📝 SummarXiv

Abstract

We refine the cycle-walk (Fourier) template of Gnacik and the author to quantify when a~$\delta$-Sule\u{\i}manova 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 \[ \delta_n \;=\; 1-\frac{1}{2\cos^2\!\Big(\frac{\pi}{4n}\rho(n)\Big)}, \quad \rho(n)\in\{0,1,2,4\}\ \text{determined by } n\bmod 8, \] which improves $1/2$ 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 \[ \delta_n^{\rm (ph)} \;=\; \begin{cases} \displaystyle 1-\dfrac{1}{2\cos^2(\pi/n)}, & n\equiv 0\pmod{4},\\[2mm] \displaystyle 1-\dfrac{1}{2\cos^2(\pi/2n)}, & n\equiv 2\pmod{4},\\[2mm] \displaystyle 1-\dfrac{1}{2\cos^2(\pi/4n)}, & n\ \text{odd}, \end{cases} \] so that $\delta_n^{\rm (ph)}<\tfrac12$ for \emph{every} $n\ge3$ and $\delta_n^{\rm (ph)}=\delta_n$ unless $8\mid n$. Next, on abelian $2$-groups, the Walsh--Hadamard basis has coherence $M=1$ and hence suffices for \emph{all} Sule\u{\i}manova lists ($\delta=0$); the same conclusion holds in every Hadamard order (\emph{e.g.}, Paley families).