We introduce a fundamentally different paradigm for contact tracing: for each positive case, do not only ask direct contacts to quarantine; instead, tell everyone how many relationships away the disease just struck (so, "2" is a close physical contact of a close physical contact). This new approach, which has already been deployed in a publicly downloadable app, brings a new tool to bear on pandemic control, powered by network theory. Like a weather satellite providing early warning of incoming hurricanes, it empowers individuals to see transmission approaching from far away, and incites behavior change to directly avoid exposure. This flipped perspective engages natural self-interested instincts of self-preservation, reducing reliance on altruism, and the resulting caution reduces pandemic spread in the social vicinity of each infection. Consequently, our new system solves the behavior coordination problem which has hampered many other app-based interventions to date. We also provide a heuristic mathematical analysis that shows how our system already achieves critical mass from the user perspective at very low adoption thresholds (likely below 10% in some common types of communities as indicated empirically in the first practical deployment); after that point, the design of our system naturally accelerates further adoption, while also alerting even non-users of the app. This article seeks to lay the theoretical foundation for our approach, and to open the area for further research along many dimensions.