Sharp Induced-Norm Paving for Symmetric Weighing Matrices

For real symmetric zero-diagonal weighing matrices \(W\) with \(W^2=dI\), we deduce induced-\(\ell_p\) \(\varepsilon\)-paving into at most \(\lceil36\varepsilon^{-q}\rceil\) parts, where \(q=\min(p,p')\) and \(q=1\) at the endpoints. The exponent \(q\) is optimal uniformly over symmetric conference matrices. The elementary compression estimate \(\|W[S]\|_p\le\|W[S]\|_2^{2/q}\) transfers the published Ravichandran–Srivastava Hilbert multi-paving theorem; a single spectral paving also gives quantitative bounds for every \(p\) simultaneously. The matching lower order follows from classical Paley conference matrices. This short, theorem-dependent note proves a precise special case, not the general arbitrary-matrix induced-norm paving question. It is self-audited and unrefereed; the observation may be folklore and no absolute priority is asserted.