Positive Planar Measures for Spectral Pinching and the Corrected BMV Conjecture
Abstract
We prove the corrected Bessis--Moussa--Villani pinching conjecture for all symmetrized word averages. For Hermitian matrices A,B and D=EA(B), we construct a positive compactly supported measure ζA,B on ℝ2 such that Tr f(xA+yB)−Tr f(xA+yD) =y2∫ℝ2f''(xa+yb) dζA,B(a,b). Its total mass is 1/2|B−D|22. This representation proves the corrected conjecture, including its equality cases, and gives a positive bivariate moment representation for the coefficient differences. We also derive complete monotonicity of normalized exponential pinching defects and a comparison with Lie--Trotter approximations. A one-dimensional second-order loss measure is obtained for trace-preserving conditional expectations on finite tracial von Neumann algebras. Finally, we relate ζA,B to Heinävaara's tracial joint spectral measure by an explicit positive radial transform.
Related
- Characterization on spectra of self-similar measures and the dual spectral set conjecture
- Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture
- Rigidity and volume pinching for the sharp gradient estimate in positive Ricci curvature
- Pinching cones for positive isotropic curvature in dimensions seven and eight
- Dimension-Corrected Hitting Times for Heavy-Tailed Spectral Emergence in Neural Optimizer Dynamics
BibTeX
@misc{dinh2026positive,
title = {{Positive Planar Measures for Spectral Pinching and the Corrected BMV Conjecture}},
author = {Trung Hoa Dinh and Cong Trinh Le and Trung Dung Vuong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31851},
eprint = {2610.31851},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31851},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive Planar Measures for Spectral Pinching and the Corrected BMV Conjecture
AU - Trung Hoa Dinh
AU - Cong Trinh Le
AU - Trung Dung Vuong
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31851
UR - https://arxiv.org/abs/2610.31851
AB - We prove the corrected Bessis--Moussa--Villani pinching conjecture for all symmetrized word averages. For Hermitian matrices A,B and D=E_A(B), we construct a positive compactly supported measure ζ_A,B on ℝ^2 such that Tr f(xA+yB)−Tr f(xA+yD) =y^2∫_ℝ^2f''(xa+yb) dζ_A,B(a,b). Its total mass is 1/2|B−D|_2^2. This representation proves the corrected conjecture, including its equality cases, and gives a positive bivariate moment representation for the coefficient differences. We also derive complete monotonicity of normalized exponential pinching defects and a comparison with Lie--Trotter approximations. A one-dimensional second-order loss measure is obtained for trace-preserving conditional expectations on finite tracial von Neumann algebras. Finally, we relate ζ_A,B to Heinävaara's tracial joint spectral measure by an explicit positive radial transform.
ER -