{"id":"openai-math/Diamond-distortion-from-midpoint-and-tree-energies","pdf":"https://github.com/openai/math/blob/main/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf","url":"https://github.com/openai/math/tree/main/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026","tags":["metric-geometry","embeddings","diamond-graphs"],"score":0,"title":"Distortion of countably branching diamonds from midpoint and tree energies","domain":"Functional analysis","family":"331","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.FA","summary":"We derive quantitative distortion bounds for countably branching diamond graphs from midpoint estimates and direct tree energies. In the dual of a finite-height segment forest, every distortion- D embedding of the depth- k diamond satisfies $`D^2\\ge1+k/4`$. The same bound holds in the infinite-height coordinate predual. We also obtain power-type distortion bounds for path-cost and recursive tree norms.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-27","reactions":{},"formalized":true,"family_title":"Reflexive midpoint convexity and diamond distortion"}