Quantitative Rigidity of Ricci Nonnegative Manifolds via the Regular Green Constant

For a Ricci non-negative non-parabolic manifold (Mn≥3,g), assuming the minimal normalized Green function admits the asymptotic expansion G=r²⁻ⁿ+Hₚ+O₁(r), we prove the scaling invariant defect HₚRⁿ⁻² controls the geometry of the geodesic ball quantitatively, 1/RdGH(Bₚ(R),B₀Rⁿ(R))≤ Cₙ (Hₚ Rⁿ⁻²)1/88(n+1)(4n+1).