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Must a reflexive Banach space with no AUC renorming contain the countably branching diamonds uniformly? (Baudier-Lancien Problem 39)

Theorem 1.1: the dual $X=J^*$ of a countably branching segment-forest space with finite, unbounded component heights is infinite-dimensional, separable and reflexive, admits no equivalent AUC norm, and its given norm has averaged midpoint modulus $\widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1$ ; every embedding of $D_k$ into $X$ has distortion at least $\sqrt{1+k/12}$ . So Problem 39 has a negative answer, and AMUC does not imply AUC renormability even for reflexive spaces (Baudier had shown the non-reflexive separation in 2026). Companions add midpoint-lens estimates, a sharper $D^2\ge1+k/4$ bound, exact moduli in a Daugavet subspace of $L_1$ , and further tree-potential and $L_1$ examples. Not shown: sharp distortion growth rates.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 27, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Must a reflexive Banach space with no AUC renorming contain the countably branching diamonds uniformly? (Baudier-Lancien Problem 39) · Reflexive diamond question (Problem 39)

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Registry verification: lean checked · announcement · candidate

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VibeMathed record: Must a reflexive Banach space with no AUC renorming contain the countably branching diamonds uniformly? (Baudier-Lancien Problem 39)

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No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the principal manuscript against Problem 39 as the paper quotes it, and the Lean statement lean/ComparatorChallenges/ForestSpace.lean (OAI.ForestSpace.main_theorem), listed in formalization.yaml; not rebuilt here. The Lean statement covers the headline: the full dual of the specified segment-norm space is infinite-dimensional, separable and reflexive, no equivalent norm is AUC, the averaged midpoint modulus is at least $\sqrt{1+t^2/12}-1$ , and every embedding of the depth- $k$ diamond has distortion at least $\sqrt{1+k/12}$ . That is exactly a reflexive non-AUC-renormable space without uniform diamond embeddings. The six companions give further AMUC non-AUC examples, each with its own Comparator statement.

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Must a reflexive Banach space with no AUC renorming contain the countably branching diamonds uniformly? (Baudier-Lancien Problem 39)

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Companion: Midpoint lenses in segment spaces

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