Comparative evidence on how Gulf Cooperation Council (GCC) states turn artificial intelligence (AI) ambitions into post--New Public Management (post-NPM) outcomes is scarce because most studies examine Western democracies. We analyze constitutional, collective-choice, and operational rules shaping AI uptake in two contrasting GCC members, the United Arab Emirates (UAE) and Kuwait, and whether they foster citizen centricity, collaborative governance, and public value creation. Anchored in Ostrom's Institutional Analysis and Development framework, the study combines a most similar/most different systems design with multiple sources: 62 public documents from 2018--2025, embedded UAE cases (Smart Dubai and MBZUAI), and 39 interviews with officials conducted Aug 2024--May 2025. Dual coding and process tracing connect rule configurations to AI performance. Cross-case analysis identifies four reinforcing mechanisms behind divergent trajectories. In the UAE, concentrated authority, credible sanctions, pro-innovation narratives, and flexible reinvestment rules scale pilots into hundreds of services and sizable recycled savings. In Kuwait, dispersed veto points, exhortative sanctions, cautious discourse, and lapsed AI budgets confine initiatives to pilot mode despite equivalent fiscal resources. The findings refine institutional theory by showing that vertical rule coherence, not wealth, determines AI's public-value yield, and temper post-NPM optimism by revealing that efficiency metrics serve societal goals only when backed by enforceable safeguards. To curb ethics washing and test transferability beyond the GCC, future work should track rule diffusion over time, develop blended legitimacy--efficiency scorecards, and examine how narrative framing shapes citizen consent for data sharing.