In this lesson, I explore how two positive quantities sharing one conserved whole can be understood through the harmonic, geometric, arithmetic, and quadratic means. When the two parts are equal, the means coincide. As imbalance grows, they separate in a precise and meaningful way.
The central identity is:
QM^2+GM^2=\frac12
for a conserved binary partition satisfying a+b=1.
This fixed relation connects the classical mean hierarchy to variance and to a Pythagorean semicircle. The geometric mean reflects shared participation between the two parts, while the quadratic mean increases with asymmetry. These are not separate patterns, but different mathematical descriptions of the same underlying structure.
The lesson also introduces a normalized response variable, an exact golden-ratio crossover, and a special point where:
\frac{QM^2}{GM^2}=3.
The goal is not to claim that these mathematical relationships prove a spiritual or physical law. Rather, they offer a rigorous way to reflect on balance, order, harmony, and the intelligibility of structure.
Read the full paper
De Jesús, Elias. (2026). Pythagoras on the Mean Hierarchy: The Identity QM^2+GM^2=1/2 and Its Structural Consequences for Conserved Binary Partitions. Zenodo.
https://doi.org/10.5281/zenodo.20113065
AI disclosure
The presenter avatar in this video is AI-generated using my likeness. The narration uses an AI-generated voice. The research, lesson content, interpretation, and final editorial responsibility are my own.
About this series
This video is part of an educational series exploring mathematics, science, independent research, and their connections to reflection, personal growth, and our appreciation of order within creation.
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