March 3, 2026Open Access

The Mathematics of Hyperbolic Grading

Key Points

Observable grades are reformulated as normalized expected payoffs based on velocity, challenging conventional grading systems.
The relationship between Elo competence and rapidity is modeled geometrically on a hyperbola, linking grading to physics.
Assessment uses rigorous mathematical frameworks, including Item Response Theory and Minkowski spacetime to redefine grading.
Highlights a novel approach to grading that combines psychometrics and physics, with implications for educational assessment.

Abstract

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 behaves like an additive rapidity on a hyperbola, the observable grade v is its stereographic projection onto a bounded segment.

The Mathematics of Hyperbolic Grading

Key Points

Abstract

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