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Prescribed-order Euclidean prefix discrepancy O(sqrt d), independent of length (Bansal-Jiang-Meka-Singla-Sinha Conjecture 6.3)

Theorem 1.1: an absolute constant $C$ such that every finite sequence in the Euclidean unit ball of $\mathbb R^d$ , in prescribed order, has one signing with every prefix of norm at most $C\sqrt d$ , independent of $N$ . This removes the $\sqrt{\log N}$ term from Banaszczyk's bound and is optimal up to the constant. It is existential: no efficient or online algorithm is given, so the algorithmic and online versions of the question remain open.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Prescribed-order Euclidean prefix discrepancy O(sqrt d), independent of length (Bansal-Jiang-Meka-Singla-Sinha Conjecture 6.3) · Euclidean prefix discrepancy sqrt d

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Prescribed-order Euclidean prefix discrepancy O(sqrt d), independent of length (Bansal-Jiang-Meka-Singla-Sinha Conjecture 6.3)

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the prescribed-order signing question as the paper cites it from Bansal et al.; the manuscript's citation was not checked against the 2021 preprint's own wording. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator SteinitzBergstrom, declaration OAI.EuclideanSteinitzBergstrom.main); its SignedPrefixBound part, read here, states exactly the length-free $C\sqrt d$ prefix-signing bound for every finite sequence in the unit ball. Not rebuilt here.

Source ↗

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

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean proof (OAI.EuclideanSteinitzBergstrom.main)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Bansal, Jiang, Meka, Singla, Sinha 2021 (arXiv 2111.07049v1)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean proof (OAI.EuclideanSteinitzBergstrom.main)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

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

VibeMathed record: Prescribed-order Euclidean prefix discrepancy O(sqrt d), independent of length (Bansal-Jiang-Meka-Singla-Sinha Conjecture 6.3) evidence for this event

Lean proof (OAI.EuclideanSteinitzBergstrom.main) evidence for this event

Prescribed-order Euclidean prefix discrepancy O(sqrt d), independent of length (Bansal-Jiang-Meka-Singla-Sinha Conjecture 6.3) parent of this event

Bansal, Jiang, Meka, Singla, Sinha 2021 (arXiv 2111.07049v1) evidence for this event

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