The H-1B Visa Reform Act of 2004 dictates an annual allocation of 85,000 visas with 20,000 reserved for advanced-degree applicants. We represent the main requirements of this legislation as formal axioms and characterize visa allocation rules consistent with the axioms. Despite the precise number reserved, we show that the range of implementations satisfying these axioms can change the allocation of advanced-degree visas by as much as 14,000 in an average year. Of all rules satisfying these axioms, the 2019 rule imposed by executive order is most favorable to advanced-degree holders. However, two earlier modifications resulted in larger changes, potentially unintentionally.