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Math-2708 · mathematics · Analysis

All-pairs diamond distortion in tree-segment Banach spaces

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let J be the real Banach-space completion of finite real arrays on the disjoint countably branching finite-height forest for the supremum $\ell _2$ norm of sums over pairwise disjoint tree segments, and $X=J*$. Let $B_infinity$ be the closed coordinate span in the dual of the analogous infinite-tree completion. Let $D_k$ be the depth-k countably branching normalized diamond, each edge replaced by countably many two-edge paths of the same total length. For every $k>=0$, every positive scale s, and every metric embedding into either X or $B_infinity$ satisfying s $d_k(u,v)<=norm(f(u)-f(v))<=D$ s $d_k(u,v)$ for all pairs, must $D^2>=1+k/4$?

Short English statement

Let J be the real Banach-space completion of finite real arrays on the disjoint countably branching finite-height forest for the supremum $\ell _2$ norm of sums over pairwise disjoint tree segments, and $X=J*$. Let $B_infinity$ be the closed coordinate span in the dual of the analogous infinite-tree completion. Let $D_k$ be the depth-k countably branching normalized diamond, each edge replaced by countably many two-edge paths of the same total length. For every $k>=0$, every positive scale s, and every metric embedding into either X or $B_infinity$ satisfying s $d_k(u,v)<=norm(f(u)-f(v))<=D$ s $d_k(u,v)$ for all pairs, must $D^2>=1+k/4$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2708-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2708

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 331; 2026-09-27).

All-pairs diamond distortion in tree-segment Banach spaces

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 331; 2026-09-27), the asserted result is that countably branching depth-$k$ diamonds incur $D^2\ge1+k/4$ in the selected segment spaces, with further power-type bounds for path and recursive norms.

Mathematical statement of the claimed result

Manuscript 1.
On a countably branching tree, give finitely supported arrays the disjoint-segment norm \[ \rho(a)=\sup_{S_1,\ldots,S_m}\left(\sum_{j=1}^m|\sum_{s\in S_j}a_s|^2\right)^{1/2}, \] where segments are ancestral intervals and are pairwise disjoint. Complete this norm on the forest $\coprod_{h\ge1}\mathbb N^{\le h}$ to obtain $J$ and set $X=J^*$. On $\mathbb N^{<\omega}$ obtain $J_{\infty}$ in the same way, and let $B_{\infty}$ be the closed span of coordinate evaluations in $J_{\infty}^*$. Form $\mathcal D_k$ by replacing each edge by countably many two-edge paths, halving the edge lengths at each stage. The claimed bound for every bi-Lipschitz embedding into either target is \[ D^2\ge1+k/4\quad(k\ge0), \] where $D$ is the ratio of upper and lower Lipschitz constants with respect to shortest-path distance.

Outline of the argument

For manuscript 1 (family 331; 2026-09-27), graph slopes select fresh coordinate windows along successive diamond levels. Atomic endpoint predictions and primal descent tests charge each layer's energy, so summation forces distortion growth; path costs give analogous estimates in the other norms.

Formal verification

For manuscript 1 (family 331; 2026-09-27), the documented Lean scope is the unconditional estimate $D^2\ge1+k/4$ for depth-$k$ diamonds in the two specified segment spaces; further graph deductions in the selected statements assume twenty-two companion inputs.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are BoundedTreePotentials.lean, DaugavetModuli.lean, DiamondDistortion.lean, ForestSpace.lean, IndependentProducts.lean, MidpointLenses.lean, RecursivePotentials.lean.

Their named declarations include OAI.BoundedTreePotentials.TreeCalculus.main_counterexample, OAI.ComparatorModel.RecursivePotentials.main, OAI.DiamondDistortion.headline_all_pairs, OAI.ExactModuli.exact_example, OAI.ExactModuli.exact_main, OAI.ExactModuli.kadets_werner_existence, OAI.ForestSpace.main_theorem, OAI.IndependentProducts.main_exponential, OAI.IndependentProducts.main_gaussian_12, OAI.IndependentProducts.main_gaussian_24, OAI.IndependentProducts.main_general, OAI.SegmentLenses.midpoint_lens_main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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