Polynomial removal fails for ordered binary matrices

OpenAI's counterexample (2026) to polynomial removal for ordered binary matrices: a fixed 66 x 66 pattern and tree-indexed matrices that are far from avoiding it but contain very few copies of it.

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

1. Ordered copies and tree matrices
2. Every copy meets the leaf diagonal
3. Distance from freeness

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Binary matrices, finite probability (union bound, independence) and induction.