The duality conjecture for metric entropy
OpenAI's counterexample (2026) to the duality conjecture for metric entropy, posed by Pietsch in 1972: covering numbers of convex bodies are not preserved by polarity up to universal constants, already when one body is a cube. The course turns a real matrix into a pair of bodies, proves the duality theorem for convexified packing, approximates all column combinations of a matrix built from partitions by a short list, and constructs the partitions from random directions for symmetric multilinear forms over a finite field.
Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
Theorems and assumptions · Reader and editable sources · Course record