Constructive Characterizations and New Families of van Dantzig Pairs

# Concise takeaways - Under symmetry, nondegeneracy and all real exponential moments, 1/M_U has automatic exponential decay. Its inverse Fourier transform is a conjugate density exactly when nonnegative everywhere; normalization still needs the Gaussian-regularization argument. - The Hankel criterion requires both matrix families at every size. Local moment bounds supply an exponential moment, then analytic continuation of M_U(z)phi_V(z)-1 gives the identity along the whole real axis. - For P(0)=1 and P>0 on [0,infinity), first choose sufficiently large b, then sufficiently large tau depending on b. Laplace convolution controls global positivity of the reciprocal density; a finite Gaussian-derivative formula controls the original density. - A finite negative Fourier or Hankel certificate can exclude existence. Finite positive tests alone do not prove the infinite criterion. - Mean-value reuse for entire functions requires the stated interchange/integrability condition. Exclusion of the cited D_P class uses its external theorem; novelty is not independently established here.