Sharp Carleman Estimates with Critical Covariant Drift for Parabolic Operators on Vector Bundles

We establish quantitative Carleman inequalities for covariant parabolic operators on thetrivial g valued bundle over Rd, with d ∈ {3, 4}, in backward cylinders. For a fixed smoothreference connection D = d + [A¯, ·] we consider∂tu + ΔDu + V ·D u +Wu = F,where V is a (time dependent) covariant drift and W is a zeroth order endomorphism term.Using the backward heat weight ϕ(x, t) = |x−x0|24ρ(t) + α log ρ(t) with ρ(t) = t0 − t + κr2,we obtain a Carleman estimate with explicit dependence on ∥A¯∥W1,∞, ∥FA¯∥L∞ and on ascale invariant local Morrey seminorm of V . The drift is absorbed under a computablesmallness threshold. The dependence on the large parameter τ is sharp, and the resultinginequality yields a covariant three cylinder inequality for propagation of smallness acrossscales. The argument is local and does not rely on divergence free structure or global gaugeconsiderations.