Coherence beyond Gaussian mixtures at low energy: sharp Wigner-positive bounds and phase-noise thresholds

Wigner-positive light can carry more vacuum–one-photon coherence than every Gaussian mixture under the same mean photon-number budget. We determine the exact Gaussian benchmark and the sharp low-energy scales of this separation. Relative to the unconstrained ceiling \(U(E)=√E(1-E)\), the Gaussian coherence shortfall is \(U(E)-M_G(E)~ E5/2\), whereas the optimum over all Wigner-positive states satisfies \(limE↓0Elog[U(E)-M_∞(E)]=-1/2\). A hard Fock cutoff changes the cost: with support on levels \(0,…,K\), the shortfall is \((KK!/2)EK+1/2[1+O(K^2E)]\), uniformly for \(K≥2\) as \(K^2E→0\). The same comparison gives a direct certification rule and its phase-noise budget. After Gaussian random phase fluctuations of variance \(Γ\), the optimized Wigner-positive source exceeds the full Gaussian coherence benchmark precisely below \(Γcert(E)=2E^2+O(E^3)\). Contact constraints determine the rank and leading mixed population of finite-support optimizers; a complete three-level frontier gives an exact test case. The exact benchmark lets an energy-and-coherence measurement exclude all Gaussian-mixture explanations and sets the phase-noise tolerance of that test.