The dyadic triangular Hilbert form
OpenAI's theorem (2026) that the dyadic triangular Hilbert form is bounded on L^3 x L^3 x L^3 uniformly in the scales and with no condition on the inputs: the absolute values of the local forms over admissible triples of dyadic intervals, summed over any finite set of scales, are at most 40 times the product of the L^3 norms. Step functions as matrices; the energy tr(P^(3/2)), its Hessian and its comparison under the matrix order; a dimension-free trace estimate for a triangle of matrices; energy increase under splitting into four children, and telescoping.
Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Dyadic triangles and the main estimate
- A matrix energy and a trace inequality
- Energy increase and telescoping
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Linear algebra over the real numbers: symmetric matrices, the spectral theorem, positive semidefinite order and the trace; the inverse function theorem in finite dimensions.
- The Lebesgue integral and Hölder's inequality, as in the core course Measure and Integration.