Mathematical Prediction of the Dysmetric Dimension: A1. Dyadic Algebra of Magnitudes v2

Dyadic Algebra of Magnitudes formalizes physical quantities not as a mere combination of two independent components—the mathematical element and the dimension—but as a unified algebraic space. Unlike the current paradigm, which fragments the mathematical and dimensional operations, Dyadic Algebra of Magnitudes unifies both aspects into a single operational entity. It achieves this coherent integration by treating the physical dyad as an indivisible whole. In this structure, the quantitative measure (an element of a number field or vector space) and the qualitative dimension (an element of a structured set of units) are intrinsically fused. Dimensional coherence is not artificially verified a posteriori, but emerges naturally as a geometric and structural property of the dyadic space itself, where generating external composition laws simultaneously govern the transformation of the mathematical entity and the change in dimension. 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. All this is achieved with a very easy-to-understand natural algebra, which unifies the mathematical and dimensional component of every physical quantity into a single operational entity, without revoking the current practical way of separating numerical calculation and dimensional analysis, and in turn highlighting the dysmetric principle of magnitudes, with its enormous potential to investigate phenomena that are hidden from isometry. This second version has been completed with section 39, Brief Idea of the Dysmetric Principle; section 40, Algebraic Foundations of Dysmetry, in sccordance with Category Theory; and section 41, Definition of Dysmetric Density, has been expanded.