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Math-2636 · mathematics · Geometry

One signing for all Euclidean prefixes

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does an absolute C exist such that every finite prescribed-order sequence $v_{1},\dots ,v_N$ in the Euclidean unit ball of $R^{d}$ admits one signing $\varepsilon _i\in {-1,1}$ for which $max_{0\leq k\leq N}\Vert \Sigma _{i\leq k}\varepsilon _i$ $v_i\Vert _{2}\leq C\surd d$? No zero-total condition is assumed, and signs may depend on the whole sequence.

Short English statement

Does an absolute C exist such that every finite prescribed-order sequence $v_{1},\dots ,v_N$ in the Euclidean unit ball of $R^{d}$ admits one signing $\varepsilon _i\in {-1,1}$ for which $max_{0\leq k\leq N}\Vert \Sigma _{i\leq k}\varepsilon _i$ $v_i\Vert _{2}\leq C\surd d$? No zero-total condition is assumed, and signs may depend on the whole sequence.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2636-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2636

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 097; 2026-09-24).

One signing for all Euclidean prefixes

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.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

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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