Beyond the Binary: A Quaternary Algebraic Framework for Inka Mathematics in the Yupana and Khipu

The decipherment of Inka mathematics stands at a crossroads. Rojas‐Gamarra and Stepanova (2015) proposed a positional, base‐10 interpretation of the Yupana and Khipu. In contrast, Overmann and Florio (2025) argue for a non‐positional, additive system based on the visual ambiguity of khipu knots. This paper resolves this contradiction by demonstrating that both interpretations capture different operational layers of a unified system grounded in Andean quaternary logic. We introduce the Quaternary Algebraic Framework , showing that the values {1,2,3,5} on the Yupana form a complete basis for representing any natural number. We prove a Unification Theorem demonstrating that the Inka system was hybrid by design : operating positionally during calculation on the Yupana and additively during recording on the Khipu. The visual ambiguity of the knots emerges as a designed feature enabling this dual functionality. The “error” in Guamán Poma's drawing is reinterpreted not as a mistake but as evidence of transition between operational modes. A computational implementation validates the framework and reveals inherent error detection. This work establishes Inka mathematics as a unique, sophisticated system that challenges Western categorical distinctions between additive and positional numeration.