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 $D=E_A(B)$, we construct a positive compactly supported measure ${\zeta }_{A,B}$ on ${\mathbb{R}}^2$ such that \[ \Tr f(xA+yB)-\Tr f(xA+yD) =y^2\int_{\mathbb R^2}f''(xa+yb)\,dζ_{A,B}(a,b). \] Its total mass is $\frac{1}{2}\Vert B-D{\Vert }_2^2$. 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 ${\zeta }_{A,B}$ to Heinävaara's tracial joint spectral measure by an explicit positive radial transform.