[#P3120] Matrix Spencer discrepancy conjecture

Problem. Does there exist an absolute constant \(C>0\) such that, for every positive integer \(n\) and all real self-adjoint matrices \(A_1,\ldots,A_n\in\mathbb R^{n\times n}\) with operator norm \(\|A_i\|_{\mathrm{op}}\le1\), there are signs \(\varepsilon_i\in\{-1,1\}\) satisfying \(\|\sum_{i=1}^n\varepsilon_iA_i\|_{\mathrm{op}}\le C\sqrt n\)?

Neutral schematic of several symmetric matrices stacked with plus-minus signs and an operator-norm gauge. — The objects and operations appearing in Matrix Spencer discrepancy conjecture.

Agent access

Work in ChatGPT

The agent reads this problem's research brief and asks you before it saves anything to TheoremDB.

Definitions and notation

1Context

Known frontier: The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor. Open boundary: Remove the logarithmic loss without rank or commutativity hypotheses.

2Problem setup

Definition 1 (Self-adjoint matrix). Here this means a real symmetric square matrix.

Definition 2 (Operator norm). \(\|A\|_{\mathrm{op}}=\sup_{\|x\|_2=1}\|Ax\|_2\).

Remark 1. This is a noncommutative analogue of Spencer vector discrepancy.

3What counts as a solution

Prove the bound with one universal \(C\), or give a family for which the minimum signed-sum norm divided by \(\sqrt n\) is unbounded.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor. Exact unresolved remainder: Remove the logarithmic loss without rank or commutativity hypotheses.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor. Exact unresolved remainder: Remove the logarithmic loss without rank or commutativity hypotheses.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor. Exact unresolved remainder: Remove the logarithmic loss without rank or commutativity hypotheses.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Matrix Spencer discrepancy conjecture. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor. The exact unresolved remainder is: Remove the logarithmic loss without rank or commutativity hypotheses. A complete resolution must meet the following acceptance conditions: - Prove the bound with one universal \(C\), or give a family for which the minimum signed-sum norm divided by \(\sqrt n\) is unbounded. Open full record →
The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor.^[1][2] Relevance to this problem Locates the present research frontier immediately below Matrix Spencer discrepancy conjecture. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor. This leaves the following boundary unresolved: Remove the logarithmic loss without rank or commutativity hypotheses. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.^[1][2] Relevance to this problem Documents why Matrix Spencer discrepancy conjecture was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across source collection and its cited primary literature, Crossref/arXiv or MathSciNet stable records, TheoremDB published, prospecting, formal-conjecture, packet, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - "Matrix Spencer" conjecture 2026 - "Resolving Matrix Spencer" rank - matrix discrepancy operator norm C sqrt n The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there. Related: Claim 1 Open full record →
Remove the logarithmic loss without rank or commutativity hypotheses.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 Remove the logarithmic loss without rank or commutativity hypotheses. Relevance to this problem Turns the remaining uncertainty in Matrix Spencer discrepancy conjecture into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: Remove the logarithmic loss without rank or commutativity hypotheses. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Prove the bound with one universal \(C\), or give a family for which the minimum signed-sum norm divided by \(\sqrt n\) is unbounded. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor. Exact unresolved remainder: Remove the logarithmic loss without rank or commutativity hypotheses.

Equivalent-formulation queries: "Matrix Spencer" conjecture 2026; "Resolving Matrix Spencer" rank; matrix discrepancy operator norm C sqrt n
Strongest checked neighboring result: The conjectured bound holds for commuting matrices and for matrices of rank at most \(n/\log^3 n\); general concentration gives an extra \(\sqrt{\log n}\) factor.
Exact unresolved remainder: Remove the logarithmic loss without rank or commutativity hypotheses.

How the 4 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemMatrix Spencer discrepancy conjecture

All 3 recorded relations between these records and the problem

2See also

Constructing a union from non-cancelling intersectionscombinatorics
Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Matrix Spencer discrepancy conjecture.” TheoremDB. P3120. Problem statement; statement text SHA-256 133dcc196ef3ff0d25887e0eb747d4d8317a00cc89367ad61d84e3fd25cb8099. https://theoremdb.org/statement/?ref=P3120

BibTeX

@misc{theoremdb-problem-133dcc196ef3ff0d25887e0eb747d4d8317a00cc89367ad61d84e3fd25cb8099,
  title = {{Matrix Spencer discrepancy conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 133dcc196ef3ff0d25887e0eb747d4d8317a00cc89367ad61d84e3fd25cb8099},
  url = {https://theoremdb.org/statement/?ref=P3120}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. Afonso S. Bandeira, Anastasia Kireeva, Antoine Maillard, and Almut Rödder, “Randomstrasse101: Open Problems of 2024,” arXiv:2504.20539, version dated 22 March 2026. Conjecture 1 and 2026 update record. ↗ open copy ↗preprint · primary source · arXiv:2504.20539, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Gives the problem statement, definitions, status discussion, and the authors’ update record.Also cited at Afonso S. Bandeira, Anastasia Kireeva, Antoine Maillard, and Almut Rödder, “Randomstrasse101: Open Problems of 2024,” arXiv:2504.20539, version dated 22 March 2026. Conjecture 1 and 2026 update record.Source used to assess the problem's recorded status.For Matrix Spencer discrepancy conjecture: This is the dated publication status for the canonical target Matrix Spencer discrepancy conjecture.Source named by the research packet.
Bansal, Nikhil, Jiang, Haotian, and Meka, Raghu, “Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank”. arXiv (2022). DOI 10.48550/arXiv.2208.11286. Abstract and main theorem. ↗ open copy ↗preprint · primary source · arXiv:2208.11286, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Proves the full conjectured order for matrices of rank at most n/log^3 n.Source used to assess the problem's recorded status.For Matrix Spencer discrepancy conjecture: Proves the full conjectured order for matrices of rank at most n/log^3 n.

Original TheoremDB statement and summary based on citation-only scholarly sources; no source prose, proof, table, code, or figure is reproduced.

[#P3120] Matrix Spencer discrepancy conjecture

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance