Simó–Yoccoz conjecture
For every choice of positive masses and every prescribed cyclic ordering of four bodies, there exists exactly one strictly convex planar four-body central configuration with that ordering, up to similarity. Equivalently, for each such mass vector and cyclic ordering, the set of strictly convex planar central configurations modulo similarities consists of exactly one element.
References
Primary source
Additional references
- Uniqueness of four-body convex central configurations for all masses — arXiv — Tejasvi Singh Tomar
Progress summary
A September 2026 preprint claims to settle the conjecture for four-body configurations, but broad mathematical review is still pending.
The Simó–Yoccoz conjecture asserts that four positive masses with any prescribed cyclic ordering have exactly one strictly convex planar central configuration up to similarity. The conjecture is presented as Problem 10 of Albouy, Cabral, and Santos.
Known results
- Existence for each prescribed cyclic ordering is classical, while general uniqueness for arbitrary positive masses was reported as unavailable on September 1, 2026.
- Partial results cover symmetric and geometrically restricted families, including kite, isosceles-trapezoidal, and cocircular configurations, plus rigorous interval verification on explicit mass regions.
September 28, 2026 claimed resolution
A report dated September 28, 2026, citing Tejasvi Singh Tomar's preprint, claims that each cyclic ordering of four positive masses has exactly one strictly convex planar central configuration up to similarity, resolving the conjecture for all positive masses. The claim is unverified: formal verification covers the encoded theorem and computation, while broader peer review remains pending.
Current status (as of September 2026): A preprint claims the conjecture is solved for all positive masses, but the claimed resolution remains unverified.
Solutions 0
No solutions have been posted yet.