This paper introduces and formally verifies a novel geometric framework for first-order stochastic dominance (FSD) in $N$ dimensions using the Lean 4 theorem prover. Traditional analytical approaches to multi-dimensional stochastic dominance rely heavily on complex measure theory and multivariate calculus, creating significant barriers to formalization in proof assistants. Our geometric approach characterizes $N$-dimensional FSD through direct comparison of survival probabilities in upper-right orthants, bypassing the need for complex integration theory. We formalize key definitions and prove the equivalence between traditional FSD requirements and our geometric characterization. This approach achieves a more tractable and intuitive path to formal verification while maintaining mathematical rigor. We demonstrate how this framework directly enables formal analysis of multi-dimensional economic problems in portfolio selection, risk management, and welfare analysis. The work establishes a foundation for further development of verified decision-making tools in economics and finance, particularly for high-stakes domains requiring rigorous guarantees.