The Sharp Sadov Constant and Local Spectral Stability for Shapiro--Diananda Cyclic Sums

We determine the sharp Sadov constant for Shapiro--Diananda cyclic sums. Sadov proved the lower bound (C\ge \log 2); we prove the matching upper bound by an explicit asymptotic construction, obtaining (C=\log 2). We also develop a local spectral stability theory for the equal point of the Shapiro--Diananda cyclic sums. The Hessian is diagonalized by Fourier modes, giving an exact local stability/saddle/quadratic-degeneracy criterion for all (n,k), periodic equality families, and explicit classifications for (k=2) and (k=3). The result determines the global infimum over all (n,k), but does not solve the separate fixed-(k) asymptotic minimization problems.