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Abstract

We compare a homogeneous real-stable count law with the multinomial distribution having the same mean. The normalized covariance deficit determines the quadratic orthogonal component of the likelihood ratio and controls its higher-order remainder through explicit spectral bounds. These estimates give a uniform quadratic approximation in chi-square distance as the trace deficit vanishes, and convergence for diffuse covariance-deficit spectra with bounded trace, allowing the degree and number of categories to vary. They also yield a sharp covariance-matching exponent, while nonfactorizable stable cubics show why a stronger Frobenius-power estimate for categorical products cannot extend to the full stable class. An optimal all-degree centered-real elementary-symmetric inequality, with equality classification, supplies the coefficient input and gives an optimal linear total-variation certificate. Separately, a sharp centered-complex inequality in six coordinates, including its complete equality orbit, yields the optimal uniform factor 3/4 in the complete-homogeneous bound for the squared L² norm of the fifth likelihood component of degree-six stable laws. Ternary determinantal representations transfer further complete-homogeneous coefficient bounds, and a fixed-law convolution theorem identifies the limiting Gaussian total variation. The arguments adapt classical tensor, polynomial and orthogonal-expansion methods; standalone exact-arithmetic code reconstructs the finite certificates used in the proofs.