Theorem: Number records conceal the source sign

For the packet family in Theorem thm:packetnormal, every joint number-diagonal probability is the same in ${\rho }_{+}$ and ${\rho }_{-}$. Each marginal occupation mean is

For one packet, its generating function is

Proof. The total parity $\Pi$ maps ${\rho }_{+}$ to ${\rho }_{-}$ and acts on each joint number vector by a scalar sign. Its diagonal probabilities are therefore unchanged, even if the baseline packet covariance has off-diagonal correlations.

Conjugation by displacement replaces $b_i$ by $b_i+{\alpha }_{s,i}$. The centered baseline gives ${\nu }_i+|{\alpha }_{s,i}{|}^2$.

For one packet the baseline is a circular coherent-state Gaussian mixture of variance ${\nu }_h$. After displacement its coherent amplitude has mean ${\alpha }_{s,h}$. Conditional number counts are Poisson. The circular Gaussian integral of $e^{-(1-t)|z{|}^2}$ gives (eq:displacedPGF); the vacuum case follows by its zero-variance limit.