Algebraic Transfer for Operator-Valued Gaussian Chaoses:Oriented Schatten Profiles and Singular Wick Multipliers

We develop an algebraic transfer calculus for the oriented Schatten profiles of kernels underlying operator-valued Gaussian chaoses.

A dimension-free link inequality propagates profile bounds through cut factorizations, tensor products, coefficient maps, and ordered contractions.

Combined with oriented-flattening Gaussian estimates, the calculus yields continuous multiplication on completed Wick chaoses with noncommuting coefficients, an associative algebra of factorially weighted analytic Wick series, and a local-to-global theorem for loop-free Peter--Weyl fusion trees.

We then apply the method to singular Wick multipliers on groups of polynomial growth.

For second-order multipliers we obtain sharp necessary and sufficient Schatten convergence thresholds; on $\mathbb Z^D$ we determine the full phase diagram at every order.

Fourier transfer gives exact Sobolev, Schatten-class, compactness, and trace-class thresholds for sandwiched Wick multiplication operators on $\mathbb T^D$, together with sharp Fourier--Galerkin rates and approximation-number decay.