Dimension-dependent bounds for the SDIEP via phase optimisation and Paley-type constructions

We refine the cycle-walk (Fourier) template of Gnacik and the author to quantify when a~$\delta$-Suleĭmanova spectrum $(1,{\lambda }_2,\dots ,{\lambda }_n)$ (with ${\lambda }_j\le 0$) is realised by a symmetric doubly stochastic matrix. For the canonical cycle basis we compute the \emph{exact} size-dependent threshold $${\delta }_n=1-\frac{1}{2{\cos }^2\text{\hspace{0.17em}⁣}(\frac{\pi }{4n}\rho (n))},\rho (n)\in \{0,1,2,4\}\text{ determined by }n8,$$ which improves $1\mathrm{/}2$ if and only if $8\nmid n$; we also prove sharpness for that template. We then introduce an \emph{optimally phase-aligned} cycle basis which removes the `$8\mid n$’ artefact and yields better sufficient bound $${\delta }_n^{(\mathrm{p}\mathrm{h})}=\{\begin{array}{ll}{\displaystyle 1-{\displaystyle \frac{1}{2{\cos }^2(\pi \mathrm{/}n)}},}& n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4),\\ {\displaystyle 1-{\displaystyle \frac{1}{2{\cos }^2(\pi \mathrm{/}2n)}},}& n\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4),\\ {\displaystyle 1-{\displaystyle \frac{1}{2{\cos }^2(\pi \mathrm{/}4n)}},}& n\text{ odd},\end{array}$$ so that ${\delta }_n^{(\mathrm{p}\mathrm{h})}<\frac{1}{2}$ for \emph{every} $n\ge 3$ and ${\delta }_n^{(\mathrm{p}\mathrm{h})}={\delta }_n$ unless $8\mid n$. Next, on abelian $2$-groups, the Walsh–Hadamard basis has coherence $M=1$ and hence suffices for \emph{all} Suleĭmanova lists ($\delta =0$); the same conclusion holds in every Hadamard order (\emph{e.g.}, Paley families).