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Breakthrough Tracker record

Matrix Spencer now has explicit constants, algorithms and a separate exposition

Akbas and Sra’s O(√n) proof claim now has a separate exposition by Bansal and Kook, while Song and Zhang claim existence below 8√n, a randomized algorithm below 12√n in n^(3+o(1)) real-arithmetic operations for dense input, and a (2−o(1))√n lower bound.

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Stable ID
math-matrix-spencer-conjecture-proof-2026
Revision
math-matrix-spencer-conjecture-proof-2026.v2
Field
Mathematics · Discrepancy theory, functional analysis and probability
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Provisional claimed proof
Last checked

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. Song and Zhang disclose several LLMs and attribute substantial proof, complexity and lower-bound development to them before human rewriting and verification.

Record details

Problem or result
The Matrix Spencer conjecture for operator-norm discrepancy of symmetric matrices
Authors
Emrullah Akbas and Suvrit Sra; Zhao Song and Lichen Zhang; Nikhil Bansal and Yunbum Kook
Institutions
Technical University of Munich; MIT; Song lists no institution in the inspected preprint
Result date
Original August 28; technical follow-ups submitted September 14 and revised through September 21, 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

All sources remain preprints. The exposition improves checkability but is not peer review. Song–Zhang runtime is stated in a real-arithmetic model, not as a practical benchmark.

Sources

  1. Primary: Akbas and Sra, claimed proof of the Matrix Spencer conjecture
  2. Primary context: Akbas and Sra, earlier structured Matrix Spencer theorem
  3. Primary: Song and Zhang, explicit Matrix Spencer constants and algorithm
  4. Independent exposition: Bansal and Kook, a simple proof of Matrix Spencer

Correction and revision history

  1. 2026-09-01 — Added after primary-source, scope, status, AI-role and limitation review.
  2. 2026-10-02 — Added explicit constants, a lower bound, a near-input-linear algorithm and a separate exposition; provisional status retained.

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