Breakthrough Tracker record

A preprint claims the optimal-order Matrix Spencer discrepancy bound

Emrullah Akbas and Suvrit Sra claim that any n symmetric n-by-n matrices of operator norm at most one admit an efficiently findable signing whose sum has operator norm O(√n). Their matrix small-ball argument removes the remaining logarithmic loss in the general case.

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Stable ID: math-matrix-spencer-conjecture-proof-2026
Revision: math-matrix-spencer-conjecture-proof-2026.v1
Field: Mathematics · Discrepancy theory, functional analysis and probability
Evidence: Tier 1 · Peer reviewed: No
Record state: Provisional · Provisional claimed proof
Last checked: 2026-09-01

AI role

The authors report extensive GPT-5.6 Sol Pro use. They attribute most calculations and the hereditary small-ball lemma to the model after human-directed exploration, and state that they simplified and verified the proof and accept responsibility for it.

Record details

Problem or result: The Matrix Spencer conjecture for operator-norm discrepancy of symmetric matrices
Authors: Emrullah Akbas and Suvrit Sra
Institutions: Technical University of Munich
Result date: First-version preprint submitted August 28, 2026

Why it matters

Matrix Spencer extends classical vector discrepancy to noncommuting matrices. The O(√n) scale matches the conjectured order and would close the general form left open by structured and moderate-rank results.

Limits

The proof is a first-version preprint with no located independent correctness assessment or formal artifact. The asymptotic statement hides a universal constant, and the claimed efficient algorithm has not been benchmarked as a practical solver.

Sources

Primary: Akbas and Sra, claimed proof of the Matrix Spencer conjecture
Primary context: Akbas and Sra, earlier structured Matrix Spencer theorem

Correction and revision history

2026-09-01 — Added after primary-source, scope, status, AI-role and limitation review.

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