arXiv:2610.31851v1 Announce Type: new 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 \[ \Tr f(xA+yB)-\Tr f(xA+yD) =y^2\int_{\mathbb R^2}f''(xa+yb)\,d\zeta_{A,B}(a,b). \] Its total mass is $\frac12\|B-D\|_2^2$. 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 $\zeta_{A,B}$ to Hein\"avaara's tracial joint spectral measure by an explicit positive radial transform.
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