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Math-2313 · mathematics · Combinatorics

Strongly Balanced Almost-Uniform Vector Permutations

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Does every finite set of $m$-dimensional vectors of Euclidean norm at most $1$, whose vector sum has norm at most $1$, admit a permutation whose every prefix sum has norm at most $\sqrt{2.7m}$ and whose block sums are uniformly bounded by an absolute constant?

Formulation references: [S1], [S2].

Short English statement

Does every finite set of $m$-dimensional vectors of Euclidean norm at most $1$, whose vector sum has norm at most $1$, admit a permutation whose every prefix sum has norm at most $\sqrt{2.7m}$ and whose block sums are uniformly bounded by an absolute constant?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2313-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2313

The Euclidean theorem supplies an unspecified absolute square-root-dimensional prefix bound after the usual zero-sum augmentation. This problem also prescribes the particular $\sqrt{2.7}$ constant and uniform block controls, which are not asserted by that estimate.

Strongly Balanced Almost-Uniform Vector Permutations

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 097; 2026-09-24), the asserted result is that one signing controls every prescribed-order prefix by $C\sqrt d$, and every zero-sum Euclidean unit-ball family has an ordering with that bound.

Mathematical statement of the claimed result

Manuscript 1.
For all integers $d,N\ge1$ and vectors $v_1,\ldots,v_N\in\mathbb R^d$ of Euclidean norm at most one, one absolute $C$ is claimed to permit signs with \[ \max_{0\le k\le N}\|\sum_{i=1}^k\varepsilon_iv_i\|_2\le C\sqrt d. \] The prescribed vector order is retained and the same signs control every prefix. If additionally $\sum_iv_i=0$, the asserted Steinitz--Bergström consequence instead gives a permutation $\pi$ with \[ \max_{0\le k\le N}\|\sum_{i=1}^kv_{\pi(i)}\|_2\le C\sqrt d. \] The constant is independent of dimension, vector count and the particular family.

Outline of the argument

For manuscript 1 (family 097; 2026-09-24), a filtered signed walk balances contractions against imposed linear constraints. Matrix trace budgets bound accumulated energy, and Brownian slab estimates turn low energy into shifted signs; a sign-to-ordering reduction gives the zero-sum conclusion.

Formal verification

For manuscript 1 (family 097; 2026-09-24), OpenAI lists Lean statements for this assertion: one signing controls every prescribed-order prefix by $C\sqrt d$, and every zero-sum Euclidean unit-ball family has an ordering with that bound.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are SteinitzBergstrom.lean.

Their named declarations include OAI.EuclideanSteinitzBergstrom.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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