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Distortion of countably branching diamonds from midpoint and tree energies
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Abstract
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.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 331: Reflexive midpoint convexity and diamond distortion (lean/docs/331.md, release of 2026-10-06). Describes the release's own Lean formalization; for this paper it covers the unconditional all-pairs bound $1+k/4\le D^2$ for embeddings of the depth-$k$ diamond into $X=J^*$ and into the closed coordinate span $B_\infty$, and notes that other graph deductions assume 22 companion inputs. [family-level]
Possibly relevant literature (found by the agent, not checked)
- F. Baudier, N. J. Kalton, G. Lancien, A new metric invariant for Banach spaces (arXiv:0912.5113; Studia Math. 199 (2010) 73–94). Gives the tree analogue (countably branching hyperbolic tree, Szlenk index) of the diamond embedding obstructions this paper quantifies in $X$ and $B_\infty$.
Digests and expositions
- Gradient Damage, Every OpenAI Math Result, Part 4 of 4: L = BPL, Integer Multiplication and 82 More (video compilation, 2026-10-14). Chapter 74 at 25:23 provides a brief visual introduction to the family’s reflexive-space claim. [family-level]
Elsewhere
- Emergent Mind: family 331
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026,
author = {{OpenAI}},
title = {{Distortion of countably branching diamonds from midpoint and tree energies}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf}{OAI:Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/331.md. The Lean statements have not been checked by people against the paper's claims.
Family 331: Reflexive midpoint convexity and diamond distortion
Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-k countably branching diamonds require distortion at least $\sqrt{1+k/12}$, so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.
Other papers in this family:
- Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space · September 27, 2026
- Midpoint lenses in segment spaces · September 27, 2026
- Exact asymptotic moduli in a Daugavet subspace of L1 · September 27, 2026
- Midpoint convexity from bounded tree potentials and path costs · September 27, 2026
- Independent products in real L1: asymptotic midpoint convexity without AUC renormings · September 27, 2026
- Midpoint convexity from two recursive potentials · September 27, 2026