Rotational Contraction and Entropy Bounds in Kerr Spacetimes

We introduce a rotational contraction ratio χ(a) that quantifies how spin deforms the Kerr horizon and reduces black hole entropy. Using analytic geometry and the Gauss–Bonnet theorem, we prove an exact bound 1/2 SSchw ≤ SKerr(M, a) ≤ SSchw, with χ(a) acting as a Lorentz-type factor that decreases monotonically with a/M. Numerical evaluation of curvature distributions confirms that rotation concentrates positive curvature near the poles and flattens the equator. We show that χ(a) rescales both the horizon area and the Hilbert-space dimension of microstates, providing a unified geometric origin for entropy reduction. This contraction principle offers a universal framework linking relativistic kinematics, differential geometry, and black hole thermodynamics, and it naturally extends to quantum and holographic entropy corrections near extremality.