look the same no matter how much you zoom in or out, much like a fractal image that repeats its shape at every level of magnification. Understanding these patterns helps researchers predict whether a smooth, calm starting point in the universe can suddenly turn into a chaotic singularity, a point where the laws of physics as we know them break down.

For years, mathematicians have been trying to count how many of these special collapse patterns exist in high-dimensional spaces. While our everyday world has three dimensions of space, the equations used to describe the strong force can be studied in spaces with many more dimensions to reveal deeper truths about their structure. In recent work, researchers narrowed down the problem to a specific algebraic puzzle: finding the roots of a particular polynomial, which is essentially a mathematical expression involving powers of a number. They had already identified two distinct solutions, or "profiles," that described how the collapse could happen smoothly. However, they suspected there was a third, hidden solution lurking in the math, one that had not yet been proven to exist. This missing piece was crucial because if it existed, it would mean there are at least three different ways a smooth system could evolve into a singularity, rather than just the two they had already found.

In a new study, physicists Piotr Bizoń and Robert Kluszczyński have finally confirmed the existence of this third solution. They focused on spaces with an odd number of dimensions, specifically starting from eleven dimensions and going higher. Their work proves that in every one of these high-dimensional spaces, there is indeed a third smooth profile that satisfies the equations. To do this, they did not rely on computer simulations alone, which can sometimes miss subtle details, but instead constructed a rigorous mathematical proof. They analyzed the behavior of the polynomial at its boundaries, showing that it changes sign in a way that guarantees a root must exist between them. They then carefully demonstrated that this new root is not just a repeat of the two solutions they already knew, but a genuinely distinct third path. This discovery settles a long-standing conjecture that had been verified only for a limited range of dimensions through calculation, extending the certainty to all relevant high dimensions.

The researchers' approach involved a clever combination of algebra and logic. First, they showed that the mathematical expression they were studying behaves in a specific way at the start and end of its range, forcing the existence of a solution in between. Then, they had to ensure this solution was unique and not just a duplicate of the known ones. For most dimensions, they used a property of numbers to show the new solution could not be the same as the old ones. For a few rare, exceptional dimensions where the numbers behave differently, they proved that the known solutions were "simple," meaning they could not be the source of the new root. This two-step process allowed them to cover every possible case. The result is a solid confirmation that the universe of these equations is richer than previously thought, containing at least three distinct ways for a smooth state to collapse into a singularity.

While the existence of this third solution is now a proven fact, the story is not entirely finished. The researchers note that it remains an open question whether there are exactly three solutions or if even more might exist. They have checked many dimensions using advanced computational tools and found only three, but a general proof for all dimensions is still needed. Furthermore, their work currently applies only to spaces with an odd number of dimensions. Numerical hints suggest that this third solution likely exists in even dimensions as well, but the mathematical tools required to prove it are different and more difficult to apply. The team plans to tackle this challenge in future work. Beyond just counting solutions, understanding the stability of these three profiles is the next big step. It appears that one of the solutions acts as a stable attractor, guiding most collapses, while the others might play more subtle roles, perhaps acting as critical thresholds that separate different types of cosmic behavior. By confirming the existence of this third path, Bizoń and Kluszczyński have provided a clearer map of the mathematical terrain, guiding future explorers toward a deeper understanding of how the fundamental forces of nature behave under extreme pressure.

Technical Summary: Existence of a Third Self-Similar Solution of the Yang–Mills Equation in High Odd Dimensions

Problem Statement
This paper addresses the classification of smooth, self-similar solutions to the Yang–Mills equation in $(d+1)$ -dimensional Minkowski spacetime, specifically focusing on high odd spatial dimensions $d \ge 11$ . Under a spherically symmetric magnetic ansatz with gauge group $SO(d)$ , the field equations reduce to a nonlinear ordinary differential equation (ODE) for the profile function $u(y)$ , where $y = r/(T-t)$ .

The central problem is determining the number of non-constant smooth profiles defined on the interval $0 \le y \le 1$ (the causal past of a blow-up point). Previous work by Bizoń, Glogić, and Wasserman (BGW) [1] established that for odd $d \ge 11$ , the existence of such profiles corresponds to finding roots of a specific "resonance polynomial" $P_m(c^2)$ in the interval $(0, 1)$ , where $m = (d-5)/2$ . It was known that two explicit roots, corresponding to explicit analytical profiles $u_\pm(y)$ , always exist within this interval. BGW conjectured the existence of exactly one additional root, implying a third distinct smooth self-similar solution, and verified this computationally for dimensions up to $d=35$ .

