Canonical Heat Covariance and Square-Preserving Averaged Nonlinearities

The heat semigroup associates with a vector field the canonical covariance tensor tau_s[u] = e^{sΔ}(u⊗u) − e^{sΔ}u ⊗ e^{sΔ}u: pointwise positive semidefinite, satisfying an exact cocycle in the filter scale, with a carré-du-champ representation as an integral of positive square fields. For the Navier–Stokes bilinear form, the commutator with the heat semigroup is exactly the Leray-projected divergence of this same covariance. The paper isolates this conjunction of same-field positivity and exact commutator compatibility as canonical heat covariance. For linear maps between finite-dimensional real Hilbert spaces, universal positivity of the symmetrized cross-covariance sym(A Σ B*) ⪰ 0 for every Σ ⪰ 0 holds if and only if A = 0 or B = cA with c ≥ 0, and the negative eigenvalue of a rank-one cross term is computed exactly. Finite generated mixtures are treated through a covariance-synthesis quotient: a positive square representation exists exactly when the affine class M + Ker Q_T meets the positive-semidefinite cone, with a constructive Cholesky representation; an explicit flip example shows that an indefinite coefficient matrix may represent the same square map as a positive one. Distributional Fourier-multiplier rigidity is proved for the canonical defect, and a finite anisotropic Gaussian frame with no common nonzero heat resonance removes distributional freedom on the isotropic resonance. This yields a precise separation boundary relative to averaged nonlinearities in the sense of Tao: termwise separation and multiplier-class separation are proved, while full non-representability of the localized blow-up operator is stated as an open gate, not a claim. Second paper of the M-series (M-I…M-V); the projective network layer that consumes this covariance is developed in M-I (doi:10.6084/m9.figshare.33150845). The package includes the governance amendment, a deterministic verification suite with SHA-256 certificate, and provenance notes. The historical origin of the broader programme (Zenodo preprints v14.8 and v7.4) is cited as prior public disclosure only, never as a theorem source.