Mathematical Prediction of the Dysmetric Dimension: A1. Dyadic Algebra of Magnitudes

From the beginning, operations with physical quantities have been based on a tacit isomorphism with the field of real numbers. This isomorphism simplifies physical operations, which by their nature are more complex. We present the Dyadic Algebra of Magnitudes, a conceptual framework wherein quantities of physical magnitudes are treated as peculiar entities (physical dyads) governed by proper composition laws. This framework defines operations combining real numbers with magnitudes (non-generating external laws that preserve the magnitude) and others that combine magnitudes with one another (external generating laws that yield new magnitudes), establishing a heterogeneous and coherent algebraic structure. With this base, we demonstrate that quantities of magnitudes are geometrically represented by a hyperbolic surface, rather than a standard real line or complex plane. Furthermore, this structure leads with mathematical naturalness to dysmetric prediction, enabling the development of mathematical models tailored to non-isometric phenomena, revealing that the current physical paradigm is the simplest and most limited case of zero dysmetric variation. Our results suggest that adherence to numerical algebra, associated with the improper isomorphism with R, limits the explanation of essential properties of the physical world. Therefore, we propose integrating dyadic and dysmetric principles into our theoretical framework. This extension significantly expands the analytical scope of the available mathematical and physical tools, enabling the systematic exploration of a wide range of previously inaccessible non-isometric scenarios.