Improbability of non-collision singularities for arbitrary particle number

Determine whether the set of initial conditions leading to non-collision singularities in the n-body problem is improbable, meaning that it is contained in a subset of phase space having Lebesgue measure zero, for arbitrary numbers of particles.

Background

A non-collision singularity is an initial condition whose solution ceases to exist in finite forward time while the system does not converge to a collision; equivalently, the moment of inertia diverges. The paper studies whether such singular behavior is exceptional in the measure-theoretic sense of being contained in a Lebesgue-null subset of phase space.

The general improbability question is known for four particles, but remains unresolved for arbitrary particle numbers. The paper proves a substantial partial result: singularities whose final cluster decomposition contains arbitrarily many colliding subsystems and non-collision subsystems consisting of exactly four particles are improbable, provided the asymptotic directions of the unbounded four-particle subsystems differ. Thus, the fully general problem remains open beyond the configurations covered by that theorem.

References

The question of whether the set of initial conditions leading to non-collision singularities in the $n$-body problem is improbable is open and is the first on Barry Simon's list of fifteen problems in mathematical physics from the year 1984.

— A general improbability result for non-collision singularities as subsystems with at most four particles  (2609.00876 - Quaschner, 1 Sep 2026) in Abstract; Section 1, Introduction

Another application of this work could be for proving the improbability of non-collision singularities in the $n$-body problem. This remains an open question from the first problem in Simon's list of open problems in mathematical physics .

— Blow-Up, Asymptotics, and Improbability of Collision Singularities in the n-Body Problem  (2609.00793 - Duignan et al., 1 Sep 2026) in Section 7, Conclusion

For more than four bodies, the possible dynamics for non-collision orbits are still unknown.

— Blow-Up, Asymptotics, and Improbability of Collision Singularities in the n-Body Problem  (2609.00793 - Duignan et al., 1 Sep 2026) in Section 7, Conclusion