The Lane-Emden conjecture

OpenAI's proof (2026) of the Lane-Emden conjecture and its weighted Henon-Lane-Emden form: below the critical hyperbola the elliptic system has no positive solution on R^n, without any condition at infinity, by a localized virial identity for Newton potentials and a comparison along parallel lines.

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Spherical means and Newton potentials
2. Positive solutions and their potentials
3. A localized virial identity
4. Intervals and source layers
5. Pressure and the energy bound

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Lebesgue integration, Fubini and Tonelli, the Laplacian, and test functions.