OpenAI · 331-05 · 2026-09-27 · Functional analysis
Midpoint convexity from bounded tree potentials and path costs
preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026
Family 331. Asymptotic midpoint uniform convexity without asymptotically uniformly convex renorming.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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- countably branching diamonds require distortion at least , so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.
Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound for , while none admits an asymptotically uniformly convex (AUC) renorming. On finite-height trees, the globally constrained norm equals the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive and asymptotically midpoint uniformly convex, and admit no equivalent AUC norm.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 331: Reflexive midpoint convexity and diamond distortion. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.