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Generative artificial intelligence systems increasingly participate in research, law, education, media, and governance. Their fluent and adaptive outputs create an experience of collaboration. However, these systems do not bear responsibility, incur liability, or share stakes in downstream consequences. This structural asymmetry has already produced sanctions, professional errors, and governance failures in high-stakes contexts We argue that stable human-AI coexistence is an institutional achievement that depends on governance infrastructure capable of distributing residual risk. Drawing on institutional analysis and evolutionary cooperation theory, we introduce a formal inequality that specifies when reliance on AI yields positive expected cooperative value. The model makes explicit how governance conditions, system policy, and accountability regimes jointly determine whether cooperation is rational or structurally defective. From this formalization we derive a cooperation ecology framework with six design principles: reciprocity contracts, visible trust infrastructure, conditional cooperation modes, defection-mitigation mechanisms, narrative literacy against authority theatre, and an Earth-first sustainability constraint. We operationalize the framework through three policy artefacts: a Human-AI Cooperation Charter, a Defection Risk Register, and a Cooperation Readiness Audit. Together, these elements shift the unit of analysis from the user-AI dyad to the institutional environment that shapes incentives, signals, accountability, and repair. The paper provides a theoretical foundation and practical toolkit for designing human-AI systems that can sustain accountable, trustworthy cooperation over time.