Result 097, Convex and metric geometry

The Euclidean Steinitz–Bergström bound

Proves that any finite sequence in the Euclidean unit ball of ℝ^d admits signs keeping every partial sum within $C\sqrt d$ , independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order $S_2(d)=\Theta(\sqrt d)$ .

Lean formalization Proof

The bigger picture

Why it matters

Adding many small vectors can produce a large excursion, even when they eventually cancel. This manuscript claims that carefully chosen signs or, for zero-sum families, a new order keep every intermediate sum within a dimension-dependent limit.

What changes?

For any finite sequence of vectors of Euclidean length at most one in d dimensions, in its prescribed order, the manuscript reports a choice of plus or minus signs keeping every prefix sum within C times the square root of d of the origin. C is an absolute constant, independent of dimension and sequence length. It also claims that any finite zero-sum family of these vectors can be reordered to obey the same bound without signs.

What does that help mathematicians do?

Together with a matching lower bound, the claimed result pins down the Euclidean Steinitz constant: the worst-case radius needed to contain all partial sums after reordering a zero-sum family. That radius grows like the square root of dimension, up to absolute factors. Researchers could therefore rule out a universally smaller growth rate, while knowing that increasing the number of vectors alone requires no larger radius.

Are there practical applications?

Its immediate value is foundational: it supplies a sharp geometric limit for controlling cumulative vector sums. The signing statement concerns a fixed input order, whereas reordering requires a zero total. These are existence claims; the supplied abstract does not establish an efficient procedure for finding the signs or ordering.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Euclidean Steinitz–Bergström theorem

September 24, 2026 29 pages Main result formalized in Lean

Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝ^d has one signing for which every signed prefix has norm at most $C\sqrt d$ , with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, $S_2(d)=\Theta(\sqrt d)$ , and resolves the Euclidean Steinitz–Bergström conjecture.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026,
  author = {{OpenAI}},
  title = {{The Euclidean Steinitz--Bergstr{\"o}m theorem}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf}{OAI:The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/097.md.

The Euclidean Steinitz–Bergström bound

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

The Euclidean Steinitz–Bergström theorem

Scope

The formalized result gives the Euclidean Steinitz–Bergström bound with one absolute constant $C$ . For every finite family of vectors in the unit ball of $\mathbb R^d$ , signs can be chosen so that every prefix in the prescribed order has norm at most $C\sqrt d$ . When the vector sum is zero, a permutation makes every unsigned prefix satisfy the same bound. Repeated and zero vectors are included. The result is existential and does not supply an online algorithm.

Comparator links

Result	Comparator statement
Euclidean signed and reordered prefix bounds	SteinitzBergstrom.lean

At a glance

Subject: Convex and metric geometry
Manuscripts: 1
Pages: 29
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Euclidean Steinitz–Bergström conjectureProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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