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Math-2662 · mathematics · Geometry

Exact ball-packing criterion in symplectic dimensions at least six

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For $n\geq 3$ and positive capacities $R,R_{1},\dots ,R_k$, do neighborhood-defined symplectic embeddings of the disjoint closed standard 2n-balls of capacities $R_i$ into the interior of the standard ball of capacity R exist exactly when $\Sigma _i$ $R_i^{n}<R^{n}$ and $R_i+R_j<R$ for every distinct pair i,j? Capacity is $\pi $ times squared Euclidean radius.

Short English statement

For $n\geq 3$ and positive capacities $R,R_{1},\dots ,R_k$, do neighborhood-defined symplectic embeddings of the disjoint closed standard 2n-balls of capacities $R_i$ into the interior of the standard ball of capacity R exist exactly when $\Sigma _i$ $R_i^{n}<R^{n}$ and $R_i+R_j<R$ for every distinct pair i,j? Capacity is $\pi $ times squared Euclidean radius.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2662-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2662

Higher-dimensional ball packing: For symplectic dimension $2n$ with $n\ge3$, the announced criterion for packing closed balls inside a larger open ball is the strict volume inequality together with each strict pairwise capacity inequality. Embeddings must be defined around the closed source balls. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Exact ball-packing criterion in symplectic dimensions at least six

OpenAI's claim

Higher-dimensional ball packing: For symplectic dimension $2n$ with $n\ge3$, the announced criterion for packing closed balls inside a larger open ball is the strict volume inequality together with each strict pairwise capacity inequality. Embeddings must be defined around the closed source balls.

Mathematical statement of the claimed result

Mathematical claim 1.
For $n\ge3$, define the closed capacity-$R$ ball $B^{2n}(R)=\{z\in\mathbb C^n:\pi\sum_j|z_j|^2\le R\}$ with the standard symplectic form. For positive $R,R_1,\ldots,R_k$, the claim is \[\bigsqcup_{i=1}^k B^{2n}(R_i)\hookrightarrow\operatorname{Int}B^{2n}(R)\quad\Longleftrightarrow\quad\sum_iR_i^n<R^n\ \text{ and }\ R_i+R_j<R\ (i<j).\] The embedding is symplectic on a neighborhood of each closed ball; compact images must be pairwise disjoint. The parameters are capacities, not Euclidean radii.

Outline of the argument

Higher-dimensional ball packing: Explicit geometric models place nearly all volume over positive-genus surfaces and provide concave area profiles. A one-handle comparison and commutator-based local deformation realize the required surface packings. The models are treated in the relevant geometric cases, with limiting area profiles controlling the passage to the closed-ball packing conditions.

Formal verification

Family 343 describes Lean coverage for the closed-ball packing equivalence and its necessary capacity inequalities.

Higher-dimensional ball packing: The manuscript links BallPacking.lean, BallPackingNecessity.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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