We study volume comparison for closed Riemannian manifolds satisfying a positive Ricci curvature lower bound together with an improved scalar curvature lower bound. Using an integral extension of the finite shuffling comparison for scalar Jacobi solutions due to Brown and Freedman BrownFreedman2022, we prove that if a closed Riemannian manifold (Nn,g) satisfies Ricg≥(n−1)g
Nearby in the stack
and the scalar curvature
Rg≥n(n−1)(1+ε)
, then its volume satisfies
∣N∣g≤1+nε1∣Sn∣.
In fact, assuming only
Ricg≥(n−1)g
, we can prove that
∣Sn∣∣N∣g≤∣N∣g1∫N(n−1Rg−(n−1))−21dvolg.
The equality holds if and only if
N
is isometric to the unit sphere. The proof combines a coefficient-adapted Jacobian comparison with the integral shuffling comparison. This yields an estimate that retains the full Ricci spectrum. The resulting volume bound agrees to first order in
ε
with the factor predicted by Bray's conjecture. A further consequence of the argument is an averaged volume comparison for metric balls involving the scalar curvature.