A Free Algebraic Model for Averaged Z-Weighted Wick Functionals

This paper constructs a universal algebraic realization model for Hermitian parameter matrices whose entries have modulus at most one and whose diagonal entries are normalized. The entries of the parameter matrix are used as weights of oriented crossings in averaged Wick-type moment formulas. The construction separates each crossing parameter into its unit complex factor and its modulus. We first construct a universal free algebraic unimodular factor model whose moments are defined by a balanced pair-oriented crossing formula. The modulus factors are then encoded by an auxiliary commutative algebra and recombined with the unimodular factors by tensorization. The resulting normalized sums converge in joint algebraic moments to the averaged weighted Wick moment functional, whose moments are given by a fully averaged pair-partition formula. In the general complex Hermitian case, the construction is purely algebraic, and no positivity, traciality, operator boundedness, or operator algebraic realization is claimed. In the real symmetric specialization, the averaged oriented formula reduces to the standard mixed Gaussian pair-partition formula with color-dependent crossing parameters, so the construction contains the known Fock-representable mixed Gaussian cases as positive Fock-space examples. Moreover, the averaged functional satisfies a uniform Gaussian-type moment growth bound.