Canonical–microcanonical positivity and phase geometry for a Bessel–₀F₁ matrix Turán determinant

Order log-concavity and the argument Turán inequality are two classical positivity directions for modified Bessel functions. We prove that they admit a sharp matrix coupling: the symmetric matrix formed from the order curvature, the mixed order–argument derivative, and the classical Turánian is positive definite throughout the full range \(ν>-1\), \(z>0\), after a trigamma correction forced by the vacuum limit. Passing to a reciprocal-gamma series identifies the coefficient matrices with Schur deficits of conditioned discrete Bessel ensembles; mixed-determinant polarization then localizes the endpoint defect to one fiber and one determinant degree. This representation separates coefficientwise and pointwise positivity into microcanonical and canonical phase boundaries controlled by the vacuum, one-particle, and thermodynamic sectors. Equivalence of ensembles then matches the high-degree coefficient walls with the large-argument Bessel boundary, with cubic agreement at a unique critical point.