The Euclidean Steinitz-Bergstrom conjecture: is the Euclidean Steinitz constant O(sqrt d)?

Let $S_2(d)$ be the least number such that every finite family of vectors $v_1,\dots,v_N$ in the Euclidean unit ball of $\mathbb R^d$ with $\sum v_i=0$ has an ordering whose partial sums all have norm at most $S_2(d)$ . Steinitz's lemma gives a bound depending only on $d$ ; Grinberg and Sevastyanov (1980) proved $S\le d$ for every norm, and the simplex example shows $S_2(d)\ge\tfrac12\sqrt d$ . The expected square-root scale in Euclidean space was discussed by Behrend (1954), and the conjecture is attributed to Bergstrom. Is $S_2(d)=O(\sqrt d)$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Discrepancy theory; rearrangements of vector sums
Posed by: Attributed to Viktor Bergstrom (attribution recorded by Ambrus and Heck, Mathematika 2026, Conjecture 5); square-root growth discussed by F. A. Behrend (1954)
Year posed: 1954
Years open: 72y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 38 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.2: there is an absolute constant $C$ such that every zero-sum family of vectors in the Euclidean unit ball of $\mathbb R^d$ (repetitions allowed) has an ordering with all partial sums of norm at most $C\sqrt d$ ; with the simplex lower bound, $\tfrac12\sqrt d\le S_2(d)\le C\sqrt d$ . It follows from the prescribed-order signing theorem (separate entry). The constant is not made explicit, and the construction is existential: no online rule or efficient algorithm is given.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 were read against the conjecture as stated in the manuscript; the ordering bound follows from the signing bound by Chobanyan's transference, proved in the introduction. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator SteinitzBergstrom, declaration OAI.EuclideanSteinitzBergstrom.main). ComparatorChallenges/SteinitzBergstrom.lean was read here: it asserts one real $C$ such that for all $d,N\ge1$ and all $v:\mathrm{Fin}\,N\to\mathbb R^d$ with $\|v_i\|\le1$ there are signs $\pm1$ keeping every prefix within $C\sqrt d$ , and every zero-sum such family has a permutation keeping every prefix within $C\sqrt d$ . This states the headline claim. Not rebuilt here.

The Euclidean Steinitz-Bergstrom conjecture: is the Euclidean Steinitz constant O(sqrt d)?

What was actually shown

What the AI did

Verification

Sources

Discussion