The Mathematics of Hyperbolic Grading

This document serves as the theoretical manual for the Hyperbolic Grading Simulator. We propose a rigorous unification of Item Response Theory (Psychometrics) and Special Relativity. By reinterpreting a grade not as a linear score but as a normalized expected payoff (velocity), we demonstrate an exact isomorphism between the Elo rating system and the geometry of Minkowski spacetime. We prove that while Elo competence $\theta$ behaves like an additive rapidity on a hyperbola, the observable grade $v$ is its stereographic projection onto a bounded segment.