trureturing — Discovering truth

bibkey: abednikolaevanovikov2026traces
authors: Sami Abdullah Abed; Irina Nikolaeva; Andrej Novikov
year: 2026
title: "How characterisations of traces and central elements fail: sharp violation constants via Michelson contrast and noncentrality"
doi: 10.48550/arXiv.2610.34388
url: https://arxiv.org/abs/2610.34388v1
claim: "Section 11.2(5) asks whether the Alhasan–Fawwaz constant is bounded on M_n(C) independently of the positive definite density."
strata_touched:
  - D5/S3/Quantum/Matrix/AbedNikolaevaNovikovAFDensityBound
license: citation-only
triage: anchor

Verified locator

• DOI: 10.48550/arXiv.2610.34388
• Version: https://arxiv.org/abs/2610.34388v1
• Section 3.1, page 8: trace, modulus, and positive definite density conventions.
• Section 5.12, Proposition 5.44, pages 32–33: the quotient and existing bounds.
• Section 11.2(5), page 59: the fixed-dimension boundedness question.

This note is citation-only; it does not reproduce the source article or grant permission to redistribute it. The original Alhasan–Fawwaz article and worldwide novelty remain unverified.

Source and domain

The source is arXiv:2610.34388v1, Sections 3.1, 5.12 (Proposition 5.44), and 11.2(5), printed pages 8, 32–33, and 59. Its PDF has SHA-256 b6d9af9cdbf570646201ecce02ef2a52bf2540f96bf8a56992a1ada924265abb.

Section 3.1 uses the ordinary matrix trace, defines $|X|=(X^{\ast }X{)}^{1/2}$, and sets ${\phi }_{\rho }(Y)=\mathrm{Tr}(\rho Y)$. The standing density assumption is $\rho >0$, without trace normalization. Section 5.12 defines

$$a(\rho )=\sup \left\{\frac{{\phi }_{\rho }(A)}{{\phi }_{\rho }(|A+iB|)}:A\ge 0,B=B^{\ast },A+iB\ne 0\right\}.$$

The selected sentence in Section 11.2(5) is:

Is $a$ bounded on $M_n(\mathbb{C})$ independently of $\rho$?

Here $n$ is fixed. Determining the entire function, the factor question, and the variant with positive $B$ are separate questions. Proposition 5.44(c) already supplies the dimension-two bound; its condition-dependent general bound does not supply a bound depending only on dimension.

Scope

The companion formalization targets $C_n=n$ for every $n\ge 1$, with the same quotient set, positive denominators, and the genuine real least upper bound. Its matrix argument diagonalizes $|X|$, including its zero eigenspaces, and does not require $X$ to be invertible. No sharpness or priority claim is made. The original Alhasan–Fawwaz article and worldwide novelty remain unverified.