Methodology
The authors prove the existence of this third root by combining analytical endpoint sign calculations with algebraic arguments regarding the structure of the polynomial $P_m$ . The methodology proceeds in three main stages:

Endpoint Sign Analysis: The authors analyze the resonance polynomial $P_m(z)$ at the boundaries of the interval $z \in [0, 1]$ . By differentiating the recurrence relations governing the Taylor expansion of the solution near the light cone ( $y=1$ ), they derive explicit formulas for the derivatives of the compatibility condition. They demonstrate that $P_m(0) < 0$ and $P_m(1) > 0$ for all $m \ge 3$ .
Factorization and Intermediate Value Theorem: Since the two explicit solutions correspond to known roots $z_\pm = u_\pm(1)^2$ , the polynomial is factored as $P_m(z) = (z-z_-)(z-z_+)S_m(z)$ . Because the quadratic factor $(z-z_-)(z-z_+)$ is positive at both endpoints, the sign change of $P_m$ implies that the remaining factor $S_m(z)$ must also change sign. By the Intermediate Value Theorem, $S_m(z)$ possesses at least one root in $(0, 1)$ .
Distinction of the Third Root: To complete the proof, the authors must show that this new root is distinct from the explicit roots $z_\pm$ $z_{\pm}$ . They divide the proof into two cases based on the arithmetic properties of the dimension $d$ $d$ :
- Non-exceptional Dimensions: When $3(d-2)(d-4)$ is not a perfect square, the quadratic factor $Q_m(z) = (z-z_-)(z-z_+)$ is irreducible over the rationals. An irreducibility argument shows that the remaining factor $S_m(z)$ cannot share roots with $Q_m(z)$ , ensuring the third root is distinct. This covers dimensions up to $d=27$ ( $m \le 11$ ).
- Exceptional Dimensions: When $3(d-2)(d-4)$ is a perfect square (e.g., $d=29, 365, \dots$ ), irreducibility does not apply. For these cases ( $m \ge 12$ ), the authors prove that the explicit roots $z_\pm$ are simple roots of $P_m$ (i.e., $P'_m(z_\pm) \neq 0$ ). This is achieved by analyzing the linearized resonance obstruction around the explicit profiles. They establish coefficient inequalities via induction on a three-term recurrence relation derived from the linearized ODE, showing that the obstruction terms are non-zero. Consequently, the explicit roots cannot be roots of the remaining factor $S_m(z)$ .

Key Contributions and Results

Proof of Existence: The paper establishes Proposition 1.1: For every integer $m \ge 3$ (corresponding to odd dimensions $d \ge 11$ ), the resonance polynomial $P_m$ possesses a root in $(0, 1)$ distinct from the two explicit roots.
Corollary on Profiles: Consequently, Corollary 1.2 confirms that in every odd dimension $d \ge 11$ , there exist at least three non-constant smooth self-similar Yang–Mills profiles (modulo reflection).
Analytical Techniques: The work provides a rigorous proof of the simplicity of the explicit roots for $m \ge 12$ by linking the derivative of the resonance polynomial to the non-vanishing of linearized resonance obstructions. This involves a detailed inductive analysis of coefficient signs in the linearized expansion.

Significance and Open Problems
The paper resolves the existence portion of the BGW conjecture, confirming that the landscape of self-similar blow-up solutions in high odd dimensions is richer than previously proven, containing at least a third distinct profile.

However, the authors explicitly state that uniqueness (i.e., proving there are exactly three roots, not more) remains an open problem. While they have verified uniqueness for $11 \le d \le 101$ using AI-assisted computations, a general proof for all $d \ge 11$ is not provided.

The paper also outlines three open problems for future research:

Uniqueness: Proving that $P_m$ has exactly three distinct roots for all $m \ge 3$ .
Even Dimensions: Extending the existence result to even dimensions $d \ge 11$ . The authors note that the algebraic reduction available in odd dimensions does not apply to even dimensions due to the presence of non-integer power terms in the light cone expansion, making it a more complex connection problem.
Stability and Dynamics: Rigorously establishing the spectral properties of the new profile (denoted $u_*$ ) and the explicit profile $u_-$ . Numerical evidence suggests $u_*$ may act as a critical solution separating blow-up from dispersion, but rigorous stability analysis is required.

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