How concentrated can a photon-number distribution be without Wigner negativity?

A sharply concentrated photon-number distribution is a basic target in the preparation of quantum light. We determine the strongest concentration compatible with a nonnegative Wigner function. As the photon number \(n\) grows, its maximal occupation probability obeys \[\omega_n:=\sup_{\rho:\,W_\rho\ge0}\langle n|\rho|n\rangle=\Theta(n^{-1/3}).\] The upper bound covers all Wigner-positive quantum states, while displaced squeezed Gaussian states attain the same order. Within the same class, the minimum photon-number variance at large mean photon number \(\mu\) is \(\Theta(\mu^{2/3})\). The concentration law also fixes a noise cost for removing the Wigner negativity of an \(n\)-photon state. Let \(R_g\) be the least noise-to-signal mixing ratio when any quantum noise state is allowed, and \(R_s\) the corresponding ratio when the added noise must itself be Wigner-positive. We obtain \[\begin{aligned}1+R_g(|n\rangle\langle n|)&=\omega_n^{-1}=\Theta(n^{1/3}),\\1+R_s(|n\rangle\langle n|)&=\Theta(n^{1/2}).\end{aligned}\] The allowed noise therefore changes the asymptotic cost of erasing the same negative Wigner function. Certified thresholds at finite photon number turn these limits into photon-counting tests of Wigner negativity.