The Jacobsthal Eigenvalue in Kerr Spacetime:

We prove that the three critical radii of Schwarzschild spacetime — the event horizon (r = 2M), photon sphere (r = 3M), and innermost stable circular orbit (r = 6M) —constitute the unique equal tripartition of the compactness interval [0,1] if and only if the governing eigenvalue equals 2, the dominant eigenvalue of the Jacobsthal recurrence. This Schwarzschild Uniqueness Theorem establishes that the Jacobsthal sequence xn = xn−1 +2xn−2 is the sole non-degenerate positive-integer second-order recurrence realising this geometric property. Extending to Kerr spacetime, we define the spin-dependent stability ratio Λ(χ) = r+ ISCO(χ)/r+ ph(χ), which decreases monotonically from Λ(0) = 2 (Jacobsthal eigenvalue, Schwarzschild) to Λ(1) = 1 (extremal degeneracy). We identify a discrete Kerr Stability Ladder: as spin increases, Λ(χ) passes through a hierarchy of classical mathematical constants at precisely defined critical spin values — the golden ratio φ at χ∗ = 0.785, the degree-3 Tower-A eigenvalue at χ∗ = 0.877, the plastic constant ϱ at χ∗ = 0.975, and higher towers converging near extremal spin. The Taylor series of the partition asymmetry δ(χ) carries Jacobsthal–Lucas structure at even orders and JacobsthalMersenne structure at odd orders through the bilinear identity Jn · jn = J2n — a discrete double-angle formula governing how spin rotates the Jacobsthal eigenspace. The extremal geometry encodes the square of the Schwarzschild self-consistency equation, with near-horizon scaling ϵ1/3 whose exponent equals the reciprocal of the Schwarzschild photon/horizon radius ratio. These results yield falsifiable predictions for spin-resolved QPO spectroscopy with eXTP and STROBE-X.