Borsuk's conjecture fails in dimension nine

OpenAI's theorem (2026) that the rank-one projections of R^4, a copy of projective three-space in R^9, cannot be covered by ten sets of smaller diameter; through admissible maps, an odd map on symmetric matrices, degree modulo two and the combinatorics of six-label triangle systems.

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

1. Diameter covers and admissible maps
2. Regular values and degree modulo two
3. Double covers and odd maps
4. An odd map on symmetric matrices
5. Supports at equality
6. Six and ten labels

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Local tools for bundles and transport (Bundles, curvature and holonomy)
• Orientations and fundamental classes (Poincaré duality on manifolds)
• Singular homology, excision and covering spaces (core course Algebraic Topology), Lebesgue measure (core course Measure and Integration), and smooth manifolds and partitions of unity.