Sharp Robin eigenvalue stability with radial scalar drift components

We study the stability of the principal Robin eigenvalue of −Δ+v·∇ on B₁ ⊂ R^d, d ≥ 2, under the pointwise Euclidean bound |v| ≤ T. Fix T > 0 and a finite Robin parameter b > 0, and take measurable drifts v(x) = a(|x|)e_r(x)+w(x) with w·e_r = 0. The eigenvalue deficit above the minimum at Te_r is comparable to the radial moment given by the average over B₁ of |x|(T−a(|x|)), and bounds the normalized L² distance from Te_r to the sharp power 2+2/d. The proof combines an exact radial residual identity, uniform positive bounds for the radial ground states, and a mass-filling estimate at the center. The Euclidean constraint converts the radial deficit into control of the full vector field, including its tangential part. A reversal of the drift in a shrinking central ball yields the leading coefficient of the deficit, proves optimality of the exponent, and rules out an L∞ distance modulus. On a fixed smooth domain, a weak-adjoint residual identity gives a corresponding weighted estimate for arbitrary bounded measurable drifts.