Random Tensor Estimates and Deterministic Diagonal Resolutions for Mixed Paracontrolled Operators

We prove an abstract operator-level convergence criterion for mixed paracontrolled random operators $$T_{\mathrm{\Lambda }}^{i;j,k}(w)=I_i(w<{\mathrm{\Psi }}_{j,\mathrm{\Lambda }})\circ {\mathrm{\Psi }}_{k,\mathrm{\Lambda }}$$ on ${\mathbb{T}}^d$. These operators are the second-level mixed blocks generated by quadratic paracontrolled expansions, and are singular in the range where the deterministic resonant-product threshold fails. At finite cutoff we separate $$T_{\mathrm{\Lambda }}^{\tau }=D_{\mathrm{\Lambda }}^{\tau }+B_{\mathrm{\Lambda }}^{\tau },$$ where $D_{\mathrm{\Lambda }}^{\tau }$ is the Fourier-diagonal covariance contraction supported on the external diagonal $n=q$, and $B_{\mathrm{\Lambda }}^{\tau }$ is a centered second homogeneous Gaussian chaos with incidence $n=q+\mathrm{\ell }+r$. The centered component is controlled by a finite second-order random tensor estimate, a Banach-valued time lift, and Sobolev–Besov dyadic summation. The deterministic component is treated through a diagonal-resolution interface: retained raw branches or model-determined orbit sums are required only to satisfy scalar Volterra dyadic bounds after the relevant finite cancellations have been formed. The result applies in arbitrary dimension, for finite color/label families, Fourier-diagonal weak covariance, random initial data, colored-in-time Gaussian profiles, and frequency-dependent amplitudes. It abstracts the reusable mixed-operator mechanism underlying concrete color-phase diagonal analyses, while leaving model-specific diagonal classification and scalar Volterra estimates as explicit inputs.