This preprint presents the Golden Window hypothesis: that the emergence of a connected, durable interstellar “relay layer” is governed by a time-limited connectivity window created by cosmic expansion, finite latency tolerance, and the lifespan/maintenance capacity of intelligent agents. The work separates connectivity (a graph/percolation property) from cooperation (a strategic/game-theoretic property), develops a minimal model for edge feasibility and percolation thresholds, and derives discriminating predictions (“relic vs. ruin” signatures, patchy connectivity, and relay artifacts). It then proposes a practical engineering playbook for building resilient sparse-edge relay networks with adversarial robustness, safety guardrails, and deep-time governance considerations. Version v4.1-preprint (8 January 2026). DOI reserved via Zenodo and registered upon publication.