Uniqueness of the Local Quadratic Probability Density

We investigate the uniqueness of local quadratic probability densities associated with complex-valued smooth functions of compact support on \(R^d\). Consider finite-order local Hermitian quadratic densities constructed from the function and its derivatives. Under assumptions of pointwise non-negativity, finite differential order, locality, and universal normalization with respect to the standard \(L^2\) norm, we establish a uniqueness result identifying the density with \(|ψ(x)|^2\) almost everywhere. The argument combines positivity of the coefficient matrix, high-frequency localized test functions, and elimination of derivative-dependent terms. We also discuss the role of the assumptions and the limitations of the result. The theorem concerns uniqueness within the specified class of local quadratic densities; it does not by itself derive the Born rule from more general physical principles or provide a theory of quantum measurement dynamics.