PNMS Case Study — Kerr Black Hole: Spin-Dependent Structure in PNMS Coordinates

Case study applying the Property-Neutral Matrixscope (PNMS) constructive operation to the Kerr metric, the rotating vacuum solution of Einstein’s field equations. The study examines how angular momentum modifies the PNMS representation of black holes at fixed mass. Starting from a Schwarzschild state, spin is introduced through the dimensionless parameter a* and the resulting Kerr quantities are re-projected through Φ_R. The analysis shows that mass (E = Mc²) remains unchanged under the addition of spin, while the horizon radius, Hawking temperature, and Bekenstein entropy shift the state in PNMS coordinates. The key structural result is that the Schwarzschild invariant (E/E_p)·(T/T_p) = 0.039789, which defines the constraint surface for non-rotating black holes, is broken by spin: increasing a* systematically lowers the product and moves Kerr states onto a family of curves below the Schwarzschild surface in the (E/E_p, T/T_p) plane. The extremal limit a* → 1 drives the Hawking temperature toward zero and the invariant toward zero. The study also shows that mass-doubling steps in PNMS coordinates remain constant across spin values, demonstrating that the scaling structure with respect to mass is spin-invariant while spin modifies the absolute position of the state in PNMS coordinate space.