Uniqueness of four-body convex central configurations for all masses

What was already known

Existence was known; uniqueness had been established only for equal, symmetric or restricted mass families.

AI involvement AI-centered

Tomar credits generative AI tools with finding and proving results and writing interval and Lean code. He explains the Hessian majorant, interval bound and covering argument.

The paper attributes discovery and proof of the results to generative AI tools, enough for central AI work but insufficient for model-specific autonomy.

Supporting source (palomar-registry.org) — v2 registry summary, 4 October 2026

Importance Major

Answers the all-positive-masses uniqueness question and establishes uniform nondegeneracy.

Limitations

The Q≥K/4 bound belongs to the later formalization; v1 had the weaker Q≥K/128. The public disclosure does not identify one model’s exact mathematical idea.

Technical verification Formal (Lean 4 with Mathlib; kernel-checked interval certificates and Comparator)

The paper and v2 registry report complete Lean formalization of A and B; v1 had only a weaker Q≥K/128 bound, so this assessment is tied to v2 and the later repository.

Independent checking

formal proof and 7,307 kernel-checked interval certificate theorems

Checker: Lean 4 kernel and Palomar Comparator
Scope: Theorems A and B, including Q≥K/4 for every positive-mass convex configuration
Basis: Reported by the linked checker

Evidence source

Human understanding Full

Tomar explains the Dziobek identity, mass-free 2×2 majorant, interval bound and covering argument in §§1.4, 3, 5 and 6.

Correction and publication history

2026-10-18 · publication