Lifetime Distributions in Galactic Civilization Networks: A Hierarchy of Moments and Connectivity Thresholds

The Drake equation uses only the mean lifetime of a communicating civilization. Do contact and connectivity depend on the mean alone? Here the single lifetime and communication range of earlier work become a joint distribution at fixed mean. Abundance depends on the first moment of lifetime and contactability on the second. At fixed contactability, the listening and observational counts are set exactly by the normalized covariance of lifetime with reachable area, which depends on the third moment when range grows with lifetime; percolation thresholds depend on still more. This covariance vanishes whenever lifetime and range are independent, so lifetime spread alone leaves these counts unchanged. When the two are coupled, the effects are large: for exponential lifetimes the covariance equals two, lifetime-integrated contact counts double, a single-epoch survey sees three times as many civilizations, and a non-detection bounds contactability three times more tightly. Simulated correlated log-normal populations lower percolation thresholds by up to 80% and move the first-contact and relay frontiers of a model Galactic disc outward by up to 4 kpc. No single effective lifetime reproduces both counts and thresholds: the mean fixes how many civilizations exist, the joint distribution how they connect.