Hamiltonian Cotangent Deformations and Closed-Form Geodesics on Non-Naturally Reductive Orbit Spheres: The Spin(9) 15-Sphere and the Cayley Plane

By the Skew-Torsion Holonomy Theorem of Olmos and Reggiani (2012), the 15-sphere $S^{15}$ admits no non-trivial naturally reductive metric under any transitive Lie group, precluding geodesic integration via canonical Ambrose-Singer connections. Simultaneously, Nikonorov (2013) established that every $Spin(9)$-invariant metric on $S^{15}$ is a geodesic orbit (GO) space. We integrate the geodesic flow on this non-naturally reductive GO manifold via the Lie Subalgebra Closure Principle: whenever the slice generators of a multi-axial reductive decomposition close into a finite-rank reductive Lie subalgebra, the geodesic vector equations decouple into an algebraic eigenvalue problem. Using the 16-dimensional spin representation and Vol'per's decomposition $\mathfrak{spin}(9)=\mathfrak{spin}(7)\oplus {\mathfrak{p}}_1\oplus {\mathfrak{p}}_2$, we prove that the slice generators $X_1\in {\mathfrak{p}}_2$ and $Y_1\in {\mathfrak{p}}_1$ close into $\mathfrak{u}(2)\cong \mathfrak{su}(2)\oplus \mathfrak{u}(1)\subset \mathfrak{so}(16)$. The generator $M=\alpha X_1+\beta Y_1$ has the four-mode spectrum $\Omega \in \{|R\pm 3\alpha |,|R\pm \alpha |\}$ ($R=\sqrt{{\alpha }^2+{\beta }^2}$), yielding a closed-form 4-projector decomposition and Euler-Cartan factorization for $\exp (\alpha X_1+\beta Y_1)$. Under the normalized Ricci flow, we prove that the Bourguignon-Karcher squashed Einstein metric at $t=3/11$ ($\lambda =64$) is an unstable repeller separating sub-Riemannian fiber collapse from the round sphere basin. Finally, on the deformed Cayley plane $\mathbb{O}{P}^2=F_4/Spin(9)$, we prove universal CROSS never-Einstein rigidity, establish the 15-dimensional three-stratum Whitney stratification of the cut locus $Cut(q_0)$ at non-polar basepoints $q_0$, and prove Blaschke and Besse rigidity for all $\lambda >0$.