Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 331-05. Midpoint convexity from bounded tree potentials and path costs
Manuscript map · lines 8815–8824
OpenAI@2026-10-07#rem:oai-p-331-05
Authors: OpenAI. Family 331.
Title: Midpoint convexity from bounded tree potentials and path costs.
Abstract: Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound ${}widehat{}delta(t){}ge{}sqrt{1+t{}2/4}-1$ for $0{}lt t{}lt 1$, 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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