Arrow's Impossibility Theorem: Quantitative Extensions via Latent Framework

We develop quantitative extensions of Arrow's impossibility theorem and the Gibbard-Satterthwaite theorem using the Latent framework. Classical social choice theory establishes that no voting rule can simultaneously satisfy unanimity, independence of irrelevant alternatives (IIA), and non-dictatorship. Our approach measures how close a voting rule can come to satisfying all axioms simultaneously. We prove that for any rule satisfying a bound on total axiom satisfaction, at least one axiom must be violated. We establish that IIA violation frequency scales as \(1/\rho^2\) and dictator-distance scales as \(1/\rho\), where \(\rho\) is the Latent number of the preference distribution. These results provide a spectral framework for optimal voting rule design: higher \(\rho\) preference distributions permit closer approximations to Arrow's ideal. The proofs are formalized in Lean 4 with 12 verified theorems and 0 axioms.