Cyclic steepest descent is $R$-superlinearly convergent for almost every initialization on two-spectrum quadratics

We completely classify cyclic steepest descent (CSD) on strictly convex quadratics whose Hessian has exactly two distinct eigenvalues. At the beginning of each cycle CSD recomputes an exact steepest-descent stepsize and reuses it for $m$ updates. For every $m\ge 2$, write $A=aP+bQ$ with $0<a<b$ and spectral projectors $P,Q$. For nonzero spectral blocks, let $r=\mathrm{\parallel }Pg{\mathrm{\parallel }}_2\mathrm{/}\mathrm{\parallel }Qg{\mathrm{\parallel }}_2$ at a cycle boundary. The exact projective dynamics is $r^{+}=r^{1-2m}$. Hence the nonzero equal-norm cone $\mathrm{\parallel }P{g}_0{\mathrm{\parallel }}_2=\mathrm{\parallel }Q{g}_0{\mathrm{\parallel }}_2>0$ is precisely the nonterminating exceptional set: there the convergence is $R$-linear with factor $(b-a)\mathrm{/}(b+a)$, whereas every other nonterminating initialization is $R$-superlinear and one-eigenspace initializations terminate in one step. The exceptional cone has Lebesgue measure zero, so $R$-superlinear convergence holds almost everywhere in this two-spectrum class. The main results of this paper were obtained through a generative-AI workflow using OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, and OpenAI GPT-6 Astra. Further details appear in the disclosure at the end of the paper.